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4398 lines
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Plaintext
4398 lines
257 KiB
Plaintext
350 BC
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PRIOR ANALYTICS
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by Aristotle
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translated by A. J. Jenkinson
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Book I
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1
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WE must first state the subject of our inquiry and the faculty to
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which it belongs: its subject is demonstration and the faculty that
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carries it out demonstrative science. We must next define a premiss, a
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term, and a syllogism, and the nature of a perfect and of an imperfect
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syllogism; and after that, the inclusion or noninclusion of one term
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in another as in a whole, and what we mean by predicating one term
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of all, or none, of another.
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A premiss then is a sentence affirming or denying one thing of
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another. This is either universal or particular or indefinite. By
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universal I mean the statement that something belongs to all or none
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of something else; by particular that it belongs to some or not to
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some or not to all; by indefinite that it does or does not belong,
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without any mark to show whether it is universal or particular, e.g.
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'contraries are subjects of the same science', or 'pleasure is not
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good'. The demonstrative premiss differs from the dialectical, because
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the demonstrative premiss is the assertion of one of two contradictory
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statements (the demonstrator does not ask for his premiss, but lays it
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down), whereas the dialectical premiss depends on the adversary's
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choice between two contradictories. But this will make no difference
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to the production of a syllogism in either case; for both the
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demonstrator and the dialectician argue syllogistically after
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stating that something does or does not belong to something else.
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Therefore a syllogistic premiss without qualification will be an
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affirmation or denial of something concerning something else in the
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way we have described; it will be demonstrative, if it is true and
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obtained through the first principles of its science; while a
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dialectical premiss is the giving of a choice between two
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contradictories, when a man is proceeding by question, but when he
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is syllogizing it is the assertion of that which is apparent and
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generally admitted, as has been said in the Topics. The nature then of
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a premiss and the difference between syllogistic, demonstrative, and
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dialectical premisses, may be taken as sufficiently defined by us in
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relation to our present need, but will be stated accurately in the
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sequel.
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I call that a term into which the premiss is resolved, i.e. both the
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predicate and that of which it is predicated, 'being' being added
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and 'not being' removed, or vice versa.
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A syllogism is discourse in which, certain things being stated,
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something other than what is stated follows of necessity from their
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being so. I mean by the last phrase that they produce the consequence,
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and by this, that no further term is required from without in order to
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make the consequence necessary.
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I call that a perfect syllogism which needs nothing other than
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what has been stated to make plain what necessarily follows; a
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syllogism is imperfect, if it needs either one or more propositions,
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which are indeed the necessary consequences of the terms set down, but
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have not been expressly stated as premisses.
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That one term should be included in another as in a whole is the
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same as for the other to be predicated of all of the first. And we say
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that one term is predicated of all of another, whenever no instance of
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the subject can be found of which the other term cannot be asserted:
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'to be predicated of none' must be understood in the same way.
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2
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Every premiss states that something either is or must be or may be
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the attribute of something else; of premisses of these three kinds
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some are affirmative, others negative, in respect of each of the three
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modes of attribution; again some affirmative and negative premisses
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are universal, others particular, others indefinite. It is necessary
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then that in universal attribution the terms of the negative premiss
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should be convertible, e.g. if no pleasure is good, then no good
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will be pleasure; the terms of the affirmative must be convertible,
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not however, universally, but in part, e.g. if every pleasure,is good,
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some good must be pleasure; the particular affirmative must convert in
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part (for if some pleasure is good, then some good will be
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pleasure); but the particular negative need not convert, for if some
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animal is not man, it does not follow that some man is not animal.
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First then take a universal negative with the terms A and B. If no B
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is A, neither can any A be B. For if some A (say C) were B, it would
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not be true that no B is A; for C is a B. But if every B is A then
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some A is B. For if no A were B, then no B could be A. But we
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assumed that every B is A. Similarly too, if the premiss is
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particular. For if some B is A, then some of the As must be B. For
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if none were, then no B would be A. But if some B is not A, there is
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no necessity that some of the As should not be B; e.g. let B stand for
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animal and A for man. Not every animal is a man; but every man is an
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animal.
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3
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The same manner of conversion will hold good also in respect of
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necessary premisses. The universal negative converts universally; each
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of the affirmatives converts into a particular. If it is necessary
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that no B is A, it is necessary also that no A is B. For if it is
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possible that some A is B, it would be possible also that some B is A.
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If all or some B is A of necessity, it is necessary also that some A
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is B: for if there were no necessity, neither would some of the Bs
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be A necessarily. But the particular negative does not convert, for
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the same reason which we have already stated.
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In respect of possible premisses, since possibility is used in
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several senses (for we say that what is necessary and what is not
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necessary and what is potential is possible), affirmative statements
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will all convert in a manner similar to those described. For if it
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is possible that all or some B is A, it will be possible that some A
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is B. For if that were not possible, then no B could possibly be A.
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This has been already proved. But in negative statements the case is
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different. Whatever is said to be possible, either because B
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necessarily is A, or because B is not necessarily A, admits of
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conversion like other negative statements, e.g. if one should say,
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it is possible that man is not horse, or that no garment is white. For
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in the former case the one term necessarily does not belong to the
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other; in the latter there is no necessity that it should: and the
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premiss converts like other negative statements. For if it is possible
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for no man to be a horse, it is also admissible for no horse to be a
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man; and if it is admissible for no garment to be white, it is also
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admissible for nothing white to be a garment. For if any white thing
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must be a garment, then some garment will necessarily be white. This
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has been already proved. The particular negative also must be
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treated like those dealt with above. But if anything is said to be
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possible because it is the general rule and natural (and it is in this
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way we define the possible), the negative premisses can no longer be
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converted like the simple negatives; the universal negative premiss
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does not convert, and the particular does. This will be plain when
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we speak about the possible. At present we may take this much as clear
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in addition to what has been said: the statement that it is possible
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that no B is A or some B is not A is affirmative in form: for the
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expression 'is possible' ranks along with 'is', and 'is' makes an
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affirmation always and in every case, whatever the terms to which it
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is added, in predication, e.g. 'it is not-good' or 'it is not-white'
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or in a word 'it is not-this'. But this also will be proved in the
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sequel. In conversion these premisses will behave like the other
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affirmative propositions.
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4
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After these distinctions we now state by what means, when, and how
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every syllogism is produced; subsequently we must speak of
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demonstration. Syllogism should be discussed before demonstration
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because syllogism is the general: the demonstration is a sort of
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syllogism, but not every syllogism is a demonstration.
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Whenever three terms are so related to one another that the last
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is contained in the middle as in a whole, and the middle is either
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contained in, or excluded from, the first as in or from a whole, the
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extremes must be related by a perfect syllogism. I call that term
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middle which is itself contained in another and contains another in
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itself: in position also this comes in the middle. By extremes I
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mean both that term which is itself contained in another and that in
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which another is contained. If A is predicated of all B, and B of
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all C, A must be predicated of all C: we have already explained what
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we mean by 'predicated of all'. Similarly also, if A is predicated
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of no B, and B of all C, it is necessary that no C will be A.
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But if the first term belongs to all the middle, but the middle to
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none of the last term, there will be no syllogism in respect of the
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extremes; for nothing necessary follows from the terms being so
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related; for it is possible that the first should belong either to all
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or to none of the last, so that neither a particular nor a universal
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conclusion is necessary. But if there is no necessary consequence,
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there cannot be a syllogism by means of these premisses. As an example
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of a universal affirmative relation between the extremes we may take
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the terms animal, man, horse; of a universal negative relation, the
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terms animal, man, stone. Nor again can syllogism be formed when
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neither the first term belongs to any of the middle, nor the middle to
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any of the last. As an example of a positive relation between the
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extremes take the terms science, line, medicine: of a negative
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relation science, line, unit.
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If then the terms are universally related, it is clear in this
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figure when a syllogism will be possible and when not, and that if a
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syllogism is possible the terms must be related as described, and if
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they are so related there will be a syllogism.
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But if one term is related universally, the other in part only, to
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its subject, there must be a perfect syllogism whenever universality
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is posited with reference to the major term either affirmatively or
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negatively, and particularity with reference to the minor term
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affirmatively: but whenever the universality is posited in relation to
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the minor term, or the terms are related in any other way, a syllogism
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is impossible. I call that term the major in which the middle is
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contained and that term the minor which comes under the middle. Let
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all B be A and some C be B. Then if 'predicated of all' means what was
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said above, it is necessary that some C is A. And if no B is A but
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some C is B, it is necessary that some C is not A. The meaning of
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'predicated of none' has also been defined. So there will be a perfect
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syllogism. This holds good also if the premiss BC should be
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indefinite, provided that it is affirmative: for we shall have the
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same syllogism whether the premiss is indefinite or particular.
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But if the universality is posited with respect to the minor term
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either affirmatively or negatively, a syllogism will not be
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possible, whether the major premiss is positive or negative,
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indefinite or particular: e.g. if some B is or is not A, and all C
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is B. As an example of a positive relation between the extremes take
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the terms good, state, wisdom: of a negative relation, good, state,
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ignorance. Again if no C is B, but some B is or is not A or not
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every B is A, there cannot be a syllogism. Take the terms white,
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horse, swan: white, horse, raven. The same terms may be taken also
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if the premiss BA is indefinite.
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Nor when the major premiss is universal, whether affirmative or
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negative, and the minor premiss is negative and particular, can
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there be a syllogism, whether the minor premiss be indefinite or
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particular: e.g. if all B is A and some C is not B, or if not all C is
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B. For the major term may be predicable both of all and of none of the
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minor, to some of which the middle term cannot be attributed.
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Suppose the terms are animal, man, white: next take some of the
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white things of which man is not predicated-swan and snow: animal is
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predicated of all of the one, but of none of the other. Consequently
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there cannot be a syllogism. Again let no B be A, but let some C not
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be B. Take the terms inanimate, man, white: then take some white
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things of which man is not predicated-swan and snow: the term
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inanimate is predicated of all of the one, of none of the other.
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Further since it is indefinite to say some C is not B, and it is
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true that some C is not B, whether no C is B, or not all C is B, and
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since if terms are assumed such that no C is B, no syllogism follows
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(this has already been stated) it is clear that this arrangement of
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terms will not afford a syllogism: otherwise one would have been
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possible with a universal negative minor premiss. A similar proof
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may also be given if the universal premiss is negative.
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Nor can there in any way be a syllogism if both the relations of
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subject and predicate are particular, either positively or negatively,
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or the one negative and the other affirmative, or one indefinite and
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the other definite, or both indefinite. Terms common to all the
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above are animal, white, horse: animal, white, stone.
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It is clear then from what has been said that if there is a
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syllogism in this figure with a particular conclusion, the terms
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must be related as we have stated: if they are related otherwise, no
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syllogism is possible anyhow. It is evident also that all the
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syllogisms in this figure are perfect (for they are all completed by
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means of the premisses originally taken) and that all conclusions
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are proved by this figure, viz. universal and particular,
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affirmative and negative. Such a figure I call the first.
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5
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Whenever the same thing belongs to all of one subject, and to none
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of another, or to all of each subject or to none of either, I call
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such a figure the second; by middle term in it I mean that which is
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predicated of both subjects, by extremes the terms of which this is
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said, by major extreme that which lies near the middle, by minor
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that which is further away from the middle. The middle term stands
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outside the extremes, and is first in position. A syllogism cannot
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be perfect anyhow in this figure, but it may be valid whether the
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terms are related universally or not.
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If then the terms are related universally a syllogism will be
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possible, whenever the middle belongs to all of one subject and to
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none of another (it does not matter which has the negative
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relation), but in no other way. Let M be predicated of no N, but of
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all O. Since, then, the negative relation is convertible, N will
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belong to no M: but M was assumed to belong to all O: consequently N
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will belong to no O. This has already been proved. Again if M
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belongs to all N, but to no O, then N will belong to no O. For if M
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belongs to no O, O belongs to no M: but M (as was said) belongs to all
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N: O then will belong to no N: for the first figure has again been
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formed. But since the negative relation is convertible, N will
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belong to no O. Thus it will be the same syllogism that proves both
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conclusions.
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It is possible to prove these results also by reductio ad
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impossibile.
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It is clear then that a syllogism is formed when the terms are so
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related, but not a perfect syllogism; for necessity is not perfectly
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established merely from the original premisses; others also are
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needed.
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But if M is predicated of every N and O, there cannot be a
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syllogism. Terms to illustrate a positive relation between the
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extremes are substance, animal, man; a negative relation, substance,
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animal, number-substance being the middle term.
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Nor is a syllogism possible when M is predicated neither of any N
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nor of any O. Terms to illustrate a positive relation are line,
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animal, man: a negative relation, line, animal, stone.
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It is clear then that if a syllogism is formed when the terms are
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universally related, the terms must be related as we stated at the
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outset: for if they are otherwise related no necessary consequence
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follows.
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If the middle term is related universally to one of the extremes,
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a particular negative syllogism must result whenever the middle term
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is related universally to the major whether positively or
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negatively, and particularly to the minor and in a manner opposite
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to that of the universal statement: by 'an opposite manner' I mean, if
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the universal statement is negative, the particular is affirmative: if
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the universal is affirmative, the particular is negative. For if M
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belongs to no N, but to some O, it is necessary that N does not belong
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to some O. For since the negative statement is convertible, N will
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belong to no M: but M was admitted to belong to some O: therefore N
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will not belong to some O: for the result is reached by means of the
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first figure. Again if M belongs to all N, but not to some O, it is
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necessary that N does not belong to some O: for if N belongs to all O,
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and M is predicated also of all N, M must belong to all O: but we
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assumed that M does not belong to some O. And if M belongs to all N
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but not to all O, we shall conclude that N does not belong to all O:
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the proof is the same as the above. But if M is predicated of all O,
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but not of all N, there will be no syllogism. Take the terms animal,
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substance, raven; animal, white, raven. Nor will there be a conclusion
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when M is predicated of no O, but of some N. Terms to illustrate a
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positive relation between the extremes are animal, substance, unit:
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a negative relation, animal, substance, science.
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If then the universal statement is opposed to the particular, we
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have stated when a syllogism will be possible and when not: but if the
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premisses are similar in form, I mean both negative or both
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affirmative, a syllogism will not be possible anyhow. First let them
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be negative, and let the major premiss be universal, e.g. let M belong
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to no N, and not to some O. It is possible then for N to belong either
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to all O or to no O. Terms to illustrate the negative relation are
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black, snow, animal. But it is not possible to find terms of which the
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extremes are related positively and universally, if M belongs to
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some O, and does not belong to some O. For if N belonged to all O, but
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M to no N, then M would belong to no O: but we assumed that it belongs
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to some O. In this way then it is not admissible to take terms: our
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point must be proved from the indefinite nature of the particular
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statement. For since it is true that M does not belong to some O, even
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if it belongs to no O, and since if it belongs to no O a syllogism
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is (as we have seen) not possible, clearly it will not be possible now
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either.
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Again let the premisses be affirmative, and let the major premiss as
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before be universal, e.g. let M belong to all N and to some O. It is
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possible then for N to belong to all O or to no O. Terms to illustrate
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the negative relation are white, swan, stone. But it is not possible
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to take terms to illustrate the universal affirmative relation, for
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the reason already stated: the point must be proved from the
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indefinite nature of the particular statement. But if the minor
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premiss is universal, and M belongs to no O, and not to some N, it
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is possible for N to belong either to all O or to no O. Terms for
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the positive relation are white, animal, raven: for the negative
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relation, white, stone, raven. If the premisses are affirmative, terms
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for the negative relation are white, animal, snow; for the positive
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relation, white, animal, swan. Evidently then, whenever the
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premisses are similar in form, and one is universal, the other
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particular, a syllogism can, not be formed anyhow. Nor is one possible
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if the middle term belongs to some of each of the extremes, or does
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not belong to some of either, or belongs to some of the one, not to
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some of the other, or belongs to neither universally, or is related to
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them indefinitely. Common terms for all the above are white, animal,
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man: white, animal, inanimate.
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It is clear then from what has been said that if the terms are related
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to one another in the way stated, a syllogism results of necessity;
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and if there is a syllogism, the terms must be so related. But it is
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evident also that all the syllogisms in this figure are imperfect: for
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all are made perfect by certain supplementary statements, which either
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are contained in the terms of necessity or are assumed as
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hypotheses, i.e. when we prove per impossibile. And it is evident that
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an affirmative conclusion is not attained by means of this figure, but
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all are negative, whether universal or particular.
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6
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But if one term belongs to all, and another to none, of a third,
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or if both belong to all, or to none, of it, I call such a figure
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the third; by middle term in it I mean that of which both the
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predicates are predicated, by extremes I mean the predicates, by the
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major extreme that which is further from the middle, by the minor that
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which is nearer to it. The middle term stands outside the extremes,
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and is last in position. A syllogism cannot be perfect in this
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figure either, but it may be valid whether the terms are related
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universally or not to the middle term.
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If they are universal, whenever both P and R belong to S, it follows
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that P will necessarily belong to some R. For, since the affirmative
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statement is convertible, S will belong to some R: consequently
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since P belongs to all S, and S to some R, P must belong to some R:
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for a syllogism in the first figure is produced. It is possible to
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demonstrate this also per impossibile and by exposition. For if both P
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and R belong to all S, should one of the Ss, e.g. N, be taken, both
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P and R will belong to this, and thus P will belong to some R.
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If R belongs to all S, and P to no S, there will be a syllogism to
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prove that P will necessarily not belong to some R. This may be
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demonstrated in the same way as before by converting the premiss RS.
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It might be proved also per impossibile, as in the former cases. But
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if R belongs to no S, P to all S, there will be no syllogism. Terms
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for the positive relation are animal, horse, man: for the negative
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relation animal, inanimate, man.
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|
|
|
Nor can there be a syllogism when both terms are asserted of no S.
|
|
Terms for the positive relation are animal, horse, inanimate; for
|
|
the negative relation man, horse, inanimate-inanimate being the middle
|
|
term.
|
|
|
|
It is clear then in this figure also when a syllogism will be
|
|
possible and when not, if the terms are related universally. For
|
|
whenever both the terms are affirmative, there will be a syllogism
|
|
to prove that one extreme belongs to some of the other; but when
|
|
they are negative, no syllogism will be possible. But when one is
|
|
negative, the other affirmative, if the major is negative, the minor
|
|
affirmative, there will be a syllogism to prove that the one extreme
|
|
does not belong to some of the other: but if the relation is reversed,
|
|
no syllogism will be possible. If one term is related universally to
|
|
the middle, the other in part only, when both are affirmative there
|
|
must be a syllogism, no matter which of the premisses is universal.
|
|
For if R belongs to all S, P to some S, P must belong to some R. For
|
|
since the affirmative statement is convertible S will belong to some
|
|
P: consequently since R belongs to all S, and S to some P, R must also
|
|
belong to some P: therefore P must belong to some R.
|
|
|
|
Again if R belongs to some S, and P to all S, P must belong to
|
|
some R. This may be demonstrated in the same way as the preceding. And
|
|
it is possible to demonstrate it also per impossibile and by
|
|
exposition, as in the former cases. But if one term is affirmative,
|
|
the other negative, and if the affirmative is universal, a syllogism
|
|
will be possible whenever the minor term is affirmative. For if R
|
|
belongs to all S, but P does not belong to some S, it is necessary
|
|
that P does not belong to some R. For if P belongs to all R, and R
|
|
belongs to all S, then P will belong to all S: but we assumed that
|
|
it did not. Proof is possible also without reduction ad impossibile,
|
|
if one of the Ss be taken to which P does not belong.
|
|
|
|
But whenever the major is affirmative, no syllogism will be
|
|
possible, e.g. if P belongs to all S and R does not belong to some
|
|
S. Terms for the universal affirmative relation are animate, man,
|
|
animal. For the universal negative relation it is not possible to
|
|
get terms, if R belongs to some S, and does not belong to some S.
|
|
For if P belongs to all S, and R to some S, then P will belong to some
|
|
R: but we assumed that it belongs to no R. We must put the matter as
|
|
before.' Since the expression 'it does not belong to some' is
|
|
indefinite, it may be used truly of that also which belongs to none.
|
|
But if R belongs to no S, no syllogism is possible, as has been shown.
|
|
Clearly then no syllogism will be possible here.
|
|
|
|
But if the negative term is universal, whenever the major is
|
|
negative and the minor affirmative there will be a syllogism. For if P
|
|
belongs to no S, and R belongs to some S, P will not belong to some R:
|
|
for we shall have the first figure again, if the premiss RS is
|
|
converted.
|
|
|
|
But when the minor is negative, there will be no syllogism. Terms
|
|
for the positive relation are animal, man, wild: for the negative
|
|
relation, animal, science, wild-the middle in both being the term
|
|
wild.
|
|
|
|
Nor is a syllogism possible when both are stated in the negative,
|
|
but one is universal, the other particular. When the minor is
|
|
related universally to the middle, take the terms animal, science,
|
|
wild; animal, man, wild. When the major is related universally to
|
|
the middle, take as terms for a negative relation raven, snow,
|
|
white. For a positive relation terms cannot be found, if R belongs
|
|
to some S, and does not belong to some S. For if P belongs to all R,
|
|
and R to some S, then P belongs to some S: but we assumed that it
|
|
belongs to no S. Our point, then, must be proved from the indefinite
|
|
nature of the particular statement.
|
|
|
|
Nor is a syllogism possible anyhow, if each of the extremes
|
|
belongs to some of the middle or does not belong, or one belongs and
|
|
the other does not to some of the middle, or one belongs to some of
|
|
the middle, the other not to all, or if the premisses are
|
|
indefinite. Common terms for all are animal, man, white: animal,
|
|
inanimate, white.
|
|
|
|
It is clear then in this figure also when a syllogism will be
|
|
possible, and when not; and that if the terms are as stated, a
|
|
syllogism results of necessity, and if there is a syllogism, the terms
|
|
must be so related. It is clear also that all the syllogisms in this
|
|
figure are imperfect (for all are made perfect by certain
|
|
supplementary assumptions), and that it will not be possible to
|
|
reach a universal conclusion by means of this figure, whether negative
|
|
or affirmative.
|
|
|
|
7
|
|
|
|
It is evident also that in all the figures, whenever a proper
|
|
syllogism does not result, if both the terms are affirmative or
|
|
negative nothing necessary follows at all, but if one is
|
|
affirmative, the other negative, and if the negative is stated
|
|
universally, a syllogism always results relating the minor to the
|
|
major term, e.g. if A belongs to all or some B, and B belongs to no C:
|
|
for if the premisses are converted it is necessary that C does not
|
|
belong to some A. Similarly also in the other figures: a syllogism
|
|
always results by means of conversion. It is evident also that the
|
|
substitution of an indefinite for a particular affirmative will effect
|
|
the same syllogism in all the figures.
|
|
|
|
It is clear too that all the imperfect syllogisms are made perfect
|
|
by means of the first figure. For all are brought to a conclusion
|
|
either ostensively or per impossibile. In both ways the first figure
|
|
is formed: if they are made perfect ostensively, because (as we saw)
|
|
all are brought to a conclusion by means of conversion, and conversion
|
|
produces the first figure: if they are proved per impossibile, because
|
|
on the assumption of the false statement the syllogism comes about
|
|
by means of the first figure, e.g. in the last figure, if A and B
|
|
belong to all C, it follows that A belongs to some B: for if A
|
|
belonged to no B, and B belongs to all C, A would belong to no C:
|
|
but (as we stated) it belongs to all C. Similarly also with the rest.
|
|
|
|
It is possible also to reduce all syllogisms to the universal
|
|
syllogisms in the first figure. Those in the second figure are clearly
|
|
made perfect by these, though not all in the same way; the universal
|
|
syllogisms are made perfect by converting the negative premiss, each
|
|
of the particular syllogisms by reductio ad impossibile. In the
|
|
first figure particular syllogisms are indeed made perfect by
|
|
themselves, but it is possible also to prove them by means of the
|
|
second figure, reducing them ad impossibile, e.g. if A belongs to
|
|
all B, and B to some C, it follows that A belongs to some C. For if it
|
|
belonged to no C, and belongs to all B, then B will belong to no C:
|
|
this we know by means of the second figure. Similarly also
|
|
demonstration will be possible in the case of the negative. For if A
|
|
belongs to no B, and B belongs to some C, A will not belong to some C:
|
|
for if it belonged to all C, and belongs to no B, then B will belong
|
|
to no C: and this (as we saw) is the middle figure. Consequently,
|
|
since all syllogisms in the middle figure can be reduced to
|
|
universal syllogisms in the first figure, and since particular
|
|
syllogisms in the first figure can be reduced to syllogisms in the
|
|
middle figure, it is clear that particular syllogisms can be reduced
|
|
to universal syllogisms in the first figure. Syllogisms in the third
|
|
figure, if the terms are universal, are directly made perfect by means
|
|
of those syllogisms; but, when one of the premisses is particular,
|
|
by means of the particular syllogisms in the first figure: and these
|
|
(we have seen) may be reduced to the universal syllogisms in the first
|
|
figure: consequently also the particular syllogisms in the third
|
|
figure may be so reduced. It is clear then that all syllogisms may
|
|
be reduced to the universal syllogisms in the first figure.
|
|
|
|
We have stated then how syllogisms which prove that something
|
|
belongs or does not belong to something else are constituted, both how
|
|
syllogisms of the same figure are constituted in themselves, and how
|
|
syllogisms of different figures are related to one another.
|
|
|
|
8
|
|
|
|
Since there is a difference according as something belongs,
|
|
necessarily belongs, or may belong to something else (for many
|
|
things belong indeed, but not necessarily, others neither
|
|
necessarily nor indeed at all, but it is possible for them to belong),
|
|
it is clear that there will be different syllogisms to prove each of
|
|
these relations, and syllogisms with differently related terms, one
|
|
syllogism concluding from what is necessary, another from what is, a
|
|
third from what is possible.
|
|
|
|
There is hardly any difference between syllogisms from necessary
|
|
premisses and syllogisms from premisses which merely assert. When
|
|
the terms are put in the same way, then, whether something belongs
|
|
or necessarily belongs (or does not belong) to something else, a
|
|
syllogism will or will not result alike in both cases, the only
|
|
difference being the addition of the expression 'necessarily' to the
|
|
terms. For the negative statement is convertible alike in both
|
|
cases, and we should give the same account of the expressions 'to be
|
|
contained in something as in a whole' and 'to be predicated of all
|
|
of something'. With the exceptions to be made below, the conclusion
|
|
will be proved to be necessary by means of conversion, in the same
|
|
manner as in the case of simple predication. But in the middle
|
|
figure when the universal statement is affirmative, and the particular
|
|
negative, and again in the third figure when the universal is
|
|
affirmative and the particular negative, the demonstration will not
|
|
take the same form, but it is necessary by the 'exposition' of a
|
|
part of the subject of the particular negative proposition, to which
|
|
the predicate does not belong, to make the syllogism in reference to
|
|
this: with terms so chosen the conclusion will necessarily follow. But
|
|
if the relation is necessary in respect of the part taken, it must
|
|
hold of some of that term in which this part is included: for the part
|
|
taken is just some of that. And each of the resulting syllogisms is in
|
|
the appropriate figure.
|
|
|
|
9
|
|
|
|
It happens sometimes also that when one premiss is necessary the
|
|
conclusion is necessary, not however when either premiss is necessary,
|
|
but only when the major is, e.g. if A is taken as necessarily
|
|
belonging or not belonging to B, but B is taken as simply belonging to
|
|
C: for if the premisses are taken in this way, A will necessarily
|
|
belong or not belong to C. For since necessarily belongs, or does
|
|
not belong, to every B, and since C is one of the Bs, it is clear that
|
|
for C also the positive or the negative relation to A will hold
|
|
necessarily. But if the major premiss is not necessary, but the
|
|
minor is necessary, the conclusion will not be necessary. For if it
|
|
were, it would result both through the first figure and through the
|
|
third that A belongs necessarily to some B. But this is false; for B
|
|
may be such that it is possible that A should belong to none of it.
|
|
Further, an example also makes it clear that the conclusion not be
|
|
necessary, e.g. if A were movement, B animal, C man: man is an
|
|
animal necessarily, but an animal does not move necessarily, nor
|
|
does man. Similarly also if the major premiss is negative; for the
|
|
proof is the same.
|
|
|
|
In particular syllogisms, if the universal premiss is necessary,
|
|
then the conclusion will be necessary; but if the particular, the
|
|
conclusion will not be necessary, whether the universal premiss is
|
|
negative or affirmative. First let the universal be necessary, and let
|
|
A belong to all B necessarily, but let B simply belong to some C: it
|
|
is necessary then that A belongs to some C necessarily: for C falls
|
|
under B, and A was assumed to belong necessarily to all B. Similarly
|
|
also if the syllogism should be negative: for the proof will be the
|
|
same. But if the particular premiss is necessary, the conclusion
|
|
will not be necessary: for from the denial of such a conclusion
|
|
nothing impossible results, just as it does not in the universal
|
|
syllogisms. The same is true of negative syllogisms. Try the terms
|
|
movement, animal, white.
|
|
|
|
10
|
|
|
|
In the second figure, if the negative premiss is necessary, then the
|
|
conclusion will be necessary, but if the affirmative, not necessary.
|
|
First let the negative be necessary; let A be possible of no B, and
|
|
simply belong to C. Since then the negative statement is
|
|
convertible, B is possible of no A. But A belongs to all C;
|
|
consequently B is possible of no C. For C falls under A. The same
|
|
result would be obtained if the minor premiss were negative: for if
|
|
A is possible be of no C, C is possible of no A: but A belongs to
|
|
all B, consequently C is possible of none of the Bs: for again we have
|
|
obtained the first figure. Neither then is B possible of C: for
|
|
conversion is possible without modifying the relation.
|
|
|
|
But if the affirmative premiss is necessary, the conclusion will not
|
|
be necessary. Let A belong to all B necessarily, but to no C simply.
|
|
If then the negative premiss is converted, the first figure results.
|
|
But it has been proved in the case of the first figure that if the
|
|
negative major premiss is not necessary the conclusion will not be
|
|
necessary either. Therefore the same result will obtain here. Further,
|
|
if the conclusion is necessary, it follows that C necessarily does not
|
|
belong to some A. For if B necessarily belongs to no C, C will
|
|
necessarily belong to no B. But B at any rate must belong to some A,
|
|
if it is true (as was assumed) that A necessarily belongs to all B.
|
|
Consequently it is necessary that C does not belong to some A. But
|
|
nothing prevents such an A being taken that it is possible for C to
|
|
belong to all of it. Further one might show by an exposition of
|
|
terms that the conclusion is not necessary without qualification,
|
|
though it is a necessary conclusion from the premisses. For example
|
|
let A be animal, B man, C white, and let the premisses be assumed to
|
|
correspond to what we had before: it is possible that animal should
|
|
belong to nothing white. Man then will not belong to anything white,
|
|
but not necessarily: for it is possible for man to be born white,
|
|
not however so long as animal belongs to nothing white. Consequently
|
|
under these conditions the conclusion will be necessary, but it is not
|
|
necessary without qualification.
|
|
|
|
Similar results will obtain also in particular syllogisms. For
|
|
whenever the negative premiss is both universal and necessary, then
|
|
the conclusion will be necessary: but whenever the affirmative premiss
|
|
is universal, the negative particular, the conclusion will not be
|
|
necessary. First then let the negative premiss be both universal and
|
|
necessary: let it be possible for no B that A should belong to it, and
|
|
let A simply belong to some C. Since the negative statement is
|
|
convertible, it will be possible for no A that B should belong to
|
|
it: but A belongs to some C; consequently B necessarily does not
|
|
belong to some of the Cs. Again let the affirmative premiss be both
|
|
universal and necessary, and let the major premiss be affirmative.
|
|
If then A necessarily belongs to all B, but does not belong to some C,
|
|
it is clear that B will not belong to some C, but not necessarily. For
|
|
the same terms can be used to demonstrate the point, which were used
|
|
in the universal syllogisms. Nor again, if the negative statement is
|
|
necessary but particular, will the conclusion be necessary. The
|
|
point can be demonstrated by means of the same terms.
|
|
|
|
11
|
|
|
|
In the last figure when the terms are related universally to the
|
|
middle, and both premisses are affirmative, if one of the two is
|
|
necessary, then the conclusion will be necessary. But if one is
|
|
negative, the other affirmative, whenever the negative is necessary
|
|
the conclusion also will be necessary, but whenever the affirmative is
|
|
necessary the conclusion will not be necessary. First let both the
|
|
premisses be affirmative, and let A and B belong to all C, and let
|
|
AC be necessary. Since then B belongs to all C, C also will belong
|
|
to some B, because the universal is convertible into the particular:
|
|
consequently if A belongs necessarily to all C, and C belongs to
|
|
some B, it is necessary that A should belong to some B also. For B
|
|
is under C. The first figure then is formed. A similar proof will be
|
|
given also if BC is necessary. For C is convertible with some A:
|
|
consequently if B belongs necessarily to all C, it will belong
|
|
necessarily also to some A.
|
|
|
|
Again let AC be negative, BC affirmative, and let the negative
|
|
premiss be necessary. Since then C is convertible with some B, but A
|
|
necessarily belongs to no C, A will necessarily not belong to some B
|
|
either: for B is under C. But if the affirmative is necessary, the
|
|
conclusion will not be necessary. For suppose BC is affirmative and
|
|
necessary, while AC is negative and not necessary. Since then the
|
|
affirmative is convertible, C also will belong to some B
|
|
necessarily: consequently if A belongs to none of the Cs, while C
|
|
belongs to some of the Bs, A will not belong to some of the Bs-but not
|
|
of necessity; for it has been proved, in the case of the first figure,
|
|
that if the negative premiss is not necessary, neither will the
|
|
conclusion be necessary. Further, the point may be made clear by
|
|
considering the terms. Let the term A be 'good', let that which B
|
|
signifies be 'animal', let the term C be 'horse'. It is possible
|
|
then that the term good should belong to no horse, and it is necessary
|
|
that the term animal should belong to every horse: but it is not
|
|
necessary that some animal should not be good, since it is possible
|
|
for every animal to be good. Or if that is not possible, take as the
|
|
term 'awake' or 'asleep': for every animal can accept these.
|
|
|
|
If, then, the premisses are universal, we have stated when the
|
|
conclusion will be necessary. But if one premiss is universal, the
|
|
other particular, and if both are affirmative, whenever the
|
|
universal is necessary the conclusion also must be necessary. The
|
|
demonstration is the same as before; for the particular affirmative
|
|
also is convertible. If then it is necessary that B should belong to
|
|
all C, and A falls under C, it is necessary that B should belong to
|
|
some A. But if B must belong to some A, then A must belong to some
|
|
B: for conversion is possible. Similarly also if AC should be
|
|
necessary and universal: for B falls under C. But if the particular
|
|
premiss is necessary, the conclusion will not be necessary. Let the
|
|
premiss BC be both particular and necessary, and let A belong to all
|
|
C, not however necessarily. If the proposition BC is converted the
|
|
first figure is formed, and the universal premiss is not necessary,
|
|
but the particular is necessary. But when the premisses were thus, the
|
|
conclusion (as we proved was not necessary: consequently it is not
|
|
here either. Further, the point is clear if we look at the terms.
|
|
Let A be waking, B biped, and C animal. It is necessary that B
|
|
should belong to some C, but it is possible for A to belong to C,
|
|
and that A should belong to B is not necessary. For there is no
|
|
necessity that some biped should be asleep or awake. Similarly and
|
|
by means of the same terms proof can be made, should the proposition
|
|
AC be both particular and necessary.
|
|
|
|
But if one premiss is affirmative, the other negative, whenever
|
|
the universal is both negative and necessary the conclusion also
|
|
will be necessary. For if it is not possible that A should belong to
|
|
any C, but B belongs to some C, it is necessary that A should not
|
|
belong to some B. But whenever the affirmative proposition is
|
|
necessary, whether universal or particular, or the negative is
|
|
particular, the conclusion will not be necessary. The proof of this by
|
|
reduction will be the same as before; but if terms are wanted, when
|
|
the universal affirmative is necessary, take the terms
|
|
'waking'-'animal'-'man', 'man' being middle, and when the
|
|
affirmative is particular and necessary, take the terms
|
|
'waking'-'animal'-'white': for it is necessary that animal should
|
|
belong to some white thing, but it is possible that waking should
|
|
belong to none, and it is not necessary that waking should not
|
|
belong to some animal. But when the negative proposition being
|
|
particular is necessary, take the terms 'biped', 'moving', 'animal',
|
|
'animal' being middle.
|
|
|
|
12
|
|
|
|
It is clear then that a simple conclusion is not reached unless both
|
|
premisses are simple assertions, but a necessary conclusion is
|
|
possible although one only of the premisses is necessary. But in
|
|
both cases, whether the syllogisms are affirmative or negative, it
|
|
is necessary that one premiss should be similar to the conclusion. I
|
|
mean by 'similar', if the conclusion is a simple assertion, the
|
|
premiss must be simple; if the conclusion is necessary, the premiss
|
|
must be necessary. Consequently this also is clear, that the
|
|
conclusion will be neither necessary nor simple unless a necessary
|
|
or simple premiss is assumed.
|
|
|
|
13
|
|
|
|
Perhaps enough has been said about the proof of necessity, how it
|
|
comes about and how it differs from the proof of a simple statement.
|
|
We proceed to discuss that which is possible, when and how and by what
|
|
means it can be proved. I use the terms 'to be possible' and 'the
|
|
possible' of that which is not necessary but, being assumed, results
|
|
in nothing impossible. We say indeed ambiguously of the necessary that
|
|
it is possible. But that my definition of the possible is correct is
|
|
clear from the phrases by which we deny or on the contrary affirm
|
|
possibility. For the expressions 'it is not possible to belong', 'it
|
|
is impossible to belong', and 'it is necessary not to belong' are
|
|
either identical or follow from one another; consequently their
|
|
opposites also, 'it is possible to belong', 'it is not impossible to
|
|
belong', and 'it is not necessary not to belong', will either be
|
|
identical or follow from one another. For of everything the
|
|
affirmation or the denial holds good. That which is possible then will
|
|
be not necessary and that which is not necessary will be possible.
|
|
It results that all premisses in the mode of possibility are
|
|
convertible into one another. I mean not that the affirmative are
|
|
convertible into the negative, but that those which are affirmative in
|
|
form admit of conversion by opposition, e.g. 'it is possible to
|
|
belong' may be converted into 'it is possible not to belong', and
|
|
'it is possible for A to belong to all B' into 'it is possible for A
|
|
to belong to no B' or 'not to all B', and 'it is possible for A to
|
|
belong to some B' into 'it is possible for A not to belong to some B'.
|
|
And similarly the other propositions in this mode can be converted.
|
|
For since that which is possible is not necessary, and that which is
|
|
not necessary may possibly not belong, it is clear that if it is
|
|
possible that A should belong to B, it is possible also that it should
|
|
not belong to B: and if it is possible that it should belong to all,
|
|
it is also possible that it should not belong to all. The same holds
|
|
good in the case of particular affirmations: for the proof is
|
|
identical. And such premisses are affirmative and not negative; for
|
|
'to be possible' is in the same rank as 'to be', as was said above.
|
|
|
|
Having made these distinctions we next point out that the expression
|
|
'to be possible' is used in two ways. In one it means to happen
|
|
generally and fall short of necessity, e.g. man's turning grey or
|
|
growing or decaying, or generally what naturally belongs to a thing
|
|
(for this has not its necessity unbroken, since man's existence is not
|
|
continuous for ever, although if a man does exist, it comes about
|
|
either necessarily or generally). In another sense the expression
|
|
means the indefinite, which can be both thus and not thus, e.g. an
|
|
animal's walking or an earthquake's taking place while it is
|
|
walking, or generally what happens by chance: for none of these
|
|
inclines by nature in the one way more than in the opposite.
|
|
|
|
That which is possible in each of its two senses is convertible into
|
|
its opposite, not however in the same way: but what is natural is
|
|
convertible because it does not necessarily belong (for in this
|
|
sense it is possible that a man should not grow grey) and what is
|
|
indefinite is convertible because it inclines this way no more than
|
|
that. Science and demonstrative syllogism are not concerned with
|
|
things which are indefinite, because the middle term is uncertain; but
|
|
they are concerned with things that are natural, and as a rule
|
|
arguments and inquiries are made about things which are possible in
|
|
this sense. Syllogisms indeed can be made about the former, but it
|
|
is unusual at any rate to inquire about them.
|
|
|
|
These matters will be treated more definitely in the sequel; our
|
|
business at present is to state the moods and nature of the
|
|
syllogism made from possible premisses. The expression 'it is possible
|
|
for this to belong to that' may be understood in two senses: 'that'
|
|
may mean either that to which 'that' belongs or that to which it may
|
|
belong; for the expression 'A is possible of the subject of B' means
|
|
that it is possible either of that of which B is stated or of that
|
|
of which B may possibly be stated. It makes no difference whether we
|
|
say, A is possible of the subject of B, or all B admits of A. It is
|
|
clear then that the expression 'A may possibly belong to all B'
|
|
might be used in two senses. First then we must state the nature and
|
|
characteristics of the syllogism which arises if B is possible of
|
|
the subject of C, and A is possible of the subject of B. For thus both
|
|
premisses are assumed in the mode of possibility; but whenever A is
|
|
possible of that of which B is true, one premiss is a simple
|
|
assertion, the other a problematic. Consequently we must start from
|
|
premisses which are similar in form, as in the other cases.
|
|
|
|
14
|
|
|
|
Whenever A may possibly belong to all B, and B to all C, there
|
|
will be a perfect syllogism to prove that A may possibly belong to all
|
|
C. This is clear from the definition: for it was in this way that we
|
|
explained 'to be possible for one term to belong to all of another'.
|
|
Similarly if it is possible for A to belong no B, and for B to
|
|
belong to all C, then it is possible for A to belong to no C. For
|
|
the statement that it is possible for A not to belong to that of which
|
|
B may be true means (as we saw) that none of those things which can
|
|
possibly fall under the term B is left out of account. But whenever
|
|
A may belong to all B, and B may belong to no C, then indeed no
|
|
syllogism results from the premisses assumed, but if the premiss BC is
|
|
converted after the manner of problematic propositions, the same
|
|
syllogism results as before. For since it is possible that B should
|
|
belong to no C, it is possible also that it should belong to all C.
|
|
This has been stated above. Consequently if B is possible for all C,
|
|
and A is possible for all B, the same syllogism again results.
|
|
Similarly if in both the premisses the negative is joined with 'it
|
|
is possible': e.g. if A may belong to none of the Bs, and B to none of
|
|
the Cs. No syllogism results from the assumed premisses, but if they
|
|
are converted we shall have the same syllogism as before. It is
|
|
clear then that if the minor premiss is negative, or if both premisses
|
|
are negative, either no syllogism results, or if one it is not
|
|
perfect. For the necessity results from the conversion.
|
|
|
|
But if one of the premisses is universal, the other particular, when
|
|
the major premiss is universal there will be a perfect syllogism.
|
|
For if A is possible for all B, and B for some C, then A is possible
|
|
for some C. This is clear from the definition of being possible. Again
|
|
if A may belong to no B, and B may belong to some of the Cs, it is
|
|
necessary that A may possibly not belong to some of the Cs. The
|
|
proof is the same as above. But if the particular premiss is negative,
|
|
and the universal is affirmative, the major still being universal
|
|
and the minor particular, e.g. A is possible for all B, B may possibly
|
|
not belong to some C, then a clear syllogism does not result from
|
|
the assumed premisses, but if the particular premiss is converted
|
|
and it is laid down that B possibly may belong to some C, we shall
|
|
have the same conclusion as before, as in the cases given at the
|
|
beginning.
|
|
|
|
But if the major premiss is the minor universal, whether both are
|
|
affirmative, or negative, or different in quality, or if both are
|
|
indefinite or particular, in no way will a syllogism be possible.
|
|
For nothing prevents B from reaching beyond A, so that as predicates
|
|
cover unequal areas. Let C be that by which B extends beyond A. To C
|
|
it is not possible that A should belong-either to all or to none or to
|
|
some or not to some, since premisses in the mode of possibility are
|
|
convertible and it is possible for B to belong to more things than A
|
|
can. Further, this is obvious if we take terms; for if the premisses
|
|
are as assumed, the major term is both possible for none of the
|
|
minor and must belong to all of it. Take as terms common to all the
|
|
cases under consideration 'animal'-'white'-'man', where the major
|
|
belongs necessarily to the minor; 'animal'-'white'-'garment', where it
|
|
is not possible that the major should belong to the minor. It is clear
|
|
then that if the terms are related in this manner, no syllogism
|
|
results. For every syllogism proves that something belongs either
|
|
simply or necessarily or possibly. It is clear that there is no
|
|
proof of the first or of the second. For the affirmative is
|
|
destroyed by the negative, and the negative by the affirmative.
|
|
There remains the proof of possibility. But this is impossible. For it
|
|
has been proved that if the terms are related in this manner it is
|
|
both necessary that the major should belong to all the minor and not
|
|
possible that it should belong to any. Consequently there cannot be
|
|
a syllogism to prove the possibility; for the necessary (as we stated)
|
|
is not possible.
|
|
|
|
It is clear that if the terms are universal in possible premisses
|
|
a syllogism always results in the first figure, whether they are
|
|
affirmative or negative, only a perfect syllogism results in the first
|
|
case, an imperfect in the second. But possibility must be understood
|
|
according to the definition laid down, not as covering necessity. This
|
|
is sometimes forgotten.
|
|
|
|
15
|
|
|
|
If one premiss is a simple proposition, the other a problematic,
|
|
whenever the major premiss indicates possibility all the syllogisms
|
|
will be perfect and establish possibility in the sense defined; but
|
|
whenever the minor premiss indicates possibility all the syllogisms
|
|
will be imperfect, and those which are negative will establish not
|
|
possibility according to the definition, but that the major does not
|
|
necessarily belong to any, or to all, of the minor. For if this is so,
|
|
we say it is possible that it should belong to none or not to all. Let
|
|
A be possible for all B, and let B belong to all C. Since C falls
|
|
under B, and A is possible for all B, clearly it is possible for all C
|
|
also. So a perfect syllogism results. Likewise if the premiss AB is
|
|
negative, and the premiss BC is affirmative, the former stating
|
|
possible, the latter simple attribution, a perfect syllogism results
|
|
proving that A possibly belongs to no C.
|
|
|
|
It is clear that perfect syllogisms result if the minor premiss
|
|
states simple belonging: but that syllogisms will result if the
|
|
modality of the premisses is reversed, must be proved per impossibile.
|
|
At the same time it will be evident that they are imperfect: for the
|
|
proof proceeds not from the premisses assumed. First we must state
|
|
that if B's being follows necessarily from A's being, B's
|
|
possibility will follow necessarily from A's possibility. Suppose, the
|
|
terms being so related, that A is possible, and B is impossible. If
|
|
then that which is possible, when it is possible for it to be, might
|
|
happen, and if that which is impossible, when it is impossible,
|
|
could not happen, and if at the same time A is possible and B
|
|
impossible, it would be possible for A to happen without B, and if
|
|
to happen, then to be. For that which has happened, when it has
|
|
happened, is. But we must take the impossible and the possible not
|
|
only in the sphere of becoming, but also in the spheres of truth and
|
|
predicability, and the various other spheres in which we speak of
|
|
the possible: for it will be alike in all. Further we must
|
|
understand the statement that B's being depends on A's being, not as
|
|
meaning that if some single thing A is, B will be: for nothing follows
|
|
of necessity from the being of some one thing, but from two at
|
|
least, i.e. when the premisses are related in the manner stated to
|
|
be that of the syllogism. For if C is predicated of D, and D of F,
|
|
then C is necessarily predicated of F. And if each is possible, the
|
|
conclusion also is possible. If then, for example, one should indicate
|
|
the premisses by A, and the conclusion by B, it would not only
|
|
result that if A is necessary B is necessary, but also that if A is
|
|
possible, B is possible.
|
|
|
|
Since this is proved it is evident that if a false and not
|
|
impossible assumption is made, the consequence of the assumption
|
|
will also be false and not impossible: e.g. if A is false, but not
|
|
impossible, and if B is the consequence of A, B also will be false but
|
|
not impossible. For since it has been proved that if B's being is
|
|
the consequence of A's being, then B's possibility will follow from
|
|
A's possibility (and A is assumed to be possible), consequently B will
|
|
be possible: for if it were impossible, the same thing would at the
|
|
same time be possible and impossible.
|
|
|
|
Since we have defined these points, let A belong to all B, and B
|
|
be possible for all C: it is necessary then that should be a
|
|
possible attribute for all C. Suppose that it is not possible, but
|
|
assume that B belongs to all C: this is false but not impossible. If
|
|
then A is not possible for C but B belongs to all C, then A is not
|
|
possible for all B: for a syllogism is formed in the third degree. But
|
|
it was assumed that A is a possible attribute for all B. It is
|
|
necessary then that A is possible for all C. For though the assumption
|
|
we made is false and not impossible, the conclusion is impossible.
|
|
It is possible also in the first figure to bring about the
|
|
impossibility, by assuming that B belongs to C. For if B belongs to
|
|
all C, and A is possible for all B, then A would be possible for all
|
|
C. But the assumption was made that A is not possible for all C.
|
|
|
|
We must understand 'that which belongs to all' with no limitation in
|
|
respect of time, e.g. to the present or to a particular period, but
|
|
simply without qualification. For it is by the help of such
|
|
premisses that we make syllogisms, since if the premiss is
|
|
understood with reference to the present moment, there cannot be a
|
|
syllogism. For nothing perhaps prevents 'man' belonging at a
|
|
particular time to everything that is moving, i.e. if nothing else
|
|
were moving: but 'moving' is possible for every horse; yet 'man' is
|
|
possible for no horse. Further let the major term be 'animal', the
|
|
middle 'moving', the the minor 'man'. The premisses then will be as
|
|
before, but the conclusion necessary, not possible. For man is
|
|
necessarily animal. It is clear then that the universal must be
|
|
understood simply, without limitation in respect of time.
|
|
|
|
Again let the premiss AB be universal and negative, and assume
|
|
that A belongs to no B, but B possibly belongs to all C. These
|
|
propositions being laid down, it is necessary that A possibly
|
|
belongs to no C. Suppose that it cannot belong, and that B belongs
|
|
to C, as above. It is necessary then that A belongs to some B: for
|
|
we have a syllogism in the third figure: but this is impossible.
|
|
Thus it will be possible for A to belong to no C; for if at is
|
|
supposed false, the consequence is an impossible one. This syllogism
|
|
then does not establish that which is possible according to the
|
|
definition, but that which does not necessarily belong to any part
|
|
of the subject (for this is the contradictory of the assumption
|
|
which was made: for it was supposed that A necessarily belongs to some
|
|
C, but the syllogism per impossibile establishes the contradictory
|
|
which is opposed to this). Further, it is clear also from an example
|
|
that the conclusion will not establish possibility. Let A be
|
|
'raven', B 'intelligent', and C 'man'. A then belongs to no B: for
|
|
no intelligent thing is a raven. But B is possible for all C: for
|
|
every man may possibly be intelligent. But A necessarily belongs to no
|
|
C: so the conclusion does not establish possibility. But neither is it
|
|
always necessary. Let A be 'moving', B 'science', C 'man'. A then will
|
|
belong to no B; but B is possible for all C. And the conclusion will
|
|
not be necessary. For it is not necessary that no man should move;
|
|
rather it is not necessary that any man should move. Clearly then
|
|
the conclusion establishes that one term does not necessarily belong
|
|
to any instance of another term. But we must take our terms better.
|
|
|
|
If the minor premiss is negative and indicates possibility, from the
|
|
actual premisses taken there can be no syllogism, but if the
|
|
problematic premiss is converted, a syllogism will be possible, as
|
|
before. Let A belong to all B, and let B possibly belong to no C. If
|
|
the terms are arranged thus, nothing necessarily follows: but if the
|
|
proposition BC is converted and it is assumed that B is possible for
|
|
all C, a syllogism results as before: for the terms are in the same
|
|
relative positions. Likewise if both the relations are negative, if
|
|
the major premiss states that A does not belong to B, and the minor
|
|
premiss indicates that B may possibly belong to no C. Through the
|
|
premisses actually taken nothing necessary results in any way; but
|
|
if the problematic premiss is converted, we shall have a syllogism.
|
|
Suppose that A belongs to no B, and B may possibly belong to no C.
|
|
Through these comes nothing necessary. But if B is assumed to be
|
|
possible for all C (and this is true) and if the premiss AB remains as
|
|
before, we shall again have the same syllogism. But if it be assumed
|
|
that B does not belong to any C, instead of possibly not belonging,
|
|
there cannot be a syllogism anyhow, whether the premiss AB is negative
|
|
or affirmative. As common instances of a necessary and positive
|
|
relation we may take the terms white-animal-snow: of a necessary and
|
|
negative relation, white-animal-pitch. Clearly then if the terms are
|
|
universal, and one of the premisses is assertoric, the other
|
|
problematic, whenever the minor premiss is problematic a syllogism
|
|
always results, only sometimes it results from the premisses that
|
|
are taken, sometimes it requires the conversion of one premiss. We
|
|
have stated when each of these happens and the reason why. But if
|
|
one of the relations is universal, the other particular, then whenever
|
|
the major premiss is universal and problematic, whether affirmative or
|
|
negative, and the particular is affirmative and assertoric, there will
|
|
be a perfect syllogism, just as when the terms are universal. The
|
|
demonstration is the same as before. But whenever the major premiss is
|
|
universal, but assertoric, not problematic, and the minor is
|
|
particular and problematic, whether both premisses are negative or
|
|
affirmative, or one is negative, the other affirmative, in all cases
|
|
there will be an imperfect syllogism. Only some of them will be proved
|
|
per impossibile, others by the conversion of the problematic
|
|
premiss, as has been shown above. And a syllogism will be possible
|
|
by means of conversion when the major premiss is universal and
|
|
assertoric, whether positive or negative, and the minor particular,
|
|
negative, and problematic, e.g. if A belongs to all B or to no B,
|
|
and B may possibly not belong to some C. For if the premiss BC is
|
|
converted in respect of possibility, a syllogism results. But whenever
|
|
the particular premiss is assertoric and negative, there cannot be a
|
|
syllogism. As instances of the positive relation we may take the terms
|
|
white-animal-snow; of the negative, white-animal-pitch. For the
|
|
demonstration must be made through the indefinite nature of the
|
|
particular premiss. But if the minor premiss is universal, and the
|
|
major particular, whether either premiss is negative or affirmative,
|
|
problematic or assertoric, nohow is a syllogism possible. Nor is a
|
|
syllogism possible when the premisses are particular or indefinite,
|
|
whether problematic or assertoric, or the one problematic, the other
|
|
assertoric. The demonstration is the same as above. As instances of
|
|
the necessary and positive relation we may take the terms
|
|
animal-white-man; of the necessary and negative relation,
|
|
animal-white-garment. It is evident then that if the major premiss
|
|
is universal, a syllogism always results, but if the minor is
|
|
universal nothing at all can ever be proved.
|
|
|
|
16
|
|
|
|
Whenever one premiss is necessary, the other problematic, there will
|
|
be a syllogism when the terms are related as before; and a perfect
|
|
syllogism when the minor premiss is necessary. If the premisses are
|
|
affirmative the conclusion will be problematic, not assertoric,
|
|
whether the premisses are universal or not: but if one is affirmative,
|
|
the other negative, when the affirmative is necessary the conclusion
|
|
will be problematic, not negative assertoric; but when the negative is
|
|
necessary the conclusion will be problematic negative, and
|
|
assertoric negative, whether the premisses are universal or not.
|
|
Possibility in the conclusion must be understood in the same manner as
|
|
before. There cannot be an inference to the necessary negative
|
|
proposition: for 'not necessarily to belong' is different from
|
|
'necessarily not to belong'.
|
|
|
|
If the premisses are affirmative, clearly the conclusion which
|
|
follows is not necessary. Suppose A necessarily belongs to all B,
|
|
and let B be possible for all C. We shall have an imperfect
|
|
syllogism to prove that A may belong to all C. That it is imperfect is
|
|
clear from the proof: for it will be proved in the same manner as
|
|
above. Again, let A be possible for all B, and let B necessarily
|
|
belong to all C. We shall then have a syllogism to prove that A may
|
|
belong to all C, not that A does belong to all C: and it is perfect,
|
|
not imperfect: for it is completed directly through the original
|
|
premisses.
|
|
|
|
But if the premisses are not similar in quality, suppose first
|
|
that the negative premiss is necessary, and let necessarily A not be
|
|
possible for any B, but let B be possible for all C. It is necessary
|
|
then that A belongs to no C. For suppose A to belong to all C or to
|
|
some C. Now we assumed that A is not possible for any B. Since then
|
|
the negative proposition is convertible, B is not possible for any
|
|
A. But A is supposed to belong to all C or to some C. Consequently B
|
|
will not be possible for any C or for all C. But it was originally
|
|
laid down that B is possible for all C. And it is clear that the
|
|
possibility of belonging can be inferred, since the fact of not
|
|
belonging is inferred. Again, let the affirmative premiss be
|
|
necessary, and let A possibly not belong to any B, and let B
|
|
necessarily belong to all C. The syllogism will be perfect, but it
|
|
will establish a problematic negative, not an assertoric negative. For
|
|
the major premiss was problematic, and further it is not possible to
|
|
prove the assertoric conclusion per impossibile. For if it were
|
|
supposed that A belongs to some C, and it is laid down that A possibly
|
|
does not belong to any B, no impossible relation between B and C
|
|
follows from these premisses. But if the minor premiss is negative,
|
|
when it is problematic a syllogism is possible by conversion, as
|
|
above; but when it is necessary no syllogism can be formed. Nor
|
|
again when both premisses are negative, and the minor is necessary.
|
|
The same terms as before serve both for the positive
|
|
relation-white-animal-snow, and for the negative
|
|
relation-white-animal-pitch.
|
|
|
|
The same relation will obtain in particular syllogisms. Whenever the
|
|
negative proposition is necessary, the conclusion will be negative
|
|
assertoric: e.g. if it is not possible that A should belong to any
|
|
B, but B may belong to some of the Cs, it is necessary that A should
|
|
not belong to some of the Cs. For if A belongs to all C, but cannot
|
|
belong to any B, neither can B belong to any A. So if A belongs to all
|
|
C, to none of the Cs can B belong. But it was laid down that B may
|
|
belong to some C. But when the particular affirmative in the
|
|
negative syllogism, e.g. BC the minor premiss, or the universal
|
|
proposition in the affirmative syllogism, e.g. AB the major premiss,
|
|
is necessary, there will not be an assertoric conclusion. The
|
|
demonstration is the same as before. But if the minor premiss is
|
|
universal, and problematic, whether affirmative or negative, and the
|
|
major premiss is particular and necessary, there cannot be a
|
|
syllogism. Premisses of this kind are possible both where the relation
|
|
is positive and necessary, e.g. animal-white-man, and where it is
|
|
necessary and negative, e.g. animal-white-garment. But when the
|
|
universal is necessary, the particular problematic, if the universal
|
|
is negative we may take the terms animal-white-raven to illustrate the
|
|
positive relation, or animal-white-pitch to illustrate the negative;
|
|
and if the universal is affirmative we may take the terms
|
|
animal-white-swan to illustrate the positive relation, and
|
|
animal-white-snow to illustrate the negative and necessary relation.
|
|
Nor again is a syllogism possible when the premisses are indefinite,
|
|
or both particular. Terms applicable in either case to illustrate
|
|
the positive relation are animal-white-man: to illustrate the
|
|
negative, animal-white-inanimate. For the relation of animal to some
|
|
white, and of white to some inanimate, is both necessary and
|
|
positive and necessary and negative. Similarly if the relation is
|
|
problematic: so the terms may be used for all cases.
|
|
|
|
Clearly then from what has been said a syllogism results or not from
|
|
similar relations of the terms whether we are dealing with simple
|
|
existence or necessity, with this exception, that if the negative
|
|
premiss is assertoric the conclusion is problematic, but if the
|
|
negative premiss is necessary the conclusion is both problematic and
|
|
negative assertoric. [It is clear also that all the syllogisms are
|
|
imperfect and are perfected by means of the figures above mentioned.]
|
|
|
|
17
|
|
|
|
In the second figure whenever both premisses are problematic, no
|
|
syllogism is possible, whether the premisses are affirmative or
|
|
negative, universal or particular. But when one premiss is assertoric,
|
|
the other problematic, if the affirmative is assertoric no syllogism
|
|
is possible, but if the universal negative is assertoric a
|
|
conclusion can always be drawn. Similarly when one premiss is
|
|
necessary, the other problematic. Here also we must understand the
|
|
term 'possible' in the conclusion, in the same sense as before.
|
|
|
|
First we must point out that the negative problematic proposition is
|
|
not convertible, e.g. if A may belong to no B, it does not follow that
|
|
B may belong to no A. For suppose it to follow and assume that B may
|
|
belong to no A. Since then problematic affirmations are convertible
|
|
with negations, whether they are contraries or contradictories, and
|
|
since B may belong to no A, it is clear that B may belong to all A.
|
|
But this is false: for if all this can be that, it does not follow
|
|
that all that can be this: consequently the negative proposition is
|
|
not convertible. Further, these propositions are not incompatible,
|
|
'A may belong to no B', 'B necessarily does not belong to some of
|
|
the As'; e.g. it is possible that no man should be white (for it is
|
|
also possible that every man should be white), but it is not true to
|
|
say that it is possible that no white thing should be a man: for
|
|
many white things are necessarily not men, and the necessary (as we
|
|
saw) other than the possible.
|
|
|
|
Moreover it is not possible to prove the convertibility of these
|
|
propositions by a reductio ad absurdum, i.e. by claiming assent to the
|
|
following argument: 'since it is false that B may belong to no A, it
|
|
is true that it cannot belong to no A, for the one statement is the
|
|
contradictory of the other. But if this is so, it is true that B
|
|
necessarily belongs to some of the As: consequently A necessarily
|
|
belongs to some of the Bs. But this is impossible.' The argument
|
|
cannot be admitted, for it does not follow that some A is
|
|
necessarily B, if it is not possible that no A should be B. For the
|
|
latter expression is used in two senses, one if A some is
|
|
necessarily B, another if some A is necessarily not B. For it is not
|
|
true to say that that which necessarily does not belong to some of the
|
|
As may possibly not belong to any A, just as it is not true to say
|
|
that what necessarily belongs to some A may possibly belong to all
|
|
A. If any one then should claim that because it is not possible for
|
|
C to belong to all D, it necessarily does not belong to some D, he
|
|
would make a false assumption: for it does belong to all D, but
|
|
because in some cases it belongs necessarily, therefore we say that it
|
|
is not possible for it to belong to all. Hence both the propositions
|
|
'A necessarily belongs to some B' and 'A necessarily does not belong
|
|
to some B' are opposed to the proposition 'A belongs to all B'.
|
|
Similarly also they are opposed to the proposition 'A may belong to no
|
|
B'. It is clear then that in relation to what is possible and not
|
|
possible, in the sense originally defined, we must assume, not that
|
|
A necessarily belongs to some B, but that A necessarily does not
|
|
belong to some B. But if this is assumed, no absurdity results:
|
|
consequently no syllogism. It is clear from what has been said that
|
|
the negative proposition is not convertible.
|
|
|
|
This being proved, suppose it possible that A may belong to no B and
|
|
to all C. By means of conversion no syllogism will result: for the
|
|
major premiss, as has been said, is not convertible. Nor can a proof
|
|
be obtained by a reductio ad absurdum: for if it is assumed that B can
|
|
belong to all C, no false consequence results: for A may belong both
|
|
to all C and to no C. In general, if there is a syllogism, it is clear
|
|
that its conclusion will be problematic because neither of the
|
|
premisses is assertoric; and this must be either affirmative or
|
|
negative. But neither is possible. Suppose the conclusion is
|
|
affirmative: it will be proved by an example that the predicate cannot
|
|
belong to the subject. Suppose the conclusion is negative: it will
|
|
be proved that it is not problematic but necessary. Let A be white,
|
|
B man, C horse. It is possible then for A to belong to all of the
|
|
one and to none of the other. But it is not possible for B to belong
|
|
nor not to belong to C. That it is not possible for it to belong, is
|
|
clear. For no horse is a man. Neither is it possible for it not to
|
|
belong. For it is necessary that no horse should be a man, but the
|
|
necessary we found to be different from the possible. No syllogism
|
|
then results. A similar proof can be given if the major premiss is
|
|
negative, the minor affirmative, or if both are affirmative or
|
|
negative. The demonstration can be made by means of the same terms.
|
|
And whenever one premiss is universal, the other particular, or both
|
|
are particular or indefinite, or in whatever other way the premisses
|
|
can be altered, the proof will always proceed through the same
|
|
terms. Clearly then, if both the premisses are problematic, no
|
|
syllogism results.
|
|
|
|
18
|
|
|
|
But if one premiss is assertoric, the other problematic, if the
|
|
affirmative is assertoric and the negative problematic no syllogism
|
|
will be possible, whether the premisses are universal or particular.
|
|
The proof is the same as above, and by means of the same terms. But
|
|
when the affirmative premiss is problematic, and the negative
|
|
assertoric, we shall have a syllogism. Suppose A belongs to no B,
|
|
but can belong to all C. If the negative proposition is converted, B
|
|
will belong to no A. But ex hypothesi can belong to all C: so a
|
|
syllogism is made, proving by means of the first figure that B may
|
|
belong to no C. Similarly also if the minor premiss is negative. But
|
|
if both premisses are negative, one being assertoric, the other
|
|
problematic, nothing follows necessarily from these premisses as
|
|
they stand, but if the problematic premiss is converted into its
|
|
complementary affirmative a syllogism is formed to prove that B may
|
|
belong to no C, as before: for we shall again have the first figure.
|
|
But if both premisses are affirmative, no syllogism will be
|
|
possible. This arrangement of terms is possible both when the relation
|
|
is positive, e.g. health, animal, man, and when it is negative, e.g.
|
|
health, horse, man.
|
|
|
|
The same will hold good if the syllogisms are particular. Whenever
|
|
the affirmative proposition is assertoric, whether universal or
|
|
particular, no syllogism is possible (this is proved similarly and
|
|
by the same examples as above), but when the negative proposition is
|
|
assertoric, a conclusion can be drawn by means of conversion, as
|
|
before. Again if both the relations are negative, and the assertoric
|
|
proposition is universal, although no conclusion follows from the
|
|
actual premisses, a syllogism can be obtained by converting the
|
|
problematic premiss into its complementary affirmative as before.
|
|
But if the negative proposition is assertoric, but particular, no
|
|
syllogism is possible, whether the other premiss is affirmative or
|
|
negative. Nor can a conclusion be drawn when both premisses are
|
|
indefinite, whether affirmative or negative, or particular. The
|
|
proof is the same and by the same terms.
|
|
|
|
19
|
|
|
|
If one of the premisses is necessary, the other problematic, then if
|
|
the negative is necessary a syllogistic conclusion can be drawn, not
|
|
merely a negative problematic but also a negative assertoric
|
|
conclusion; but if the affirmative premiss is necessary, no conclusion
|
|
is possible. Suppose that A necessarily belongs to no B, but may
|
|
belong to all C. If the negative premiss is converted B will belong to
|
|
no A: but A ex hypothesi is capable of belonging to all C: so once
|
|
more a conclusion is drawn by the first figure that B may belong to no
|
|
C. But at the same time it is clear that B will not belong to any C.
|
|
For assume that it does: then if A cannot belong to any B, and B
|
|
belongs to some of the Cs, A cannot belong to some of the Cs: but ex
|
|
hypothesi it may belong to all. A similar proof can be given if the
|
|
minor premiss is negative. Again let the affirmative proposition be
|
|
necessary, and the other problematic; i.e. suppose that A may belong
|
|
to no B, but necessarily belongs to all C. When the terms are arranged
|
|
in this way, no syllogism is possible. For (1) it sometimes turns
|
|
out that B necessarily does not belong to C. Let A be white, B man,
|
|
C swan. White then necessarily belongs to swan, but may belong to no
|
|
man; and man necessarily belongs to no swan; Clearly then we cannot
|
|
draw a problematic conclusion; for that which is necessary is
|
|
admittedly distinct from that which is possible. (2) Nor again can
|
|
we draw a necessary conclusion: for that presupposes that both
|
|
premisses are necessary, or at any rate the negative premiss. (3)
|
|
Further it is possible also, when the terms are so arranged, that B
|
|
should belong to C: for nothing prevents C falling under B, A being
|
|
possible for all B, and necessarily belonging to C; e.g. if C stands
|
|
for 'awake', B for 'animal', A for 'motion'. For motion necessarily
|
|
belongs to what is awake, and is possible for every animal: and
|
|
everything that is awake is animal. Clearly then the conclusion cannot
|
|
be the negative assertion, if the relation must be positive when the
|
|
terms are related as above. Nor can the opposite affirmations be
|
|
established: consequently no syllogism is possible. A similar proof is
|
|
possible if the major premiss is affirmative.
|
|
|
|
But if the premisses are similar in quality, when they are
|
|
negative a syllogism can always be formed by converting the
|
|
problematic premiss into its complementary affirmative as before.
|
|
Suppose A necessarily does not belong to B, and possibly may not
|
|
belong to C: if the premisses are converted B belongs to no A, and A
|
|
may possibly belong to all C: thus we have the first figure. Similarly
|
|
if the minor premiss is negative. But if the premisses are affirmative
|
|
there cannot be a syllogism. Clearly the conclusion cannot be a
|
|
negative assertoric or a negative necessary proposition because no
|
|
negative premiss has been laid down either in the assertoric or in the
|
|
necessary mode. Nor can the conclusion be a problematic negative
|
|
proposition. For if the terms are so related, there are cases in which
|
|
B necessarily will not belong to C; e.g. suppose that A is white, B
|
|
swan, C man. Nor can the opposite affirmations be established, since
|
|
we have shown a case in which B necessarily does not belong to C. A
|
|
syllogism then is not possible at all.
|
|
|
|
Similar relations will obtain in particular syllogisms. For whenever
|
|
the negative proposition is universal and necessary, a syllogism
|
|
will always be possible to prove both a problematic and a negative
|
|
assertoric proposition (the proof proceeds by conversion); but when
|
|
the affirmative proposition is universal and necessary, no syllogistic
|
|
conclusion can be drawn. This can be proved in the same way as for
|
|
universal propositions, and by the same terms. Nor is a syllogistic
|
|
conclusion possible when both premisses are affirmative: this also may
|
|
be proved as above. But when both premisses are negative, and the
|
|
premiss that definitely disconnects two terms is universal and
|
|
necessary, though nothing follows necessarily from the premisses as
|
|
they are stated, a conclusion can be drawn as above if the problematic
|
|
premiss is converted into its complementary affirmative. But if both
|
|
are indefinite or particular, no syllogism can be formed. The same
|
|
proof will serve, and the same terms.
|
|
|
|
It is clear then from what has been said that if the universal and
|
|
negative premiss is necessary, a syllogism is always possible, proving
|
|
not merely a negative problematic, but also a negative assertoric
|
|
proposition; but if the affirmative premiss is necessary no conclusion
|
|
can be drawn. It is clear too that a syllogism is possible or not
|
|
under the same conditions whether the mode of the premisses is
|
|
assertoric or necessary. And it is clear that all the syllogisms are
|
|
imperfect, and are completed by means of the figures mentioned.
|
|
|
|
20
|
|
|
|
In the last figure a syllogism is possible whether both or only
|
|
one of the premisses is problematic. When the premisses are
|
|
problematic the conclusion will be problematic; and also when one
|
|
premiss is problematic, the other assertoric. But when the other
|
|
premiss is necessary, if it is affirmative the conclusion will be
|
|
neither necessary or assertoric; but if it is negative the syllogism
|
|
will result in a negative assertoric proposition, as above. In these
|
|
also we must understand the expression 'possible' in the conclusion in
|
|
the same way as before.
|
|
|
|
First let the premisses be problematic and suppose that both A and B
|
|
may possibly belong to every C. Since then the affirmative proposition
|
|
is convertible into a particular, and B may possibly belong to every
|
|
C, it follows that C may possibly belong to some B. So, if A is
|
|
possible for every C, and C is possible for some of the Bs, then A
|
|
is possible for some of the Bs. For we have got the first figure.
|
|
And A if may possibly belong to no C, but B may possibly belong to all
|
|
C, it follows that A may possibly not belong to some B: for we shall
|
|
have the first figure again by conversion. But if both premisses
|
|
should be negative no necessary consequence will follow from them as
|
|
they are stated, but if the premisses are converted into their
|
|
corresponding affirmatives there will be a syllogism as before. For if
|
|
A and B may possibly not belong to C, if 'may possibly belong' is
|
|
substituted we shall again have the first figure by means of
|
|
conversion. But if one of the premisses is universal, the other
|
|
particular, a syllogism will be possible, or not, under the
|
|
arrangement of the terms as in the case of assertoric propositions.
|
|
Suppose that A may possibly belong to all C, and B to some C. We shall
|
|
have the first figure again if the particular premiss is converted.
|
|
For if A is possible for all C, and C for some of the Bs, then A is
|
|
possible for some of the Bs. Similarly if the proposition BC is
|
|
universal. Likewise also if the proposition AC is negative, and the
|
|
proposition BC affirmative: for we shall again have the first figure
|
|
by conversion. But if both premisses should be negative-the one
|
|
universal and the other particular-although no syllogistic
|
|
conclusion will follow from the premisses as they are put, it will
|
|
follow if they are converted, as above. But when both premisses are
|
|
indefinite or particular, no syllogism can be formed: for A must
|
|
belong sometimes to all B and sometimes to no B. To illustrate the
|
|
affirmative relation take the terms animal-man-white; to illustrate
|
|
the negative, take the terms horse-man-white--white being the middle
|
|
term.
|
|
|
|
21
|
|
|
|
If one premiss is pure, the other problematic, the conclusion will
|
|
be problematic, not pure; and a syllogism will be possible under the
|
|
same arrangement of the terms as before. First let the premisses be
|
|
affirmative: suppose that A belongs to all C, and B may possibly
|
|
belong to all C. If the proposition BC is converted, we shall have the
|
|
first figure, and the conclusion that A may possibly belong to some of
|
|
the Bs. For when one of the premisses in the first figure is
|
|
problematic, the conclusion also (as we saw) is problematic. Similarly
|
|
if the proposition BC is pure, AC problematic; or if AC is negative,
|
|
BC affirmative, no matter which of the two is pure; in both cases
|
|
the conclusion will be problematic: for the first figure is obtained
|
|
once more, and it has been proved that if one premiss is problematic
|
|
in that figure the conclusion also will be problematic. But if the
|
|
minor premiss BC is negative, or if both premisses are negative, no
|
|
syllogistic conclusion can be drawn from the premisses as they
|
|
stand, but if they are converted a syllogism is obtained as before.
|
|
|
|
If one of the premisses is universal, the other particular, then
|
|
when both are affirmative, or when the universal is negative, the
|
|
particular affirmative, we shall have the same sort of syllogisms: for
|
|
all are completed by means of the first figure. So it is clear that we
|
|
shall have not a pure but a problematic syllogistic conclusion. But if
|
|
the affirmative premiss is universal, the negative particular, the
|
|
proof will proceed by a reductio ad impossibile. Suppose that B
|
|
belongs to all C, and A may possibly not belong to some C: it
|
|
follows that may possibly not belong to some B. For if A necessarily
|
|
belongs to all B, and B (as has been assumed) belongs to all C, A will
|
|
necessarily belong to all C: for this has been proved before. But it
|
|
was assumed at the outset that A may possibly not belong to some C.
|
|
|
|
Whenever both premisses are indefinite or particular, no syllogism
|
|
will be possible. The demonstration is the same as was given in the
|
|
case of universal premisses, and proceeds by means of the same terms.
|
|
|
|
22
|
|
|
|
If one of the premisses is necessary, the other problematic, when
|
|
the premisses are affirmative a problematic affirmative conclusion can
|
|
always be drawn; when one proposition is affirmative, the other
|
|
negative, if the affirmative is necessary a problematic negative can
|
|
be inferred; but if the negative proposition is necessary both a
|
|
problematic and a pure negative conclusion are possible. But a
|
|
necessary negative conclusion will not be possible, any more than in
|
|
the other figures. Suppose first that the premisses are affirmative,
|
|
i.e. that A necessarily belongs to all C, and B may possibly belong to
|
|
all C. Since then A must belong to all C, and C may belong to some
|
|
B, it follows that A may (not does) belong to some B: for so it
|
|
resulted in the first figure. A similar proof may be given if the
|
|
proposition BC is necessary, and AC is problematic. Again suppose
|
|
one proposition is affirmative, the other negative, the affirmative
|
|
being necessary: i.e. suppose A may possibly belong to no C, but B
|
|
necessarily belongs to all C. We shall have the first figure once
|
|
more: and-since the negative premiss is problematic-it is clear that
|
|
the conclusion will be problematic: for when the premisses stand
|
|
thus in the first figure, the conclusion (as we found) is problematic.
|
|
But if the negative premiss is necessary, the conclusion will be not
|
|
only that A may possibly not belong to some B but also that it does
|
|
not belong to some B. For suppose that A necessarily does not belong
|
|
to C, but B may belong to all C. If the affirmative proposition BC
|
|
is converted, we shall have the first figure, and the negative premiss
|
|
is necessary. But when the premisses stood thus, it resulted that A
|
|
might possibly not belong to some C, and that it did not belong to
|
|
some C; consequently here it follows that A does not belong to some B.
|
|
But when the minor premiss is negative, if it is problematic we
|
|
shall have a syllogism by altering the premiss into its
|
|
complementary affirmative, as before; but if it is necessary no
|
|
syllogism can be formed. For A sometimes necessarily belongs to all B,
|
|
and sometimes cannot possibly belong to any B. To illustrate the
|
|
former take the terms sleep-sleeping horse-man; to illustrate the
|
|
latter take the terms sleep-waking horse-man.
|
|
|
|
Similar results will obtain if one of the terms is related
|
|
universally to the middle, the other in part. If both premisses are
|
|
affirmative, the conclusion will be problematic, not pure; and also
|
|
when one premiss is negative, the other affirmative, the latter
|
|
being necessary. But when the negative premiss is necessary, the
|
|
conclusion also will be a pure negative proposition; for the same kind
|
|
of proof can be given whether the terms are universal or not. For
|
|
the syllogisms must be made perfect by means of the first figure, so
|
|
that a result which follows in the first figure follows also in the
|
|
third. But when the minor premiss is negative and universal, if it
|
|
is problematic a syllogism can be formed by means of conversion; but
|
|
if it is necessary a syllogism is not possible. The proof will
|
|
follow the same course as where the premisses are universal; and the
|
|
same terms may be used.
|
|
|
|
It is clear then in this figure also when and how a syllogism can be
|
|
formed, and when the conclusion is problematic, and when it is pure.
|
|
It is evident also that all syllogisms in this figure are imperfect,
|
|
and that they are made perfect by means of the first figure.
|
|
|
|
23
|
|
|
|
It is clear from what has been said that the syllogisms in these
|
|
figures are made perfect by means of universal syllogisms in the first
|
|
figure and are reduced to them. That every syllogism without
|
|
qualification can be so treated, will be clear presently, when it
|
|
has been proved that every syllogism is formed through one or other of
|
|
these figures.
|
|
|
|
It is necessary that every demonstration and every syllogism
|
|
should prove either that something belongs or that it does not, and
|
|
this either universally or in part, and further either ostensively
|
|
or hypothetically. One sort of hypothetical proof is the reductio ad
|
|
impossibile. Let us speak first of ostensive syllogisms: for after
|
|
these have been pointed out the truth of our contention will be
|
|
clear with regard to those which are proved per impossibile, and in
|
|
general hypothetically.
|
|
|
|
If then one wants to prove syllogistically A of B, either as an
|
|
attribute of it or as not an attribute of it, one must assert
|
|
something of something else. If now A should be asserted of B, the
|
|
proposition originally in question will have been assumed. But if A
|
|
should be asserted of C, but C should not be asserted of anything, nor
|
|
anything of it, nor anything else of A, no syllogism will be possible.
|
|
For nothing necessarily follows from the assertion of some one thing
|
|
concerning some other single thing. Thus we must take another
|
|
premiss as well. If then A be asserted of something else, or something
|
|
else of A, or something different of C, nothing prevents a syllogism
|
|
being formed, but it will not be in relation to B through the
|
|
premisses taken. Nor when C belongs to something else, and that to
|
|
something else and so on, no connexion however being made with B, will
|
|
a syllogism be possible concerning A in its relation to B. For in
|
|
general we stated that no syllogism can establish the attribution of
|
|
one thing to another, unless some middle term is taken, which is
|
|
somehow related to each by way of predication. For the syllogism in
|
|
general is made out of premisses, and a syllogism referring to this
|
|
out of premisses with the same reference, and a syllogism relating
|
|
this to that proceeds through premisses which relate this to that. But
|
|
it is impossible to take a premiss in reference to B, if we neither
|
|
affirm nor deny anything of it; or again to take a premiss relating
|
|
A to B, if we take nothing common, but affirm or deny peculiar
|
|
attributes of each. So we must take something midway between the
|
|
two, which will connect the predications, if we are to have a
|
|
syllogism relating this to that. If then we must take something common
|
|
in relation to both, and this is possible in three ways (either by
|
|
predicating A of C, and C of B, or C of both, or both of C), and these
|
|
are the figures of which we have spoken, it is clear that every
|
|
syllogism must be made in one or other of these figures. The
|
|
argument is the same if several middle terms should be necessary to
|
|
establish the relation to B; for the figure will be the same whether
|
|
there is one middle term or many.
|
|
|
|
It is clear then that the ostensive syllogisms are effected by means
|
|
of the aforesaid figures; these considerations will show that
|
|
reductiones ad also are effected in the same way. For all who effect
|
|
an argument per impossibile infer syllogistically what is false, and
|
|
prove the original conclusion hypothetically when something impossible
|
|
results from the assumption of its contradictory; e.g. that the
|
|
diagonal of the square is incommensurate with the side, because odd
|
|
numbers are equal to evens if it is supposed to be commensurate. One
|
|
infers syllogistically that odd numbers come out equal to evens, and
|
|
one proves hypothetically the incommensurability of the diagonal,
|
|
since a falsehood results through contradicting this. For this we
|
|
found to be reasoning per impossibile, viz. proving something
|
|
impossible by means of an hypothesis conceded at the beginning.
|
|
Consequently, since the falsehood is established in reductions ad
|
|
impossibile by an ostensive syllogism, and the original conclusion
|
|
is proved hypothetically, and we have already stated that ostensive
|
|
syllogisms are effected by means of these figures, it is evident
|
|
that syllogisms per impossibile also will be made through these
|
|
figures. Likewise all the other hypothetical syllogisms: for in
|
|
every case the syllogism leads up to the proposition that is
|
|
substituted for the original thesis; but the original thesis is
|
|
reached by means of a concession or some other hypothesis. But if this
|
|
is true, every demonstration and every syllogism must be formed by
|
|
means of the three figures mentioned above. But when this has been
|
|
shown it is clear that every syllogism is perfected by means of the
|
|
first figure and is reducible to the universal syllogisms in this
|
|
figure.
|
|
|
|
24
|
|
|
|
Further in every syllogism one of the premisses must be affirmative,
|
|
and universality must be present: unless one of the premisses is
|
|
universal either a syllogism will not be possible, or it will not
|
|
refer to the subject proposed, or the original position will be
|
|
begged. Suppose we have to prove that pleasure in music is good. If
|
|
one should claim as a premiss that pleasure is good without adding
|
|
'all', no syllogism will be possible; if one should claim that some
|
|
pleasure is good, then if it is different from pleasure in music, it
|
|
is not relevant to the subject proposed; if it is this very
|
|
pleasure, one is assuming that which was proposed at the outset to
|
|
be proved. This is more obvious in geometrical proofs, e.g. that the
|
|
angles at the base of an isosceles triangle are equal. Suppose the
|
|
lines A and B have been drawn to the centre. If then one should assume
|
|
that the angle AC is equal to the angle BD, without claiming generally
|
|
that angles of semicircles are equal; and again if one should assume
|
|
that the angle C is equal to the angle D, without the additional
|
|
assumption that every angle of a segment is equal to every other angle
|
|
of the same segment; and further if one should assume that when
|
|
equal angles are taken from the whole angles, which are themselves
|
|
equal, the remainders E and F are equal, he will beg the thing to be
|
|
proved, unless he also states that when equals are taken from equals
|
|
the remainders are equal.
|
|
|
|
It is clear then that in every syllogism there must be a universal
|
|
premiss, and that a universal statement is proved only when all the
|
|
premisses are universal, while a particular statement is proved both
|
|
from two universal premisses and from one only: consequently if the
|
|
conclusion is universal, the premisses also must be universal, but
|
|
if the premisses are universal it is possible that the conclusion
|
|
may not be universal. And it is clear also that in every syllogism
|
|
either both or one of the premisses must be like the conclusion. I
|
|
mean not only in being affirmative or negative, but also in being
|
|
necessary, pure, problematic. We must consider also the other forms of
|
|
predication.
|
|
|
|
It is clear also when a syllogism in general can be made and when it
|
|
cannot; and when a valid, when a perfect syllogism can be formed;
|
|
and that if a syllogism is formed the terms must be arranged in one of
|
|
the ways that have been mentioned.
|
|
|
|
25
|
|
|
|
It is clear too that every demonstration will proceed through
|
|
three terms and no more, unless the same conclusion is established
|
|
by different pairs of propositions; e.g. the conclusion E may be
|
|
established through the propositions A and B, and through the
|
|
propositions C and D, or through the propositions A and B, or A and C,
|
|
or B and C. For nothing prevents there being several middles for the
|
|
same terms. But in that case there is not one but several
|
|
syllogisms. Or again when each of the propositions A and B is obtained
|
|
by syllogistic inference, e.g. by means of D and E, and again B by
|
|
means of F and G. Or one may be obtained by syllogistic, the other
|
|
by inductive inference. But thus also the syllogisms are many; for the
|
|
conclusions are many, e.g. A and B and C. But if this can be called
|
|
one syllogism, not many, the same conclusion may be reached by more
|
|
than three terms in this way, but it cannot be reached as C is
|
|
established by means of A and B. Suppose that the proposition E is
|
|
inferred from the premisses A, B, C, and D. It is necessary then
|
|
that of these one should be related to another as whole to part: for
|
|
it has already been proved that if a syllogism is formed some of its
|
|
terms must be related in this way. Suppose then that A stands in
|
|
this relation to B. Some conclusion then follows from them. It must
|
|
either be E or one or other of C and D, or something other than these.
|
|
|
|
(1) If it is E the syllogism will have A and B for its sole
|
|
premisses. But if C and D are so related that one is whole, the
|
|
other part, some conclusion will follow from them also; and it must be
|
|
either E, or one or other of the propositions A and B, or something
|
|
other than these. And if it is (i) E, or (ii) A or B, either (i) the
|
|
syllogisms will be more than one, or (ii) the same thing happens to be
|
|
inferred by means of several terms only in the sense which we saw to
|
|
be possible. But if (iii) the conclusion is other than E or A or B,
|
|
the syllogisms will be many, and unconnected with one another. But
|
|
if C is not so related to D as to make a syllogism, the propositions
|
|
will have been assumed to no purpose, unless for the sake of induction
|
|
or of obscuring the argument or something of the sort.
|
|
|
|
(2) But if from the propositions A and B there follows not E but
|
|
some other conclusion, and if from C and D either A or B follows or
|
|
something else, then there are several syllogisms, and they do not
|
|
establish the conclusion proposed: for we assumed that the syllogism
|
|
proved E. And if no conclusion follows from C and D, it turns out that
|
|
these propositions have been assumed to no purpose, and the
|
|
syllogism does not prove the original proposition.
|
|
|
|
So it is clear that every demonstration and every syllogism will
|
|
proceed through three terms only.
|
|
|
|
This being evident, it is clear that a syllogistic conclusion
|
|
follows from two premisses and not from more than two. For the three
|
|
terms make two premisses, unless a new premiss is assumed, as was said
|
|
at the beginning, to perfect the syllogisms. It is clear therefore
|
|
that in whatever syllogistic argument the premisses through which
|
|
the main conclusion follows (for some of the preceding conclusions
|
|
must be premisses) are not even in number, this argument either has
|
|
not been drawn syllogistically or it has assumed more than was
|
|
necessary to establish its thesis.
|
|
|
|
If then syllogisms are taken with respect to their main premisses,
|
|
every syllogism will consist of an even number of premisses and an odd
|
|
number of terms (for the terms exceed the premisses by one), and the
|
|
conclusions will be half the number of the premisses. But whenever a
|
|
conclusion is reached by means of prosyllogisms or by means of several
|
|
continuous middle terms, e.g. the proposition AB by means of the
|
|
middle terms C and D, the number of the terms will similarly exceed
|
|
that of the premisses by one (for the extra term must either be
|
|
added outside or inserted: but in either case it follows that the
|
|
relations of predication are one fewer than the terms related), and
|
|
the premisses will be equal in number to the relations of predication.
|
|
The premisses however will not always be even, the terms odd; but they
|
|
will alternate-when the premisses are even, the terms must be odd;
|
|
when the terms are even, the premisses must be odd: for along with one
|
|
term one premiss is added, if a term is added from any quarter.
|
|
Consequently since the premisses were (as we saw) even, and the
|
|
terms odd, we must make them alternately even and odd at each
|
|
addition. But the conclusions will not follow the same arrangement
|
|
either in respect to the terms or to the premisses. For if one term is
|
|
added, conclusions will be added less by one than the pre-existing
|
|
terms: for the conclusion is drawn not in relation to the single
|
|
term last added, but in relation to all the rest, e.g. if to ABC the
|
|
term D is added, two conclusions are thereby added, one in relation to
|
|
A, the other in relation to B. Similarly with any further additions.
|
|
And similarly too if the term is inserted in the middle: for in
|
|
relation to one term only, a syllogism will not be constructed.
|
|
Consequently the conclusions will be much more numerous than the terms
|
|
or the premisses.
|
|
|
|
26
|
|
|
|
Since we understand the subjects with which syllogisms are
|
|
concerned, what sort of conclusion is established in each figure,
|
|
and in how many moods this is done, it is evident to us both what sort
|
|
of problem is difficult and what sort is easy to prove. For that which
|
|
is concluded in many figures and through many moods is easier; that
|
|
which is concluded in few figures and through few moods is more
|
|
difficult to attempt. The universal affirmative is proved by means
|
|
of the first figure only and by this in only one mood; the universal
|
|
negative is proved both through the first figure and through the
|
|
second, through the first in one mood, through the second in two.
|
|
The particular affirmative is proved through the first and through the
|
|
last figure, in one mood through the first, in three moods through the
|
|
last. The particular negative is proved in all the figures, but once
|
|
in the first, in two moods in the second, in three moods in the third.
|
|
It is clear then that the universal affirmative is most difficult to
|
|
establish, most easy to overthrow. In general, universals are easier
|
|
game for the destroyer than particulars: for whether the predicate
|
|
belongs to none or not to some, they are destroyed: and the particular
|
|
negative is proved in all the figures, the universal negative in
|
|
two. Similarly with universal negatives: the original statement is
|
|
destroyed, whether the predicate belongs to all or to some: and this
|
|
we found possible in two figures. But particular statements can be
|
|
refuted in one way only-by proving that the predicate belongs either
|
|
to all or to none. But particular statements are easier to
|
|
establish: for proof is possible in more figures and through more
|
|
moods. And in general we must not forget that it is possible to refute
|
|
statements by means of one another, I mean, universal statements by
|
|
means of particular, and particular statements by means of
|
|
universal: but it is not possible to establish universal statements by
|
|
means of particular, though it is possible to establish particular
|
|
statements by means of universal. At the same time it is evident
|
|
that it is easier to refute than to establish.
|
|
|
|
The manner in which every syllogism is produced, the number of the
|
|
terms and premisses through which it proceeds, the relation of the
|
|
premisses to one another, the character of the problem proved in
|
|
each figure, and the number of the figures appropriate to each
|
|
problem, all these matters are clear from what has been said.
|
|
|
|
27
|
|
|
|
We must now state how we may ourselves always have a supply of
|
|
syllogisms in reference to the problem proposed and by what road we
|
|
may reach the principles relative to the problem: for perhaps we ought
|
|
not only to investigate the construction of syllogisms, but also to
|
|
have the power of making them.
|
|
|
|
Of all the things which exist some are such that they cannot be
|
|
predicated of anything else truly and universally, e.g. Cleon and
|
|
Callias, i.e. the individual and sensible, but other things may be
|
|
predicated of them (for each of these is both man and animal); and
|
|
some things are themselves predicated of others, but nothing prior
|
|
is predicated of them; and some are predicated of others, and yet
|
|
others of them, e.g. man of Callias and animal of man. It is clear
|
|
then that some things are naturally not stated of anything: for as a
|
|
rule each sensible thing is such that it cannot be predicated of
|
|
anything, save incidentally: for we sometimes say that that white
|
|
object is Socrates, or that that which approaches is Callias. We shall
|
|
explain in another place that there is an upward limit also to the
|
|
process of predicating: for the present we must assume this. Of
|
|
these ultimate predicates it is not possible to demonstrate another
|
|
predicate, save as a matter of opinion, but these may be predicated of
|
|
other things. Neither can individuals be predicated of other things,
|
|
though other things can be predicated of them. Whatever lies between
|
|
these limits can be spoken of in both ways: they may be stated of
|
|
others, and others stated of them. And as a rule arguments and
|
|
inquiries are concerned with these things. We must select the
|
|
premisses suitable to each problem in this manner: first we must lay
|
|
down the subject and the definitions and the properties of the
|
|
thing; next we must lay down those attributes which follow the
|
|
thing, and again those which the thing follows, and those which cannot
|
|
belong to it. But those to which it cannot belong need not be
|
|
selected, because the negative statement implied above is convertible.
|
|
Of the attributes which follow we must distinguish those which fall
|
|
within the definition, those which are predicated as properties, and
|
|
those which are predicated as accidents, and of the latter those which
|
|
apparently and those which really belong. The larger the supply a
|
|
man has of these, the more quickly will he reach a conclusion; and
|
|
in proportion as he apprehends those which are truer, the more
|
|
cogently will he demonstrate. But he must select not those which
|
|
follow some particular but those which follow the thing as a whole,
|
|
e.g. not what follows a particular man but what follows every man: for
|
|
the syllogism proceeds through universal premisses. If the statement
|
|
is indefinite, it is uncertain whether the premiss is universal, but
|
|
if the statement is definite, the matter is clear. Similarly one
|
|
must select those attributes which the subject follows as wholes,
|
|
for the reason given. But that which follows one must not suppose to
|
|
follow as a whole, e.g. that every animal follows man or every science
|
|
music, but only that it follows, without qualification, and indeed
|
|
we state it in a proposition: for the other statement is useless and
|
|
impossible, e.g. that every man is every animal or justice is all
|
|
good. But that which something follows receives the mark 'every'.
|
|
Whenever the subject, for which we must obtain the attributes that
|
|
follow, is contained by something else, what follows or does not
|
|
follow the highest term universally must not be selected in dealing
|
|
with the subordinate term (for these attributes have been taken in
|
|
dealing with the superior term; for what follows animal also follows
|
|
man, and what does not belong to animal does not belong to man); but
|
|
we must choose those attributes which are peculiar to each subject.
|
|
For some things are peculiar to the species as distinct from the
|
|
genus; for species being distinct there must be attributes peculiar to
|
|
each. Nor must we take as things which the superior term follows,
|
|
those things which the inferior term follows, e.g. take as subjects of
|
|
the predicate 'animal' what are really subjects of the predicate
|
|
'man'. It is necessary indeed, if animal follows man, that it should
|
|
follow all these also. But these belong more properly to the choice of
|
|
what concerns man. One must apprehend also normal consequents and
|
|
normal antecedents-, for propositions which obtain normally are
|
|
established syllogistically from premisses which obtain normally, some
|
|
if not all of them having this character of normality. For the
|
|
conclusion of each syllogism resembles its principles. We must not
|
|
however choose attributes which are consequent upon all the terms: for
|
|
no syllogism can be made out of such premisses. The reason why this is
|
|
so will be clear in the sequel.
|
|
|
|
28
|
|
|
|
If men wish to establish something about some whole, they must
|
|
look to the subjects of that which is being established (the
|
|
subjects of which it happens to be asserted), and the attributes which
|
|
follow that of which it is to be predicated. For if any of these
|
|
subjects is the same as any of these attributes, the attribute
|
|
originally in question must belong to the subject originally in
|
|
question. But if the purpose is to establish not a universal but a
|
|
particular proposition, they must look for the terms of which the
|
|
terms in question are predicable: for if any of these are identical,
|
|
the attribute in question must belong to some of the subject in
|
|
question. Whenever the one term has to belong to none of the other,
|
|
one must look to the consequents of the subject, and to those
|
|
attributes which cannot possibly be present in the predicate in
|
|
question: or conversely to the attributes which cannot possibly be
|
|
present in the subject, and to the consequents of the predicate. If
|
|
any members of these groups are identical, one of the terms in
|
|
question cannot possibly belong to any of the other. For sometimes a
|
|
syllogism in the first figure results, sometimes a syllogism in the
|
|
second. But if the object is to establish a particular negative
|
|
proposition, we must find antecedents of the subject in question and
|
|
attributes which cannot possibly belong to the predicate in
|
|
question. If any members of these two groups are identical, it follows
|
|
that one of the terms in question does not belong to some of the
|
|
other. Perhaps each of these statements will become clearer in the
|
|
following way. Suppose the consequents of A are designated by B, the
|
|
antecedents of A by C, attributes which cannot possibly belong to A by
|
|
D. Suppose again that the attributes of E are designated by F, the
|
|
antecedents of E by G, and attributes which cannot belong to E by H.
|
|
If then one of the Cs should be identical with one of the Fs, A must
|
|
belong to all E: for F belongs to all E, and A to all C,
|
|
consequently A belongs to all E. If C and G are identical, A must
|
|
belong to some of the Es: for A follows C, and E follows all G. If F
|
|
and D are identical, A will belong to none of the Es by a
|
|
prosyllogism: for since the negative proposition is convertible, and F
|
|
is identical with D, A will belong to none of the Fs, but F belongs to
|
|
all E. Again, if B and H are identical, A will belong to none of the
|
|
Es: for B will belong to all A, but to no E: for it was assumed to
|
|
be identical with H, and H belonged to none of the Es. If D and G
|
|
are identical, A will not belong to some of the Es: for it will not
|
|
belong to G, because it does not belong to D: but G falls under E:
|
|
consequently A will not belong to some of the Es. If B is identical
|
|
with G, there will be a converted syllogism: for E will belong to
|
|
all A since B belongs to A and E to B (for B was found to be identical
|
|
with G): but that A should belong to all E is not necessary, but it
|
|
must belong to some E because it is possible to convert the
|
|
universal statement into a particular.
|
|
|
|
It is clear then that in every proposition which requires proof we
|
|
must look to the aforesaid relations of the subject and predicate in
|
|
question: for all syllogisms proceed through these. But if we are
|
|
seeking consequents and antecedents we must look for those which are
|
|
primary and most universal, e.g. in reference to E we must look to
|
|
KF rather than to F alone, and in reference to A we must look to KC
|
|
rather than to C alone. For if A belongs to KF, it belongs both to F
|
|
and to E: but if it does not follow KF, it may yet follow F. Similarly
|
|
we must consider the antecedents of A itself: for if a term follows
|
|
the primary antecedents, it will follow those also which are
|
|
subordinate, but if it does not follow the former, it may yet follow
|
|
the latter.
|
|
|
|
It is clear too that the inquiry proceeds through the three terms
|
|
and the two premisses, and that all the syllogisms proceed through the
|
|
aforesaid figures. For it is proved that A belongs to all E,
|
|
whenever an identical term is found among the Cs and Fs. This will
|
|
be the middle term; A and E will be the extremes. So the first
|
|
figure is formed. And A will belong to some E, whenever C and G are
|
|
apprehended to be the same. This is the last figure: for G becomes the
|
|
middle term. And A will belong to no E, when D and F are identical.
|
|
Thus we have both the first figure and the middle figure; the first,
|
|
because A belongs to no F, since the negative statement is
|
|
convertible, and F belongs to all E: the middle figure because D
|
|
belongs to no A, and to all E. And A will not belong to some E,
|
|
whenever D and G are identical. This is the last figure: for A will
|
|
belong to no G, and E will belong to all G. Clearly then all
|
|
syllogisms proceed through the aforesaid figures, and we must not
|
|
select consequents of all the terms, because no syllogism is
|
|
produced from them. For (as we saw) it is not possible at all to
|
|
establish a proposition from consequents, and it is not possible to
|
|
refute by means of a consequent of both the terms in question: for the
|
|
middle term must belong to the one, and not belong to the other.
|
|
|
|
It is clear too that other methods of inquiry by selection of middle
|
|
terms are useless to produce a syllogism, e.g. if the consequents of
|
|
the terms in question are identical, or if the antecedents of A are
|
|
identical with those attributes which cannot possibly belong to E,
|
|
or if those attributes are identical which cannot belong to either
|
|
term: for no syllogism is produced by means of these. For if the
|
|
consequents are identical, e.g. B and F, we have the middle figure
|
|
with both premisses affirmative: if the antecedents of A are identical
|
|
with attributes which cannot belong to E, e.g. C with H, we have the
|
|
first figure with its minor premiss negative. If attributes which
|
|
cannot belong to either term are identical, e.g. C and H, both
|
|
premisses are negative, either in the first or in the middle figure.
|
|
But no syllogism is possible in this way.
|
|
|
|
It is evident too that we must find out which terms in this
|
|
inquiry are identical, not which are different or contrary, first
|
|
because the object of our investigation is the middle term, and the
|
|
middle term must be not diverse but identical. Secondly, wherever it
|
|
happens that a syllogism results from taking contraries or terms which
|
|
cannot belong to the same thing, all arguments can be reduced to the
|
|
aforesaid moods, e.g. if B and F are contraries or cannot belong to
|
|
the same thing. For if these are taken, a syllogism will be formed
|
|
to prove that A belongs to none of the Es, not however from the
|
|
premisses taken but in the aforesaid mood. For B will belong to all
|
|
A and to no E. Consequently B must be identical with one of the Hs.
|
|
Again, if B and G cannot belong to the same thing, it follows that A
|
|
will not belong to some of the Es: for then too we shall have the
|
|
middle figure: for B will belong to all A and to no G. Consequently
|
|
B must be identical with some of the Hs. For the fact that B and G
|
|
cannot belong to the same thing differs in no way from the fact that B
|
|
is identical with some of the Hs: for that includes everything which
|
|
cannot belong to E.
|
|
|
|
It is clear then that from the inquiries taken by themselves no
|
|
syllogism results; but if B and F are contraries B must be identical
|
|
with one of the Hs, and the syllogism results through these terms.
|
|
It turns out then that those who inquire in this manner are looking
|
|
gratuitously for some other way than the necessary way because they
|
|
have failed to observe the identity of the Bs with the Hs.
|
|
|
|
29
|
|
|
|
Syllogisms which lead to impossible conclusions are similar to
|
|
ostensive syllogisms; they also are formed by means of the consequents
|
|
and antecedents of the terms in question. In both cases the same
|
|
inquiry is involved. For what is proved ostensively may also be
|
|
concluded syllogistically per impossibile by means of the same
|
|
terms; and what is proved per impossibile may also be proved
|
|
ostensively, e.g. that A belongs to none of the Es. For suppose A to
|
|
belong to some E: then since B belongs to all A and A to some of the
|
|
Es, B will belong to some of the Es: but it was assumed that it
|
|
belongs to none. Again we may prove that A belongs to some E: for if A
|
|
belonged to none of the Es, and E belongs to all G, A will belong to
|
|
none of the Gs: but it was assumed to belong to all. Similarly with
|
|
the other propositions requiring proof. The proof per impossibile will
|
|
always and in all cases be from the consequents and antecedents of the
|
|
terms in question. Whatever the problem the same inquiry is
|
|
necessary whether one wishes to use an ostensive syllogism or a
|
|
reduction to impossibility. For both the demonstrations start from the
|
|
same terms, e.g. suppose it has been proved that A belongs to no E,
|
|
because it turns out that otherwise B belongs to some of the Es and
|
|
this is impossible-if now it is assumed that B belongs to no E and
|
|
to all A, it is clear that A will belong to no E. Again if it has been
|
|
proved by an ostensive syllogism that A belongs to no E, assume that A
|
|
belongs to some E and it will be proved per impossibile to belong to
|
|
no E. Similarly with the rest. In all cases it is necessary to find
|
|
some common term other than the subjects of inquiry, to which the
|
|
syllogism establishing the false conclusion may relate, so that if
|
|
this premiss is converted, and the other remains as it is, the
|
|
syllogism will be ostensive by means of the same terms. For the
|
|
ostensive syllogism differs from the reductio ad impossibile in
|
|
this: in the ostensive syllogism both remisses are laid down in
|
|
accordance with the truth, in the reductio ad impossibile one of the
|
|
premisses is assumed falsely.
|
|
|
|
These points will be made clearer by the sequel, when we discuss the
|
|
reduction to impossibility: at present this much must be clear, that
|
|
we must look to terms of the kinds mentioned whether we wish to use an
|
|
ostensive syllogism or a reduction to impossibility. In the other
|
|
hypothetical syllogisms, I mean those which proceed by substitution,
|
|
or by positing a certain quality, the inquiry will be directed to
|
|
the terms of the problem to be proved-not the terms of the original
|
|
problem, but the new terms introduced; and the method of the inquiry
|
|
will be the same as before. But we must consider and determine in
|
|
how many ways hypothetical syllogisms are possible.
|
|
|
|
Each of the problems then can be proved in the manner described; but
|
|
it is possible to establish some of them syllogistically in another
|
|
way, e.g. universal problems by the inquiry which leads up to a
|
|
particular conclusion, with the addition of an hypothesis. For if
|
|
the Cs and the Gs should be identical, but E should be assumed to
|
|
belong to the Gs only, then A would belong to every E: and again if
|
|
the Ds and the Gs should be identical, but E should be predicated of
|
|
the Gs only, it follows that A will belong to none of the Es.
|
|
Clearly then we must consider the matter in this way also. The
|
|
method is the same whether the relation is necessary or possible.
|
|
For the inquiry will be the same, and the syllogism will proceed
|
|
through terms arranged in the same order whether a possible or a
|
|
pure proposition is proved. We must find in the case of possible
|
|
relations, as well as terms that belong, terms which can belong though
|
|
they actually do not: for we have proved that the syllogism which
|
|
establishes a possible relation proceeds through these terms as
|
|
well. Similarly also with the other modes of predication.
|
|
|
|
It is clear then from what has been said not only that all
|
|
syllogisms can be formed in this way, but also that they cannot be
|
|
formed in any other. For every syllogism has been proved to be
|
|
formed through one of the aforementioned figures, and these cannot
|
|
be composed through other terms than the consequents and antecedents
|
|
of the terms in question: for from these we obtain the premisses and
|
|
find the middle term. Consequently a syllogism cannot be formed by
|
|
means of other terms.
|
|
|
|
30
|
|
|
|
The method is the same in all cases, in philosophy, in any art or
|
|
study. We must look for the attributes and the subjects of both our
|
|
terms, and we must supply ourselves with as many of these as possible,
|
|
and consider them by means of the three terms, refuting statements
|
|
in one way, confirming them in another, in the pursuit of truth
|
|
starting from premisses in which the arrangement of the terms is in
|
|
accordance with truth, while if we look for dialectical syllogisms
|
|
we must start from probable premisses. The principles of syllogisms
|
|
have been stated in general terms, both how they are characterized and
|
|
how we must hunt for them, so as not to look to everything that is
|
|
said about the terms of the problem or to the same points whether we
|
|
are confirming or refuting, or again whether we are confirming of
|
|
all or of some, and whether we are refuting of all or some. we must
|
|
look to fewer points and they must be definite. We have also stated
|
|
how we must select with reference to everything that is, e.g. about
|
|
good or knowledge. But in each science the principles which are
|
|
peculiar are the most numerous. Consequently it is the business of
|
|
experience to give the principles which belong to each subject. I mean
|
|
for example that astronomical experience supplies the principles of
|
|
astronomical science: for once the phenomena were adequately
|
|
apprehended, the demonstrations of astronomy were discovered.
|
|
Similarly with any other art or science. Consequently, if the
|
|
attributes of the thing are apprehended, our business will then be
|
|
to exhibit readily the demonstrations. For if none of the true
|
|
attributes of things had been omitted in the historical survey, we
|
|
should be able to discover the proof and demonstrate everything
|
|
which admitted of proof, and to make that clear, whose nature does not
|
|
admit of proof.
|
|
|
|
In general then we have explained fairly well how we must select
|
|
premisses: we have discussed the matter accurately in the treatise
|
|
concerning dialectic.
|
|
|
|
31
|
|
|
|
It is easy to see that division into classes is a small part of
|
|
the method we have described: for division is, so to speak, a weak
|
|
syllogism; for what it ought to prove, it begs, and it always
|
|
establishes something more general than the attribute in question.
|
|
First, this very point had escaped all those who used the method of
|
|
division; and they attempted to persuade men that it was possible to
|
|
make a demonstration of substance and essence. Consequently they did
|
|
not understand what it is possible to prove syllogistically by
|
|
division, nor did they understand that it was possible to prove
|
|
syllogistically in the manner we have described. In demonstrations,
|
|
when there is a need to prove a positive statement, the middle term
|
|
through which the syllogism is formed must always be inferior to and
|
|
not comprehend the first of the extremes. But division has a
|
|
contrary intention: for it takes the universal as middle. Let animal
|
|
be the term signified by A, mortal by B, and immortal by C, and let
|
|
man, whose definition is to be got, be signified by D. The man who
|
|
divides assumes that every animal is either mortal or immortal: i.e.
|
|
whatever is A is all either B or C. Again, always dividing, he lays it
|
|
down that man is an animal, so he assumes A of D as belonging to it.
|
|
Now the true conclusion is that every D is either B or C, consequently
|
|
man must be either mortal or immortal, but it is not necessary that
|
|
man should be a mortal animal-this is begged: and this is what ought
|
|
to have been proved syllogistically. And again, taking A as mortal
|
|
animal, B as footed, C as footless, and D as man, he assumes in the
|
|
same way that A inheres either in B or in C (for every mortal animal
|
|
is either footed or footless), and he assumes A of D (for he assumed
|
|
man, as we saw, to be a mortal animal); consequently it is necessary
|
|
that man should be either a footed or a footless animal; but it is not
|
|
necessary that man should be footed: this he assumes: and it is just
|
|
this again which he ought to have demonstrated. Always dividing then
|
|
in this way it turns out that these logicians assume as middle the
|
|
universal term, and as extremes that which ought to have been the
|
|
subject of demonstration and the differentiae. In conclusion, they
|
|
do not make it clear, and show it to be necessary, that this is man or
|
|
whatever the subject of inquiry may be: for they pursue the other
|
|
method altogether, never even suspecting the presence of the rich
|
|
supply of evidence which might be used. It is clear that it is neither
|
|
possible to refute a statement by this method of division, nor to draw
|
|
a conclusion about an accident or property of a thing, nor about its
|
|
genus, nor in cases in which it is unknown whether it is thus or thus,
|
|
e.g. whether the diagonal is incommensurate. For if he assumes that
|
|
every length is either commensurate or incommensurate, and the
|
|
diagonal is a length, he has proved that the diagonal is either
|
|
incommensurate or commensurate. But if he should assume that it is
|
|
incommensurate, he will have assumed what he ought to have proved.
|
|
He cannot then prove it: for this is his method, but proof is not
|
|
possible by this method. Let A stand for 'incommensurate or
|
|
commensurate', B for 'length', C for 'diagonal'. It is clear then that
|
|
this method of investigation is not suitable for every inquiry, nor is
|
|
it useful in those cases in which it is thought to be most suitable.
|
|
|
|
From what has been said it is clear from what elements
|
|
demonstrations are formed and in what manner, and to what points we
|
|
must look in each problem.
|
|
|
|
32
|
|
|
|
Our next business is to state how we can reduce syllogisms to the
|
|
aforementioned figures: for this part of the inquiry still remains. If
|
|
we should investigate the production of the syllogisms and had the
|
|
power of discovering them, and further if we could resolve the
|
|
syllogisms produced into the aforementioned figures, our original
|
|
problem would be brought to a conclusion. It will happen at the same
|
|
time that what has been already said will be confirmed and its truth
|
|
made clearer by what we are about to say. For everything that is
|
|
true must in every respect agree with itself First then we must
|
|
attempt to select the two premisses of the syllogism (for it is easier
|
|
to divide into large parts than into small, and the composite parts
|
|
are larger than the elements out of which they are made); next we must
|
|
inquire which are universal and which particular, and if both
|
|
premisses have not been stated, we must ourselves assume the one which
|
|
is missing. For sometimes men put forward the universal premiss, but
|
|
do not posit the premiss which is contained in it, either in writing
|
|
or in discussion: or men put forward the premisses of the principal
|
|
syllogism, but omit those through which they are inferred, and
|
|
invite the concession of others to no purpose. We must inquire then
|
|
whether anything unnecessary has been assumed, or anything necessary
|
|
has been omitted, and we must posit the one and take away the other,
|
|
until we have reached the two premisses: for unless we have these,
|
|
we cannot reduce arguments put forward in the way described. In some
|
|
arguments it is easy to see what is wanting, but some escape us, and
|
|
appear to be syllogisms, because something necessary results from what
|
|
has been laid down, e.g. if the assumptions were made that substance
|
|
is not annihilated by the annihilation of what is not substance, and
|
|
that if the elements out of which a thing is made are annihilated,
|
|
then that which is made out of them is destroyed: these propositions
|
|
being laid down, it is necessary that any part of substance is
|
|
substance; this has not however been drawn by syllogism from the
|
|
propositions assumed, but premisses are wanting. Again if it is
|
|
necessary that animal should exist, if man does, and that substance
|
|
should exist, if animal does, it is necessary that substance should
|
|
exist if man does: but as yet the conclusion has not been drawn
|
|
syllogistically: for the premisses are not in the shape we required.
|
|
We are deceived in such cases because something necessary results from
|
|
what is assumed, since the syllogism also is necessary. But that which
|
|
is necessary is wider than the syllogism: for every syllogism is
|
|
necessary, but not everything which is necessary is a syllogism.
|
|
Consequently, though something results when certain propositions are
|
|
assumed, we must not try to reduce it directly, but must first state
|
|
the two premisses, then divide them into their terms. We must take
|
|
that term as middle which is stated in both the remisses: for it is
|
|
necessary that the middle should be found in both premisses in all the
|
|
figures.
|
|
|
|
If then the middle term is a predicate and a subject of predication,
|
|
or if it is a predicate, and something else is denied of it, we
|
|
shall have the first figure: if it both is a predicate and is denied
|
|
of something, the middle figure: if other things are predicated of it,
|
|
or one is denied, the other predicated, the last figure. For it was
|
|
thus that we found the middle term placed in each figure. It is placed
|
|
similarly too if the premisses are not universal: for the middle
|
|
term is determined in the same way. Clearly then, if the same term
|
|
is not stated more than once in the course of an argument, a syllogism
|
|
cannot be made: for a middle term has not been taken. Since we know
|
|
what sort of thesis is established in each figure, and in which the
|
|
universal, in what sort the particular is described, clearly we must
|
|
not look for all the figures, but for that which is appropriate to the
|
|
thesis in hand. If the thesis is established in more figures than one,
|
|
we shall recognize the figure by the position of the middle term.
|
|
|
|
33
|
|
|
|
Men are frequently deceived about syllogisms because the inference
|
|
is necessary, as has been said above; sometimes they are deceived by
|
|
the similarity in the positing of the terms; and this ought not to
|
|
escape our notice. E.g. if A is stated of B, and B of C: it would seem
|
|
that a syllogism is possible since the terms stand thus: but nothing
|
|
necessary results, nor does a syllogism. Let A represent the term
|
|
'being eternal', B 'Aristomenes as an object of thought', C
|
|
'Aristomenes'. It is true then that A belongs to B. For Aristomenes as
|
|
an object of thought is eternal. But B also belongs to C: for
|
|
Aristomenes is Aristomenes as an object of thought. But A does not
|
|
belong to C: for Aristomenes is perishable. For no syllogism was
|
|
made although the terms stood thus: that required that the premiss
|
|
AB should be stated universally. But this is false, that every
|
|
Aristomenes who is an object of thought is eternal, since
|
|
Aristomenes is perishable. Again let C stand for 'Miccalus', B for
|
|
'musical Miccalus', A for 'perishing to-morrow'. It is true to
|
|
predicate B of C: for Miccalus is musical Miccalus. Also A can be
|
|
predicated of B: for musical Miccalus might perish to-morrow. But to
|
|
state A of C is false at any rate. This argument then is identical
|
|
with the former; for it is not true universally that musical
|
|
Miccalus perishes to-morrow: but unless this is assumed, no
|
|
syllogism (as we have shown) is possible.
|
|
|
|
This deception then arises through ignoring a small distinction. For
|
|
if we accept the conclusion as though it made no difference whether we
|
|
said 'This belong to that' or 'This belongs to all of that'.
|
|
|
|
34
|
|
|
|
Men will frequently fall into fallacies through not setting out
|
|
the terms of the premiss well, e.g. suppose A to be health, B disease,
|
|
C man. It is true to say that A cannot belong to any B (for health
|
|
belongs to no disease) and again that B belongs to every C (for
|
|
every man is capable of disease). It would seem to follow that
|
|
health cannot belong to any man. The reason for this is that the terms
|
|
are not set out well in the statement, since if the things which are
|
|
in the conditions are substituted, no syllogism can be made, e.g. if
|
|
'healthy' is substituted for 'health' and 'diseased' for 'disease'.
|
|
For it is not true to say that being healthy cannot belong to one
|
|
who is diseased. But unless this is assumed no conclusion results,
|
|
save in respect of possibility: but such a conclusion is not
|
|
impossible: for it is possible that health should belong to no man.
|
|
Again the fallacy may occur in a similar way in the middle figure: 'it
|
|
is not possible that health should belong to any disease, but it is
|
|
possible that health should belong to every man, consequently it is
|
|
not possible that disease should belong to any man'. In the third
|
|
figure the fallacy results in reference to possibility. For health and
|
|
diseae and knowledge and ignorance, and in general contraries, may
|
|
possibly belong to the same thing, but cannot belong to one another.
|
|
This is not in agreement with what was said before: for we stated that
|
|
when several things could belong to the same thing, they could
|
|
belong to one another.
|
|
|
|
It is evident then that in all these cases the fallacy arises from
|
|
the setting out of the terms: for if the things that are in the
|
|
conditions are substituted, no fallacy arises. It is clear then that
|
|
in such premisses what possesses the condition ought always to be
|
|
substituted for the condition and taken as the term.
|
|
|
|
35
|
|
|
|
We must not always seek to set out the terms a single word: for we
|
|
shall often have complexes of words to which a single name is not
|
|
given. Hence it is difficult to reduce syllogisms with such terms.
|
|
Sometimes too fallacies will result from such a search, e.g. the
|
|
belief that syllogism can establish that which has no mean. Let A
|
|
stand for two right angles, B for triangle, C for isosceles
|
|
triangle. A then belongs to C because of B: but A belongs to B without
|
|
the mediation of another term: for the triangle in virtue of its own
|
|
nature contains two right angles, consequently there will be no middle
|
|
term for the proposition AB, although it is demonstrable. For it is
|
|
clear that the middle must not always be assumed to be an individual
|
|
thing, but sometimes a complex of words, as happens in the case
|
|
mentioned.
|
|
|
|
36
|
|
|
|
That the first term belongs to the middle, and the middle to the
|
|
extreme, must not be understood in the sense that they can always be
|
|
predicated of one another or that the first term will be predicated of
|
|
the middle in the same way as the middle is predicated of the last
|
|
term. The same holds if the premisses are negative. But we must
|
|
suppose the verb 'to belong' to have as many meanings as the senses in
|
|
which the verb 'to be' is used, and in which the assertion that a
|
|
thing 'is' may be said to be true. Take for example the statement that
|
|
there is a single science of contraries. Let A stand for 'there
|
|
being a single science', and B for things which are contrary to one
|
|
another. Then A belongs to B, not in the sense that contraries are the
|
|
fact of there being a single science of them, but in the sense that it
|
|
is true to say of the contraries that there is a single science of
|
|
them.
|
|
|
|
It happens sometimes that the first term is stated of the middle,
|
|
but the middle is not stated of the third term, e.g. if wisdom is
|
|
knowledge, and wisdom is of the good, the conclusion is that there
|
|
is knowledge of the good. The good then is not knowledge, though
|
|
wisdom is knowledge. Sometimes the middle term is stated of the third,
|
|
but the first is not stated of the middle, e.g. if there is a
|
|
science of everything that has a quality, or is a contrary, and the
|
|
good both is a contrary and has a quality, the conclusion is that
|
|
there is a science of the good, but the good is not science, nor is
|
|
that which has a quality or is a contrary, though the good is both
|
|
of these. Sometimes neither the first term is stated of the middle,
|
|
nor the middle of the third, while the first is sometimes stated of
|
|
the third, and sometimes not: e.g. if there is a genus of that of
|
|
which there is a science, and if there is a science of the good, we
|
|
conclude that there is a genus of the good. But nothing is
|
|
predicated of anything. And if that of which there is a science is a
|
|
genus, and if there is a science of the good, we conclude that the
|
|
good is a genus. The first term then is predicated of the extreme, but
|
|
in the premisses one thing is not stated of another.
|
|
|
|
The same holds good where the relation is negative. For 'that does
|
|
not belong to this' does not always mean that 'this is not that',
|
|
but sometimes that 'this is not of that' or 'for that', e.g. 'there is
|
|
not a motion of a motion or a becoming of a becoming, but there is a
|
|
becoming of pleasure: so pleasure is not a becoming.' Or again it
|
|
may be said that there is a sign of laughter, but there is not a
|
|
sign of a sign, consequently laughter is not a sign. This holds in the
|
|
other cases too, in which the thesis is refuted because the genus is
|
|
asserted in a particular way, in relation to the terms of the
|
|
thesis. Again take the inference 'opportunity is not the right time:
|
|
for opportunity belongs to God, but the right time does not, since
|
|
nothing is useful to God'. We must take as terms opportunity-right
|
|
time-God: but the premiss must be understood according to the case
|
|
of the noun. For we state this universally without qualification, that
|
|
the terms ought always to be stated in the nominative, e.g. man, good,
|
|
contraries, not in oblique cases, e.g. of man, of a good, of
|
|
contraries, but the premisses ought to be understood with reference to
|
|
the cases of each term-either the dative, e.g. 'equal to this', or the
|
|
genitive, e.g. 'double of this', or the accusative, e.g. 'that which
|
|
strikes or sees this', or the nominative, e.g. 'man is an animal',
|
|
or in whatever other way the word falls in the premiss.
|
|
|
|
37
|
|
|
|
The expressions 'this belongs to that' and 'this holds true of that'
|
|
must be understood in as many ways as there are different
|
|
categories, and these categories must be taken either with or
|
|
without qualification, and further as simple or compound: the same
|
|
holds good of the corresponding negative expressions. We must consider
|
|
these points and define them better.
|
|
|
|
38
|
|
|
|
A term which is repeated in the premisses ought to be joined to
|
|
the first extreme, not to the middle. I mean for example that if a
|
|
syllogism should be made proving that there is knowledge of justice,
|
|
that it is good, the expression 'that it is good' (or 'qua good')
|
|
should be joined to the first term. Let A stand for 'knowledge that it
|
|
is good', B for good, C for justice. It is true to predicate A of B.
|
|
For of the good there is knowledge that it is good. Also it is true to
|
|
predicate B of C. For justice is identical with a good. In this way an
|
|
analysis of the argument can be made. But if the expression 'that it
|
|
is good' were added to B, the conclusion will not follow: for A will
|
|
be true of B, but B will not be true of C. For to predicate of justice
|
|
the term 'good that it is good' is false and not intelligible.
|
|
Similarly if it should be proved that the healthy is an object of
|
|
knowledge qua good, of goat-stag an object of knowledge qua not
|
|
existing, or man perishable qua an object of sense: in every case in
|
|
which an addition is made to the predicate, the addition must be
|
|
joined to the extreme.
|
|
|
|
The position of the terms is not the same when something is
|
|
established without qualification and when it is qualified by some
|
|
attribute or condition, e.g. when the good is proved to be an object
|
|
of knowledge and when it is proved to be an object of knowledge that
|
|
it is good. If it has been proved to be an object of knowledge without
|
|
qualification, we must put as middle term 'that which is', but if we
|
|
add the qualification 'that it is good', the middle term must be 'that
|
|
which is something'. Let A stand for 'knowledge that it is something',
|
|
B stand for 'something', and C stand for 'good'. It is true to
|
|
predicate A of B: for ex hypothesi there is a science of that which is
|
|
something, that it is something. B too is true of C: for that which
|
|
C represents is something. Consequently A is true of C: there will
|
|
then be knowledge of the good, that it is good: for ex hypothesi the
|
|
term 'something' indicates the thing's special nature. But if
|
|
'being' were taken as middle and 'being' simply were joined to the
|
|
extreme, not 'being something', we should not have had a syllogism
|
|
proving that there is knowledge of the good, that it is good, but that
|
|
it is; e.g. let A stand for knowledge that it is, B for being, C for
|
|
good. Clearly then in syllogisms which are thus limited we must take
|
|
the terms in the way stated.
|
|
|
|
39
|
|
|
|
We ought also to exchange terms which have the same value, word
|
|
for word, and phrase for phrase, and word and phrase, and always
|
|
take a word in preference to a phrase: for thus the setting out of the
|
|
terms will be easier. For example if it makes no difference whether we
|
|
say that the supposable is not the genus of the opinable or that the
|
|
opinable is not identical with a particular kind of supposable (for
|
|
what is meant is the same in both statements), it is better to take as
|
|
the terms the supposable and the opinable in preference to the
|
|
phrase suggested.
|
|
|
|
40
|
|
|
|
Since the expressions 'pleasure is good' and 'pleasure is the
|
|
good' are not identical, we must not set out the terms in the same
|
|
way; but if the syllogism is to prove that pleasure is the good, the
|
|
term must be 'the good', but if the object is to prove that pleasure
|
|
is good, the term will be 'good'. Similarly in all other cases.
|
|
|
|
41
|
|
|
|
It is not the same, either in fact or in speech, that A belongs to
|
|
all of that to which B belongs, and that A belongs to all of that to
|
|
all of which B belongs: for nothing prevents B from belonging to C,
|
|
though not to all C: e.g. let B stand for beautiful, and C for
|
|
white. If beauty belongs to something white, it is true to say that
|
|
beauty belongs to that which is white; but not perhaps to everything
|
|
that is white. If then A belongs to B, but not to everything of
|
|
which B is predicated, then whether B belongs to all C or merely
|
|
belongs to C, it is not necessary that A should belong, I do not say
|
|
to all C, but even to C at all. But if A belongs to everything of
|
|
which B is truly stated, it will follow that A can be said of all of
|
|
that of all of which B is said. If however A is said of that of all of
|
|
which B may be said, nothing prevents B belonging to C, and yet A
|
|
not belonging to all C or to any C at all. If then we take three terms
|
|
it is clear that the expression 'A is said of all of which B is
|
|
said' means this, 'A is said of all the things of which B is said'.
|
|
And if B is said of all of a third term, so also is A: but if B is not
|
|
said of all of the third term, there is no necessity that A should
|
|
be said of all of it.
|
|
|
|
We must not suppose that something absurd results through setting
|
|
out the terms: for we do not use the existence of this particular
|
|
thing, but imitate the geometrician who says that 'this line a foot
|
|
long' or 'this straight line' or 'this line without breadth' exists
|
|
although it does not, but does not use the diagrams in the sense
|
|
that he reasons from them. For in general, if two things are not
|
|
related as whole to part and part to whole, the prover does not
|
|
prove from them, and so no syllogism a is formed. We (I mean the
|
|
learner) use the process of setting out terms like perception by
|
|
sense, not as though it were impossible to demonstrate without these
|
|
illustrative terms, as it is to demonstrate without the premisses of
|
|
the syllogism.
|
|
|
|
42
|
|
|
|
We should not forget that in the same syllogism not all
|
|
conclusions are reached through one figure, but one through one
|
|
figure, another through another. Clearly then we must analyse
|
|
arguments in accordance with this. Since not every problem is proved
|
|
in every figure, but certain problems in each figure, it is clear from
|
|
the conclusion in what figure the premisses should be sought.
|
|
|
|
43
|
|
|
|
In reference to those arguments aiming at a definition which have
|
|
been directed to prove some part of the definition, we must take as
|
|
a term the point to which the argument has been directed, not the
|
|
whole definition: for so we shall be less likely to be disturbed by
|
|
the length of the term: e.g. if a man proves that water is a drinkable
|
|
liquid, we must take as terms drinkable and water.
|
|
|
|
44
|
|
|
|
Further we must not try to reduce hypothetical syllogisms; for
|
|
with the given premisses it is not possible to reduce them. For they
|
|
have not been proved by syllogism, but assented to by agreement. For
|
|
instance if a man should suppose that unless there is one faculty of
|
|
contraries, there cannot be one science, and should then argue that
|
|
not every faculty is of contraries, e.g. of what is healthy and what
|
|
is sickly: for the same thing will then be at the same time healthy
|
|
and sickly. He has shown that there is not one faculty of all
|
|
contraries, but he has not proved that there is not a science. And yet
|
|
one must agree. But the agreement does not come from a syllogism,
|
|
but from an hypothesis. This argument cannot be reduced: but the proof
|
|
that there is not a single faculty can. The latter argument perhaps
|
|
was a syllogism: but the former was an hypothesis.
|
|
|
|
The same holds good of arguments which are brought to a conclusion
|
|
per impossibile. These cannot be analysed either; but the reduction to
|
|
what is impossible can be analysed since it is proved by syllogism,
|
|
though the rest of the argument cannot, because the conclusion is
|
|
reached from an hypothesis. But these differ from the previous
|
|
arguments: for in the former a preliminary agreement must be reached
|
|
if one is to accept the conclusion; e.g. an agreement that if there is
|
|
proved to be one faculty of contraries, then contraries fall under the
|
|
same science; whereas in the latter, even if no preliminary
|
|
agreement has been made, men still accept the reasoning, because the
|
|
falsity is patent, e.g. the falsity of what follows from the
|
|
assumption that the diagonal is commensurate, viz. that then odd
|
|
numbers are equal to evens.
|
|
|
|
Many other arguments are brought to a conclusion by the help of an
|
|
hypothesis; these we ought to consider and mark out clearly. We
|
|
shall describe in the sequel their differences, and the various ways
|
|
in which hypothetical arguments are formed: but at present this much
|
|
must be clear, that it is not possible to resolve such arguments
|
|
into the figures. And we have explained the reason.
|
|
|
|
45
|
|
|
|
Whatever problems are proved in more than one figure, if they have
|
|
been established in one figure by syllogism, can be reduced to another
|
|
figure, e.g. a negative syllogism in the first figure can be reduced
|
|
to the second, and a syllogism in the middle figure to the first,
|
|
not all however but some only. The point will be clear in the
|
|
sequel. If A belongs to no B, and B to all C, then A belongs to no
|
|
C. Thus the first figure; but if the negative statement is
|
|
converted, we shall have the middle figure. For B belongs to no A, and
|
|
to all C. Similarly if the syllogism is not universal but
|
|
particular, e.g. if A belongs to no B, and B to some C. Convert the
|
|
negative statement and you will have the middle figure.
|
|
|
|
The universal syllogisms in the second figure can be reduced to
|
|
the first, but only one of the two particular syllogisms. Let A belong
|
|
to no B and to all C. Convert the negative statement, and you will
|
|
have the first figure. For B will belong to no A and A to all C. But
|
|
if the affirmative statement concerns B, and the negative C, C must be
|
|
made first term. For C belongs to no A, and A to all B: therefore C
|
|
belongs to no B. B then belongs to no C: for the negative statement is
|
|
convertible.
|
|
|
|
But if the syllogism is particular, whenever the negative
|
|
statement concerns the major extreme, reduction to the first figure
|
|
will be possible, e.g. if A belongs to no B and to some C: convert the
|
|
negative statement and you will have the first figure. For B will
|
|
belong to no A and A to some C. But when the affirmative statement
|
|
concerns the major extreme, no resolution will be possible, e.g. if
|
|
A belongs to all B, but not to all C: for the statement AB does not
|
|
admit of conversion, nor would there be a syllogism if it did.
|
|
|
|
Again syllogisms in the third figure cannot all be resolved into the
|
|
first, though all syllogisms in the first figure can be resolved
|
|
into the third. Let A belong to all B and B to some C. Since the
|
|
particular affirmative is convertible, C will belong to some B: but
|
|
A belonged to all B: so that the third figure is formed. Similarly
|
|
if the syllogism is negative: for the particular affirmative is
|
|
convertible: therefore A will belong to no B, and to some C.
|
|
|
|
Of the syllogisms in the last figure one only cannot be resolved
|
|
into the first, viz. when the negative statement is not universal: all
|
|
the rest can be resolved. Let A and B be affirmed of all C: then C can
|
|
be converted partially with either A or B: C then belongs to some B.
|
|
Consequently we shall get the first figure, if A belongs to all C, and
|
|
C to some of the Bs. If A belongs to all C and B to some C, the
|
|
argument is the same: for B is convertible in reference to C. But if B
|
|
belongs to all C and A to some C, the first term must be B: for B
|
|
belongs to all C, and C to some A, therefore B belongs to some A.
|
|
But since the particular statement is convertible, A will belong to
|
|
some B. If the syllogism is negative, when the terms are universal
|
|
we must take them in a similar way. Let B belong to all C, and A to no
|
|
C: then C will belong to some B, and A to no C; and so C will be
|
|
middle term. Similarly if the negative statement is universal, the
|
|
affirmative particular: for A will belong to no C, and C to some of
|
|
the Bs. But if the negative statement is particular, no resolution
|
|
will be possible, e.g. if B belongs to all C, and A not belong to some
|
|
C: convert the statement BC and both premisses will be particular.
|
|
|
|
It is clear that in order to resolve the figures into one another
|
|
the premiss which concerns the minor extreme must be converted in both
|
|
the figures: for when this premiss is altered, the transition to the
|
|
other figure is made.
|
|
|
|
One of the syllogisms in the middle figure can, the other cannot, be
|
|
resolved into the third figure. Whenever the universal statement is
|
|
negative, resolution is possible. For if A belongs to no B and to some
|
|
C, both B and C alike are convertible in relation to A, so that B
|
|
belongs to no A and C to some A. A therefore is middle term. But
|
|
when A belongs to all B, and not to some C, resolution will not be
|
|
possible: for neither of the premisses is universal after conversion.
|
|
|
|
Syllogisms in the third figure can be resolved into the middle
|
|
figure, whenever the negative statement is universal, e.g. if A
|
|
belongs to no C, and B to some or all C. For C then will belong to
|
|
no A and to some B. But if the negative statement is particular, no
|
|
resolution will be possible: for the particular negative does not
|
|
admit of conversion.
|
|
|
|
It is clear then that the same syllogisms cannot be resolved in
|
|
these figures which could not be resolved into the first figure, and
|
|
that when syllogisms are reduced to the first figure these alone are
|
|
confirmed by reduction to what is impossible.
|
|
|
|
It is clear from what we have said how we ought to reduce
|
|
syllogisms, and that the figures may be resolved into one another.
|
|
|
|
46
|
|
|
|
In establishing or refuting, it makes some difference whether we
|
|
suppose the expressions 'not to be this' and 'to be not-this' are
|
|
identical or different in meaning, e.g. 'not to be white' and 'to be
|
|
not-white'. For they do not mean the same thing, nor is 'to be
|
|
not-white' the negation of 'to be white', but 'not to be white'. The
|
|
reason for this is as follows. The relation of 'he can walk' to 'he
|
|
can not-walk' is similar to the relation of 'it is white' to 'it is
|
|
not-white'; so is that of 'he knows what is good' to 'he knows what is
|
|
not-good'. For there is no difference between the expressions 'he
|
|
knows what is good' and 'he is knowing what is good', or 'he can walk'
|
|
and 'he is able to walk': therefore there is no difference between
|
|
their contraries 'he cannot walk'-'he is not able to walk'. If then
|
|
'he is not able to walk' means the same as 'he is able not to walk',
|
|
capacity to walk and incapacity to walk will belong at the same time
|
|
to the same person (for the same man can both walk and not-walk, and
|
|
is possessed of knowledge of what is good and of what is not-good),
|
|
but an affirmation and a denial which are opposed to one another do
|
|
not belong at the same time to the same thing. As then 'not to know
|
|
what is good' is not the same as 'to know what is not good', so 'to be
|
|
not-good' is not the same as 'not to be good'. For when two pairs
|
|
correspond, if the one pair are different from one another, the
|
|
other pair also must be different. Nor is 'to be not-equal' the same
|
|
as 'not to be equal': for there is something underlying the one,
|
|
viz. that which is not-equal, and this is the unequal, but there is
|
|
nothing underlying the other. Wherefore not everything is either equal
|
|
or unequal, but everything is equal or is not equal. Further the
|
|
expressions 'it is a not-white log' and 'it is not a white log' do not
|
|
imply one another's truth. For if 'it is a not-white log', it must
|
|
be a log: but that which is not a white log need not be a log at
|
|
all. Therefore it is clear that 'it is not-good' is not the denial
|
|
of 'it is good'. If then every single statement may truly be said to
|
|
be either an affirmation or a negation, if it is not a negation
|
|
clearly it must in a sense be an affirmation. But every affirmation
|
|
has a corresponding negation. The negation then of 'it is not-good' is
|
|
'it is not not-good'. The relation of these statements to one
|
|
another is as follows. Let A stand for 'to be good', B for 'not to
|
|
be good', let C stand for 'to be not-good' and be placed under B,
|
|
and let D stand for not to be not-good' and be placed under A. Then
|
|
either A or B will belong to everything, but they will never belong to
|
|
the same thing; and either C or D will belong to everything, but
|
|
they will never belong to the same thing. And B must belong to
|
|
everything to which C belongs. For if it is true to say 'it is a
|
|
not-white', it is true also to say 'it is not white': for it is
|
|
impossible that a thing should simultaneously be white and be
|
|
not-white, or be a not-white log and be a white log; consequently if
|
|
the affirmation does not belong, the denial must belong. But C does
|
|
not always belong to B: for what is not a log at all, cannot be a
|
|
not-white log either. On the other hand D belongs to everything to
|
|
which A belongs. For either C or D belongs to everything to which A
|
|
belongs. But since a thing cannot be simultaneously not-white and
|
|
white, D must belong to everything to which A belongs. For of that
|
|
which is white it is true to say that it is not not-white. But A is
|
|
not true of all D. For of that which is not a log at all it is not
|
|
true to say A, viz. that it is a white log. Consequently D is true,
|
|
but A is not true, i.e. that it is a white log. It is clear also
|
|
that A and C cannot together belong to the same thing, and that B
|
|
and D may possibly belong to the same thing.
|
|
|
|
Privative terms are similarly related positive ter terms respect
|
|
of this arrangement. Let A stand for 'equal', B for 'not equal', C for
|
|
'unequal', D for 'not unequal'.
|
|
|
|
In many things also, to some of which something belongs which does
|
|
not belong to others, the negation may be true in a similar way,
|
|
viz. that all are not white or that each is not white, while that each
|
|
is not-white or all are not-white is false. Similarly also 'every
|
|
animal is not-white' is not the negation of 'every animal is white'
|
|
(for both are false): the proper negation is 'every animal is not
|
|
white'. Since it is clear that 'it is not-white' and 'it is not white'
|
|
mean different things, and one is an affirmation, the other a
|
|
denial, it is evident that the method of proving each cannot be the
|
|
same, e.g. that whatever is an animal is not white or may not be
|
|
white, and that it is true to call it not-white; for this means that
|
|
it is not-white. But we may prove that it is true to call it white
|
|
or not-white in the same way for both are proved constructively by
|
|
means of the first figure. For the expression 'it is true' stands on a
|
|
similar footing to 'it is'. For the negation of 'it is true to call it
|
|
white' is not 'it is true to call it not-white' but 'it is not true to
|
|
call it white'. If then it is to be true to say that whatever is a man
|
|
is musical or is not-musical, we must assume that whatever is an
|
|
animal either is musical or is not-musical; and the proof has been
|
|
made. That whatever is a man is not musical is proved destructively in
|
|
the three ways mentioned.
|
|
|
|
In general whenever A and B are such that they cannot belong at
|
|
the same time to the same thing, and one of the two necessarily
|
|
belongs to everything, and again C and D are related in the same
|
|
way, and A follows C but the relation cannot be reversed, then D
|
|
must follow B and the relation cannot be reversed. And A and D may
|
|
belong to the same thing, but B and C cannot. First it is clear from
|
|
the following consideration that D follows B. For since either C or
|
|
D necessarily belongs to everything; and since C cannot belong to that
|
|
to which B belongs, because it carries A along with it and A and B
|
|
cannot belong to the same thing; it is clear that D must follow B.
|
|
Again since C does not reciprocate with but A, but C or D belongs to
|
|
everything, it is possible that A and D should belong to the same
|
|
thing. But B and C cannot belong to the same thing, because A
|
|
follows C; and so something impossible results. It is clear then
|
|
that B does not reciprocate with D either, since it is possible that D
|
|
and A should belong at the same time to the same thing.
|
|
|
|
It results sometimes even in such an arrangement of terms that one
|
|
is deceived through not apprehending the opposites rightly, one of
|
|
which must belong to everything, e.g. we may reason that 'if A and B
|
|
cannot belong at the same time to the same thing, but it is
|
|
necessary that one of them should belong to whatever the other does
|
|
not belong to: and again C and D are related in the same way, and
|
|
follows everything which C follows: it will result that B belongs
|
|
necessarily to everything to which D belongs': but this is false.
|
|
'Assume that F stands for the negation of A and B, and again that H
|
|
stands for the negation of C and D. It is necessary then that either A
|
|
or F should belong to everything: for either the affirmation or the
|
|
denial must belong. And again either C or H must belong to everything:
|
|
for they are related as affirmation and denial. And ex hypothesi A
|
|
belongs to everything ever thing to which C belongs. Therefore H
|
|
belongs to everything to which F belongs. Again since either F or B
|
|
belongs to everything, and similarly either H or D, and since H
|
|
follows F, B must follow D: for we know this. If then A follows C, B
|
|
must follow D'. But this is false: for as we proved the sequence is
|
|
reversed in terms so constituted. The fallacy arises because perhaps
|
|
it is not necessary that A or F should belong to everything, or that F
|
|
or B should belong to everything: for F is not the denial of A. For
|
|
not good is the negation of good: and not-good is not identical with
|
|
'neither good nor not-good'. Similarly also with C and D. For two
|
|
negations have been assumed in respect to one term.
|
|
|
|
Book II
|
|
|
|
1
|
|
|
|
WE have already explained the number of the figures, the character
|
|
and number of the premisses, when and how a syllogism is formed;
|
|
further what we must look for when a refuting and establishing
|
|
propositions, and how we should investigate a given problem in any
|
|
branch of inquiry, also by what means we shall obtain principles
|
|
appropriate to each subject. Since some syllogisms are universal,
|
|
others particular, all the universal syllogisms give more than one
|
|
result, and of particular syllogisms the affirmative yield more than
|
|
one, the negative yield only the stated conclusion. For all
|
|
propositions are convertible save only the particular negative: and
|
|
the conclusion states one definite thing about another definite thing.
|
|
Consequently all syllogisms save the particular negative yield more
|
|
than one conclusion, e.g. if A has been proved to to all or to some B,
|
|
then B must belong to some A: and if A has been proved to belong to no
|
|
B, then B belongs to no A. This is a different conclusion from the
|
|
former. But if A does not belong to some B, it is not necessary that B
|
|
should not belong to some A: for it may possibly belong to all A.
|
|
|
|
This then is the reason common to all syllogisms whether universal
|
|
or particular. But it is possible to give another reason concerning
|
|
those which are universal. For all the things that are subordinate
|
|
to the middle term or to the conclusion may be proved by the same
|
|
syllogism, if the former are placed in the middle, the latter in the
|
|
conclusion; e.g. if the conclusion AB is proved through C, whatever is
|
|
subordinate to B or C must accept the predicate A: for if D is
|
|
included in B as in a whole, and B is included in A, then D will be
|
|
included in A. Again if E is included in C as in a whole, and C is
|
|
included in A, then E will be included in A. Similarly if the
|
|
syllogism is negative. In the second figure it will be possible to
|
|
infer only that which is subordinate to the conclusion, e.g. if A
|
|
belongs to no B and to all C; we conclude that B belongs to no C. If
|
|
then D is subordinate to C, clearly B does not belong to it. But
|
|
that B does not belong to what is subordinate to A is not clear by
|
|
means of the syllogism. And yet B does not belong to E, if E is
|
|
subordinate to A. But while it has been proved through the syllogism
|
|
that B belongs to no C, it has been assumed without proof that B
|
|
does not belong to A, consequently it does not result through the
|
|
syllogism that B does not belong to E.
|
|
|
|
But in particular syllogisms there will be no necessity of inferring
|
|
what is subordinate to the conclusion (for a syllogism does not result
|
|
when this premiss is particular), but whatever is subordinate to the
|
|
middle term may be inferred, not however through the syllogism, e.g.
|
|
if A belongs to all B and B to some C. Nothing can be inferred about
|
|
that which is subordinate to C; something can be inferred about that
|
|
which is subordinate to B, but not through the preceding syllogism.
|
|
Similarly in the other figures. That which is subordinate to the
|
|
conclusion cannot be proved; the other subordinate can be proved, only
|
|
not through the syllogism, just as in the universal syllogisms what is
|
|
subordinate to the middle term is proved (as we saw) from a premiss
|
|
which is not demonstrated: consequently either a conclusion is not
|
|
possible in the case of universal syllogisms or else it is possible
|
|
also in the case of particular syllogisms.
|
|
|
|
2
|
|
|
|
It is possible for the premisses of the syllogism to be true, or
|
|
to be false, or to be the one true, the other false. The conclusion is
|
|
either true or false necessarily. From true premisses it is not
|
|
possible to draw a false conclusion, but a true conclusion may be
|
|
drawn from false premisses, true however only in respect to the
|
|
fact, not to the reason. The reason cannot be established from false
|
|
premisses: why this is so will be explained in the sequel.
|
|
|
|
First then that it is not possible to draw a false conclusion from
|
|
true premisses, is made clear by this consideration. If it is
|
|
necessary that B should be when A is, it is necessary that A should
|
|
not be when B is not. If then A is true, B must be true: otherwise
|
|
it will turn out that the same thing both is and is not at the same
|
|
time. But this is impossible. Let it not, because A is laid down as
|
|
a single term, be supposed that it is possible, when a single fact
|
|
is given, that something should necessarily result. For that is not
|
|
possible. For what results necessarily is the conclusion, and the
|
|
means by which this comes about are at the least three terms, and
|
|
two relations of subject and predicate or premisses. If then it is
|
|
true that A belongs to all that to which B belongs, and that B belongs
|
|
to all that to which C belongs, it is necessary that A should belong
|
|
to all that to which C belongs, and this cannot be false: for then the
|
|
same thing will belong and not belong at the same time. So A is
|
|
posited as one thing, being two premisses taken together. The same
|
|
holds good of negative syllogisms: it is not possible to prove a false
|
|
conclusion from true premisses.
|
|
|
|
But from what is false a true conclusion may be drawn, whether
|
|
both the premisses are false or only one, provided that this is not
|
|
either of the premisses indifferently, if it is taken as wholly false:
|
|
but if the premiss is not taken as wholly false, it does not matter
|
|
which of the two is false. (1) Let A belong to the whole of C, but
|
|
to none of the Bs, neither let B belong to C. This is possible, e.g.
|
|
animal belongs to no stone, nor stone to any man. If then A is taken
|
|
to belong to all B and B to all C, A will belong to all C;
|
|
consequently though both the premisses are false the conclusion is
|
|
true: for every man is an animal. Similarly with the negative. For
|
|
it is possible that neither A nor B should belong to any C, although A
|
|
belongs to all B, e.g. if the same terms are taken and man is put as
|
|
middle: for neither animal nor man belongs to any stone, but animal
|
|
belongs to every man. Consequently if one term is taken to belong to
|
|
none of that to which it does belong, and the other term is taken to
|
|
belong to all of that to which it does not belong, though both the
|
|
premisses are false the conclusion will be true. (2) A similar proof
|
|
may be given if each premiss is partially false.
|
|
|
|
(3) But if one only of the premisses is false, when the first
|
|
premiss is wholly false, e.g. AB, the conclusion will not be true, but
|
|
if the premiss BC is wholly false, a true conclusion will be possible.
|
|
I mean by 'wholly false' the contrary of the truth, e.g. if what
|
|
belongs to none is assumed to belong to all, or if what belongs to all
|
|
is assumed to belong to none. Let A belong to no B, and B to all C. If
|
|
then the premiss BC which I take is true, and the premiss AB is wholly
|
|
false, viz. that A belongs to all B, it is impossible that the
|
|
conclusion should be true: for A belonged to none of the Cs, since A
|
|
belonged to nothing to which B belonged, and B belonged to all C.
|
|
Similarly there cannot be a true conclusion if A belongs to all B, and
|
|
B to all C, but while the true premiss BC is assumed, the wholly false
|
|
premiss AB is also assumed, viz. that A belongs to nothing to which
|
|
B belongs: here the conclusion must be false. For A will belong to all
|
|
C, since A belongs to everything to which B belongs, and B to all C.
|
|
It is clear then that when the first premiss is wholly false,
|
|
whether affirmative or negative, and the other premiss is true, the
|
|
conclusion cannot be true.
|
|
|
|
(4) But if the premiss is not wholly false, a true conclusion is
|
|
possible. For if A belongs to all C and to some B, and if B belongs to
|
|
all C, e.g. animal to every swan and to some white thing, and white to
|
|
every swan, then if we take as premisses that A belongs to all B,
|
|
and B to all C, A will belong to all C truly: for every swan is an
|
|
animal. Similarly if the statement AB is negative. For it is
|
|
possible that A should belong to some B and to no C, and that B should
|
|
belong to all C, e.g. animal to some white thing, but to no snow,
|
|
and white to all snow. If then one should assume that A belongs to
|
|
no B, and B to all C, then will belong to no C.
|
|
|
|
(5) But if the premiss AB, which is assumed, is wholly true, and the
|
|
premiss BC is wholly false, a true syllogism will be possible: for
|
|
nothing prevents A belonging to all B and to all C, though B belongs
|
|
to no C, e.g. these being species of the same genus which are not
|
|
subordinate one to the other: for animal belongs both to horse and
|
|
to man, but horse to no man. If then it is assumed that A belongs to
|
|
all B and B to all C, the conclusion will be true, although the
|
|
premiss BC is wholly false. Similarly if the premiss AB is negative.
|
|
For it is possible that A should belong neither to any B nor to any C,
|
|
and that B should not belong to any C, e.g. a genus to species of
|
|
another genus: for animal belongs neither to music nor to the art of
|
|
healing, nor does music belong to the art of healing. If then it is
|
|
assumed that A belongs to no B, and B to all C, the conclusion will be
|
|
true.
|
|
|
|
(6) And if the premiss BC is not wholly false but in part only, even
|
|
so the conclusion may be true. For nothing prevents A belonging to the
|
|
whole of B and of C, while B belongs to some C, e.g. a genus to its
|
|
species and difference: for animal belongs to every man and to every
|
|
footed thing, and man to some footed things though not to all. If then
|
|
it is assumed that A belongs to all B, and B to all C, A will belong
|
|
to all C: and this ex hypothesi is true. Similarly if the premiss AB
|
|
is negative. For it is possible that A should neither belong to any
|
|
B nor to any C, though B belongs to some C, e.g. a genus to the
|
|
species of another genus and its difference: for animal neither
|
|
belongs to any wisdom nor to any instance of 'speculative', but wisdom
|
|
belongs to some instance of 'speculative'. If then it should be
|
|
assumed that A belongs to no B, and B to all C, will belong to no C:
|
|
and this ex hypothesi is true.
|
|
|
|
In particular syllogisms it is possible when the first premiss is
|
|
wholly false, and the other true, that the conclusion should be
|
|
true; also when the first premiss is false in part, and the other
|
|
true; and when the first is true, and the particular is false; and
|
|
when both are false. (7) For nothing prevents A belonging to no B, but
|
|
to some C, and B to some C, e.g. animal belongs to no snow, but to
|
|
some white thing, and snow to some white thing. If then snow is
|
|
taken as middle, and animal as first term, and it is assumed that A
|
|
belongs to the whole of B, and B to some C, then the premiss BC is
|
|
wholly false, the premiss BC true, and the conclusion true.
|
|
Similarly if the premiss AB is negative: for it is possible that A
|
|
should belong to the whole of B, but not to some C, although B belongs
|
|
to some C, e.g. animal belongs to every man, but does not follow
|
|
some white, but man belongs to some white; consequently if man be
|
|
taken as middle term and it is assumed that A belongs to no B but B
|
|
belongs to some C, the conclusion will be true although the premiss AB
|
|
is wholly false. (If the premiss AB is false in part, the conclusion
|
|
may be true. For nothing prevents A belonging both to B and to some C,
|
|
and B belonging to some C, e.g. animal to something beautiful and to
|
|
something great, and beautiful belonging to something great. If then A
|
|
is assumed to belong to all B, and B to some C, the a premiss AB
|
|
will be partially false, the premiss BC will be true, and the
|
|
conclusion true. Similarly if the premiss AB is negative. For the same
|
|
terms will serve, and in the same positions, to prove the point.
|
|
|
|
(9) Again if the premiss AB is true, and the premiss BC is false,
|
|
the conclusion may be true. For nothing prevents A belonging to the
|
|
whole of B and to some C, while B belongs to no C, e.g. animal to
|
|
every swan and to some black things, though swan belongs to no black
|
|
thing. Consequently if it should be assumed that A belongs to all B,
|
|
and B to some C, the conclusion will be true, although the statement
|
|
BC is false. Similarly if the premiss AB is negative. For it is
|
|
possible that A should belong to no B, and not to some C, while B
|
|
belongs to no C, e.g. a genus to the species of another genus and to
|
|
the accident of its own species: for animal belongs to no number and
|
|
not to some white things, and number belongs to nothing white. If then
|
|
number is taken as middle, and it is assumed that A belongs to no B,
|
|
and B to some C, then A will not belong to some C, which ex
|
|
hypothesi is true. And the premiss AB is true, the premiss BC false.
|
|
|
|
(10) Also if the premiss AB is partially false, and the premiss BC
|
|
is false too, the conclusion may be true. For nothing prevents A
|
|
belonging to some B and to some C, though B belongs to no C, e.g. if B
|
|
is the contrary of C, and both are accidents of the same genus: for
|
|
animal belongs to some white things and to some black things, but
|
|
white belongs to no black thing. If then it is assumed that A
|
|
belongs to all B, and B to some C, the conclusion will be true.
|
|
Similarly if the premiss AB is negative: for the same terms arranged
|
|
in the same way will serve for the proof.
|
|
|
|
(11) Also though both premisses are false the conclusion may be
|
|
true. For it is possible that A may belong to no B and to some C,
|
|
while B belongs to no C, e.g. a genus in relation to the species of
|
|
another genus, and to the accident of its own species: for animal
|
|
belongs to no number, but to some white things, and number to
|
|
nothing white. If then it is assumed that A belongs to all B and B
|
|
to some C, the conclusion will be true, though both premisses are
|
|
false. Similarly also if the premiss AB is negative. For nothing
|
|
prevents A belonging to the whole of B, and not to some C, while B
|
|
belongs to no C, e.g. animal belongs to every swan, and not to some
|
|
black things, and swan belongs to nothing black. Consequently if it is
|
|
assumed that A belongs to no B, and B to some C, then A does not
|
|
belong to some C. The conclusion then is true, but the premisses arc
|
|
false.
|
|
|
|
3
|
|
|
|
In the middle figure it is possible in every way to reach a true
|
|
conclusion through false premisses, whether the syllogisms are
|
|
universal or particular, viz. when both premisses are wholly false;
|
|
when each is partially false; when one is true, the other wholly false
|
|
(it does not matter which of the two premisses is false); if both
|
|
premisses are partially false; if one is quite true, the other
|
|
partially false; if one is wholly false, the other partially true. For
|
|
(1) if A belongs to no B and to all C, e.g. animal to no stone and
|
|
to every horse, then if the premisses are stated contrariwise and it
|
|
is assumed that A belongs to all B and to no C, though the premisses
|
|
are wholly false they will yield a true conclusion. Similarly if A
|
|
belongs to all B and to no C: for we shall have the same syllogism.
|
|
|
|
(2) Again if one premiss is wholly false, the other wholly true: for
|
|
nothing prevents A belonging to all B and to all C, though B belongs
|
|
to no C, e.g. a genus to its co-ordinate species. For animal belongs
|
|
to every horse and man, and no man is a horse. If then it is assumed
|
|
that animal belongs to all of the one, and none of the other, the
|
|
one premiss will be wholly false, the other wholly true, and the
|
|
conclusion will be true whichever term the negative statement
|
|
concerns.
|
|
|
|
(3) Also if one premiss is partially false, the other wholly true.
|
|
For it is possible that A should belong to some B and to all C, though
|
|
B belongs to no C, e.g. animal to some white things and to every
|
|
raven, though white belongs to no raven. If then it is assumed that
|
|
A belongs to no B, but to the whole of C, the premiss AB is
|
|
partially false, the premiss AC wholly true, and the conclusion
|
|
true. Similarly if the negative statement is transposed: the proof can
|
|
be made by means of the same terms. Also if the affirmative premiss is
|
|
partially false, the negative wholly true, a true conclusion is
|
|
possible. For nothing prevents A belonging to some B, but not to C
|
|
as a whole, while B belongs to no C, e.g. animal belongs to some white
|
|
things, but to no pitch, and white belongs to no pitch. Consequently
|
|
if it is assumed that A belongs to the whole of B, but to no C, the
|
|
premiss AB is partially false, the premiss AC is wholly true, and
|
|
the conclusion is true.
|
|
|
|
(4) And if both the premisses are partially false, the conclusion
|
|
may be true. For it is possible that A should belong to some B and
|
|
to some C, and B to no C, e.g. animal to some white things and to some
|
|
black things, though white belongs to nothing black. If then it is
|
|
assumed that A belongs to all B and to no C, both premisses are
|
|
partially false, but the conclusion is true. Similarly, if the
|
|
negative premiss is transposed, the proof can be made by means of
|
|
the same terms.
|
|
|
|
It is clear also that our thesis holds in particular syllogisms. For
|
|
(5) nothing prevents A belonging to all B and to some C, though B does
|
|
not belong to some C, e.g. animal to every man and to some white
|
|
things, though man will not belong to some white things. If then it is
|
|
stated that A belongs to no B and to some C, the universal premiss
|
|
is wholly false, the particular premiss is true, and the conclusion is
|
|
true. Similarly if the premiss AB is affirmative: for it is possible
|
|
that A should belong to no B, and not to some C, though B does not
|
|
belong to some C, e.g. animal belongs to nothing lifeless, and does
|
|
not belong to some white things, and lifeless will not belong to
|
|
some white things. If then it is stated that A belongs to all B and
|
|
not to some C, the premiss AB which is universal is wholly false,
|
|
the premiss AC is true, and the conclusion is true. Also a true
|
|
conclusion is possible when the universal premiss is true, and the
|
|
particular is false. For nothing prevents A following neither B nor
|
|
C at all, while B does not belong to some C, e.g. animal belongs to no
|
|
number nor to anything lifeless, and number does not follow some
|
|
lifeless things. If then it is stated that A belongs to no B and to
|
|
some C, the conclusion will be true, and the universal premiss true,
|
|
but the particular false. Similarly if the premiss which is stated
|
|
universally is affirmative. For it is possible that should A belong
|
|
both to B and to C as wholes, though B does not follow some C, e.g.
|
|
a genus in relation to its species and difference: for animal
|
|
follows every man and footed things as a whole, but man does not
|
|
follow every footed thing. Consequently if it is assumed that A
|
|
belongs to the whole of B, but does not belong to some C, the
|
|
universal premiss is true, the particular false, and the conclusion
|
|
true.
|
|
|
|
(6) It is clear too that though both premisses are false they may
|
|
yield a true conclusion, since it is possible that A should belong
|
|
both to B and to C as wholes, though B does not follow some C. For
|
|
if it is assumed that A belongs to no B and to some C, the premisses
|
|
are both false, but the conclusion is true. Similarly if the universal
|
|
premiss is affirmative and the particular negative. For it is possible
|
|
that A should follow no B and all C, though B does not belong to
|
|
some C, e.g. animal follows no science but every man, though science
|
|
does not follow every man. If then A is assumed to belong to the whole
|
|
of B, and not to follow some C, the premisses are false but the
|
|
conclusion is true.
|
|
|
|
4
|
|
|
|
In the last figure a true conclusion may come through what is false,
|
|
alike when both premisses are wholly false, when each is partly false,
|
|
when one premiss is wholly true, the other false, when one premiss
|
|
is partly false, the other wholly true, and vice versa, and in every
|
|
other way in which it is possible to alter the premisses. For (1)
|
|
nothing prevents neither A nor B from belonging to any C, while A
|
|
belongs to some B, e.g. neither man nor footed follows anything
|
|
lifeless, though man belongs to some footed things. If then it is
|
|
assumed that A and B belong to all C, the premisses will be wholly
|
|
false, but the conclusion true. Similarly if one premiss is
|
|
negative, the other affirmative. For it is possible that B should
|
|
belong to no C, but A to all C, and that should not belong to some
|
|
B, e.g. black belongs to no swan, animal to every swan, and animal not
|
|
to everything black. Consequently if it is assumed that B belongs to
|
|
all C, and A to no C, A will not belong to some B: and the
|
|
conclusion is true, though the premisses are false.
|
|
|
|
(2) Also if each premiss is partly false, the conclusion may be
|
|
true. For nothing prevents both A and B from belonging to some C while
|
|
A belongs to some B, e.g. white and beautiful belong to some
|
|
animals, and white to some beautiful things. If then it is stated that
|
|
A and B belong to all C, the premisses are partially false, but the
|
|
conclusion is true. Similarly if the premiss AC is stated as negative.
|
|
For nothing prevents A from not belonging, and B from belonging, to
|
|
some C, while A does not belong to all B, e.g. white does not belong
|
|
to some animals, beautiful belongs to some animals, and white does not
|
|
belong to everything beautiful. Consequently if it is assumed that A
|
|
belongs to no C, and B to all C, both premisses are partly false,
|
|
but the conclusion is true.
|
|
|
|
(3) Similarly if one of the premisses assumed is wholly false, the
|
|
other wholly true. For it is possible that both A and B should
|
|
follow all C, though A does not belong to some B, e.g. animal and
|
|
white follow every swan, though animal does not belong to everything
|
|
white. Taking these then as terms, if one assumes that B belongs to
|
|
the whole of C, but A does not belong to C at all, the premiss BC will
|
|
be wholly true, the premiss AC wholly false, and the conclusion
|
|
true. Similarly if the statement BC is false, the statement AC true,
|
|
the conclusion may be true. The same terms will serve for the proof.
|
|
Also if both the premisses assumed are affirmative, the conclusion may
|
|
be true. For nothing prevents B from following all C, and A from not
|
|
belonging to C at all, though A belongs to some B, e.g. animal belongs
|
|
to every swan, black to no swan, and black to some animals.
|
|
Consequently if it is assumed that A and B belong to every C, the
|
|
premiss BC is wholly true, the premiss AC is wholly false, and the
|
|
conclusion is true. Similarly if the premiss AC which is assumed is
|
|
true: the proof can be made through the same terms.
|
|
|
|
(4) Again if one premiss is wholly true, the other partly false, the
|
|
conclusion may be true. For it is possible that B should belong to all
|
|
C, and A to some C, while A belongs to some B, e.g. biped belongs to
|
|
every man, beautiful not to every man, and beautiful to some bipeds.
|
|
If then it is assumed that both A and B belong to the whole of C,
|
|
the premiss BC is wholly true, the premiss AC partly false, the
|
|
conclusion true. Similarly if of the premisses assumed AC is true
|
|
and BC partly false, a true conclusion is possible: this can be
|
|
proved, if the same terms as before are transposed. Also the
|
|
conclusion may be true if one premiss is negative, the other
|
|
affirmative. For since it is possible that B should belong to the
|
|
whole of C, and A to some C, and, when they are so, that A should
|
|
not belong to all B, therefore it is assumed that B belongs to the
|
|
whole of C, and A to no C, the negative premiss is partly false, the
|
|
other premiss wholly true, and the conclusion is true. Again since
|
|
it has been proved that if A belongs to no C and B to some C, it is
|
|
possible that A should not belong to some C, it is clear that if the
|
|
premiss AC is wholly true, and the premiss BC partly false, it is
|
|
possible that the conclusion should be true. For if it is assumed that
|
|
A belongs to no C, and B to all C, the premiss AC is wholly true,
|
|
and the premiss BC is partly false.
|
|
|
|
(5) It is clear also in the case of particular syllogisms that a
|
|
true conclusion may come through what is false, in every possible way.
|
|
For the same terms must be taken as have been taken when the premisses
|
|
are universal, positive terms in positive syllogisms, negative terms
|
|
in negative. For it makes no difference to the setting out of the
|
|
terms, whether one assumes that what belongs to none belongs to all or
|
|
that what belongs to some belongs to all. The same applies to negative
|
|
statements.
|
|
|
|
It is clear then that if the conclusion is false, the premisses of
|
|
the argument must be false, either all or some of them; but when the
|
|
conclusion is true, it is not necessary that the premisses should be
|
|
true, either one or all, yet it is possible, though no part of the
|
|
syllogism is true, that the conclusion may none the less be true;
|
|
but it is not necessitated. The reason is that when two things are
|
|
so related to one another, that if the one is, the other necessarily
|
|
is, then if the latter is not, the former will not be either, but if
|
|
the latter is, it is not necessary that the former should be. But it
|
|
is impossible that the same thing should be necessitated by the
|
|
being and by the not-being of the same thing. I mean, for example,
|
|
that it is impossible that B should necessarily be great since A is
|
|
white and that B should necessarily be great since A is not white. For
|
|
whenever since this, A, is white it is necessary that that, B,
|
|
should be great, and since B is great that C should not be white, then
|
|
it is necessary if is white that C should not be white. And whenever
|
|
it is necessary, since one of two things is, that the other should be,
|
|
it is necessary, if the latter is not, that the former (viz. A) should
|
|
not be. If then B is not great A cannot be white. But if, when A is
|
|
not white, it is necessary that B should be great, it necessarily
|
|
results that if B is not great, B itself is great. (But this is
|
|
impossible.) For if B is not great, A will necessarily not be white.
|
|
If then when this is not white B must be great, it results that if B
|
|
is not great, it is great, just as if it were proved through three
|
|
terms.
|
|
|
|
5
|
|
|
|
Circular and reciprocal proof means proof by means of the
|
|
conclusion, i.e. by converting one of the premisses simply and
|
|
inferring the premiss which was assumed in the original syllogism:
|
|
e.g. suppose it has been necessary to prove that A belongs to all C,
|
|
and it has been proved through B; suppose that A should now be
|
|
proved to belong to B by assuming that A belongs to C, and C to B-so A
|
|
belongs to B: but in the first syllogism the converse was assumed,
|
|
viz. that B belongs to C. Or suppose it is necessary to prove that B
|
|
belongs to C, and A is assumed to belong to C, which was the
|
|
conclusion of the first syllogism, and B to belong to A but the
|
|
converse was assumed in the earlier syllogism, viz. that A belongs
|
|
to B. In no other way is reciprocal proof possible. If another term is
|
|
taken as middle, the proof is not circular: for neither of the
|
|
propositions assumed is the same as before: if one of the accepted
|
|
terms is taken as middle, only one of the premisses of the first
|
|
syllogism can be assumed in the second: for if both of them are
|
|
taken the same conclusion as before will result: but it must be
|
|
different. If the terms are not convertible, one of the premisses from
|
|
which the syllogism results must be undemonstrated: for it is not
|
|
possible to demonstrate through these terms that the third belongs
|
|
to the middle or the middle to the first. If the terms are
|
|
convertible, it is possible to demonstrate everything reciprocally,
|
|
e.g. if A and B and C are convertible with one another. Suppose the
|
|
proposition AC has been demonstrated through B as middle term, and
|
|
again the proposition AB through the conclusion and the premiss BC
|
|
converted, and similarly the proposition BC through the conclusion and
|
|
the premiss AB converted. But it is necessary to prove both the
|
|
premiss CB, and the premiss BA: for we have used these alone without
|
|
demonstrating them. If then it is assumed that B belongs to all C, and
|
|
C to all A, we shall have a syllogism relating B to A. Again if it
|
|
is assumed that C belongs to all A, and A to all B, C must belong to
|
|
all B. In both these syllogisms the premiss CA has been assumed
|
|
without being demonstrated: the other premisses had ex hypothesi
|
|
been proved. Consequently if we succeed in demonstrating this premiss,
|
|
all the premisses will have been proved reciprocally. If then it is
|
|
assumed that C belongs to all B, and B to all A, both the premisses
|
|
assumed have been proved, and C must belong to A. It is clear then
|
|
that only if the terms are convertible is circular and reciprocal
|
|
demonstration possible (if the terms are not convertible, the matter
|
|
stands as we said above). But it turns out in these also that we use
|
|
for the demonstration the very thing that is being proved: for C is
|
|
proved of B, and B of by assuming that C is said of and C is proved of
|
|
A through these premisses, so that we use the conclusion for the
|
|
demonstration.
|
|
|
|
In negative syllogisms reciprocal proof is as follows. Let B
|
|
belong to all C, and A to none of the Bs: we conclude that A belongs
|
|
to none of the Cs. If again it is necessary to prove that A belongs to
|
|
none of the Bs (which was previously assumed) A must belong to no C,
|
|
and C to all B: thus the previous premiss is reversed. If it is
|
|
necessary to prove that B belongs to C, the proposition AB must no
|
|
longer be converted as before: for the premiss 'B belongs to no A'
|
|
is identical with the premiss 'A belongs to no B'. But we must
|
|
assume that B belongs to all of that to none of which longs. Let A
|
|
belong to none of the Cs (which was the previous conclusion) and
|
|
assume that B belongs to all of that to none of which A belongs. It is
|
|
necessary then that B should belong to all C. Consequently each of the
|
|
three propositions has been made a conclusion, and this is circular
|
|
demonstration, to assume the conclusion and the converse of one of the
|
|
premisses, and deduce the remaining premiss.
|
|
|
|
In particular syllogisms it is not possible to demonstrate the
|
|
universal premiss through the other propositions, but the particular
|
|
premiss can be demonstrated. Clearly it is impossible to demonstrate
|
|
the universal premiss: for what is universal is proved through
|
|
propositions which are universal, but the conclusion is not universal,
|
|
and the proof must start from the conclusion and the other premiss.
|
|
Further a syllogism cannot be made at all if the other premiss is
|
|
converted: for the result is that both premisses are particular. But
|
|
the particular premiss may be proved. Suppose that A has been proved
|
|
of some C through B. If then it is assumed that B belongs to all A and
|
|
the conclusion is retained, B will belong to some C: for we obtain the
|
|
first figure and A is middle. But if the syllogism is negative, it
|
|
is not possible to prove the universal premiss, for the reason given
|
|
above. But it is possible to prove the particular premiss, if the
|
|
proposition AB is converted as in the universal syllogism, i.e 'B
|
|
belongs to some of that to some of which A does not belong': otherwise
|
|
no syllogism results because the particular premiss is negative.
|
|
|
|
6
|
|
|
|
In the second figure it is not possible to prove an affirmative
|
|
proposition in this way, but a negative proposition may be proved.
|
|
An affirmative proposition is not proved because both premisses of the
|
|
new syllogism are not affirmative (for the conclusion is negative) but
|
|
an affirmative proposition is (as we saw) proved from premisses
|
|
which are both affirmative. The negative is proved as follows. Let A
|
|
belong to all B, and to no C: we conclude that B belongs to no C. If
|
|
then it is assumed that B belongs to all A, it is necessary that A
|
|
should belong to no C: for we get the second figure, with B as middle.
|
|
But if the premiss AB was negative, and the other affirmative, we
|
|
shall have the first figure. For C belongs to all A and B to no C,
|
|
consequently B belongs to no A: neither then does A belong to B.
|
|
Through the conclusion, therefore, and one premiss, we get no
|
|
syllogism, but if another premiss is assumed in addition, a
|
|
syllogism will be possible. But if the syllogism not universal, the
|
|
universal premiss cannot be proved, for the same reason as we gave
|
|
above, but the particular premiss can be proved whenever the universal
|
|
statement is affirmative. Let A belong to all B, and not to all C: the
|
|
conclusion is BC. If then it is assumed that B belongs to all A, but
|
|
not to all C, A will not belong to some C, B being middle. But if
|
|
the universal premiss is negative, the premiss AC will not be
|
|
demonstrated by the conversion of AB: for it turns out that either
|
|
both or one of the premisses is negative; consequently a syllogism
|
|
will not be possible. But the proof will proceed as in the universal
|
|
syllogisms, if it is assumed that A belongs to some of that to some of
|
|
which B does not belong.
|
|
|
|
7
|
|
|
|
In the third figure, when both premisses are taken universally, it
|
|
is not possible to prove them reciprocally: for that which is
|
|
universal is proved through statements which are universal, but the
|
|
conclusion in this figure is always particular, so that it is clear
|
|
that it is not possible at all to prove through this figure the
|
|
universal premiss. But if one premiss is universal, the other
|
|
particular, proof of the latter will sometimes be possible,
|
|
sometimes not. When both the premisses assumed are affirmative, and
|
|
the universal concerns the minor extreme, proof will be possible,
|
|
but when it concerns the other extreme, impossible. Let A belong to
|
|
all C and B to some C: the conclusion is the statement AB. If then
|
|
it is assumed that C belongs to all A, it has been proved that C
|
|
belongs to some B, but that B belongs to some C has not been proved.
|
|
And yet it is necessary, if C belongs to some B, that B should
|
|
belong to some C. But it is not the same that this should belong to
|
|
that, and that to this: but we must assume besides that if this
|
|
belongs to some of that, that belongs to some of this. But if this
|
|
is assumed the syllogism no longer results from the conclusion and the
|
|
other premiss. But if B belongs to all C, and A to some C, it will
|
|
be possible to prove the proposition AC, when it is assumed that C
|
|
belongs to all B, and A to some B. For if C belongs to all B and A
|
|
to some B, it is necessary that A should belong to some C, B being
|
|
middle. And whenever one premiss is affirmative the other negative,
|
|
and the affirmative is universal, the other premiss can be proved. Let
|
|
B belong to all C, and A not to some C: the conclusion is that A
|
|
does not belong to some B. If then it is assumed further that C
|
|
belongs to all B, it is necessary that A should not belong to some
|
|
C, B being middle. But when the negative premiss is universal, the
|
|
other premiss is not except as before, viz. if it is assumed that that
|
|
belongs to some of that, to some of which this does not belong, e.g.
|
|
if A belongs to no C, and B to some C: the conclusion is that A does
|
|
not belong to some B. If then it is assumed that C belongs to some
|
|
of that to some of which does not belong, it is necessary that C
|
|
should belong to some of the Bs. In no other way is it possible by
|
|
converting the universal premiss to prove the other: for in no other
|
|
way can a syllogism be formed.
|
|
|
|
It is clear then that in the first figure reciprocal proof is made
|
|
both through the third and through the first figure-if the
|
|
conclusion is affirmative through the first; if the conclusion is
|
|
negative through the last. For it is assumed that that belongs to
|
|
all of that to none of which this belongs. In the middle figure,
|
|
when the syllogism is universal, proof is possible through the
|
|
second figure and through the first, but when particular through the
|
|
second and the last. In the third figure all proofs are made through
|
|
itself. It is clear also that in the third figure and in the middle
|
|
figure those syllogisms which are not made through those figures
|
|
themselves either are not of the nature of circular proof or are
|
|
imperfect.
|
|
|
|
8
|
|
|
|
To convert a syllogism means to alter the conclusion and make
|
|
another syllogism to prove that either the extreme cannot belong to
|
|
the middle or the middle to the last term. For it is necessary, if the
|
|
conclusion has been changed into its opposite and one of the premisses
|
|
stands, that the other premiss should be destroyed. For if it should
|
|
stand, the conclusion also must stand. It makes a difference whether
|
|
the conclusion is converted into its contradictory or into its
|
|
contrary. For the same syllogism does not result whichever form the
|
|
conversion takes. This will be made clear by the sequel. By
|
|
contradictory opposition I mean the opposition of 'to all' to 'not
|
|
to all', and of 'to some' to 'to none'; by contrary opposition I
|
|
mean the opposition of 'to all' to 'to none', and of 'to some' to 'not
|
|
to some'. Suppose that A been proved of C, through B as middle term.
|
|
If then it should be assumed that A belongs to no C, but to all B, B
|
|
will belong to no C. And if A belongs to no C, and B to all C, A
|
|
will belong, not to no B at all, but not to all B. For (as we saw) the
|
|
universal is not proved through the last figure. In a word it is not
|
|
possible to refute universally by conversion the premiss which
|
|
concerns the major extreme: for the refutation always proceeds through
|
|
the third since it is necessary to take both premisses in reference to
|
|
the minor extreme. Similarly if the syllogism is negative. Suppose
|
|
it has been proved that A belongs to no C through B. Then if it is
|
|
assumed that A belongs to all C, and to no B, B will belong to none of
|
|
the Cs. And if A and B belong to all C, A will belong to some B: but
|
|
in the original premiss it belonged to no B.
|
|
|
|
If the conclusion is converted into its contradictory, the
|
|
syllogisms will be contradictory and not universal. For one premiss is
|
|
particular, so that the conclusion also will be particular. Let the
|
|
syllogism be affirmative, and let it be converted as stated. Then if A
|
|
belongs not to all C, but to all B, B will belong not to all C. And if
|
|
A belongs not to all C, but B belongs to all C, A will belong not to
|
|
all B. Similarly if the syllogism is negative. For if A belongs to
|
|
some C, and to no B, B will belong, not to no C at all, but-not to
|
|
some C. And if A belongs to some C, and B to all C, as was
|
|
originally assumed, A will belong to some B.
|
|
|
|
In particular syllogisms when the conclusion is converted into its
|
|
contradictory, both premisses may be refuted, but when it is converted
|
|
into its contrary, neither. For the result is no longer, as in the
|
|
universal syllogisms, refutation in which the conclusion reached by O,
|
|
conversion lacks universality, but no refutation at all. Suppose
|
|
that A has been proved of some C. If then it is assumed that A belongs
|
|
to no C, and B to some C, A will not belong to some B: and if A
|
|
belongs to no C, but to all B, B will belong to no C. Thus both
|
|
premisses are refuted. But neither can be refuted if the conclusion is
|
|
converted into its contrary. For if A does not belong to some C, but
|
|
to all B, then B will not belong to some C. But the original premiss
|
|
is not yet refuted: for it is possible that B should belong to some C,
|
|
and should not belong to some C. The universal premiss AB cannot be
|
|
affected by a syllogism at all: for if A does not belong to some of
|
|
the Cs, but B belongs to some of the Cs, neither of the premisses is
|
|
universal. Similarly if the syllogism is negative: for if it should be
|
|
assumed that A belongs to all C, both premisses are refuted: but if
|
|
the assumption is that A belongs to some C, neither premiss is
|
|
refuted. The proof is the same as before.
|
|
|
|
9
|
|
|
|
In the second figure it is not possible to refute the premiss
|
|
which concerns the major extreme by establishing something contrary to
|
|
it, whichever form the conversion of the conclusion may take. For
|
|
the conclusion of the refutation will always be in the third figure,
|
|
and in this figure (as we saw) there is no universal syllogism. The
|
|
other premiss can be refuted in a manner similar to the conversion:
|
|
I mean, if the conclusion of the first syllogism is converted into its
|
|
contrary, the conclusion of the refutation will be the contrary of the
|
|
minor premiss of the first, if into its contradictory, the
|
|
contradictory. Let A belong to all B and to no C: conclusion BC. If
|
|
then it is assumed that B belongs to all C, and the proposition AB
|
|
stands, A will belong to all C, since the first figure is produced. If
|
|
B belongs to all C, and A to no C, then A belongs not to all B: the
|
|
figure is the last. But if the conclusion BC is converted into its
|
|
contradictory, the premiss AB will be refuted as before, the
|
|
premiss, AC by its contradictory. For if B belongs to some C, and A to
|
|
no C, then A will not belong to some B. Again if B belongs to some
|
|
C, and A to all B, A will belong to some C, so that the syllogism
|
|
results in the contradictory of the minor premiss. A similar proof can
|
|
be given if the premisses are transposed in respect of their quality.
|
|
|
|
If the syllogism is particular, when the conclusion is converted
|
|
into its contrary neither premiss can be refuted, as also happened
|
|
in the first figure,' if the conclusion is converted into its
|
|
contradictory, both premisses can be refuted. Suppose that A belongs
|
|
to no B, and to some C: the conclusion is BC. If then it is assumed
|
|
that B belongs to some C, and the statement AB stands, the
|
|
conclusion will be that A does not belong to some C. But the
|
|
original statement has not been refuted: for it is possible that A
|
|
should belong to some C and also not to some C. Again if B belongs
|
|
to some C and A to some C, no syllogism will be possible: for
|
|
neither of the premisses taken is universal. Consequently the
|
|
proposition AB is not refuted. But if the conclusion is converted into
|
|
its contradictory, both premisses can be refuted. For if B belongs
|
|
to all C, and A to no B, A will belong to no C: but it was assumed
|
|
to belong to some C. Again if B belongs to all C and A to some C, A
|
|
will belong to some B. The same proof can be given if the universal
|
|
statement is affirmative.
|
|
|
|
10
|
|
|
|
In the third figure when the conclusion is converted into its
|
|
contrary, neither of the premisses can be refuted in any of the
|
|
syllogisms, but when the conclusion is converted into its
|
|
contradictory, both premisses may be refuted and in all the moods.
|
|
Suppose it has been proved that A belongs to some B, C being taken
|
|
as middle, and the premisses being universal. If then it is assumed
|
|
that A does not belong to some B, but B belongs to all C, no syllogism
|
|
is formed about A and C. Nor if A does not belong to some B, but
|
|
belongs to all C, will a syllogism be possible about B and C. A
|
|
similar proof can be given if the premisses are not universal. For
|
|
either both premisses arrived at by the conversion must be particular,
|
|
or the universal premiss must refer to the minor extreme. But we found
|
|
that no syllogism is possible thus either in the first or in the
|
|
middle figure. But if the conclusion is converted into its
|
|
contradictory, both the premisses can be refuted. For if A belongs
|
|
to no B, and B to all C, then A belongs to no C: again if A belongs to
|
|
no B, and to all C, B belongs to no C. And similarly if one of the
|
|
premisses is not universal. For if A belongs to no B, and B to some C,
|
|
A will not belong to some C: if A belongs to no B, and to C, B will
|
|
belong to no C.
|
|
|
|
Similarly if the original syllogism is negative. Suppose it has been
|
|
proved that A does not belong to some B, BC being affirmative, AC
|
|
being negative: for it was thus that, as we saw, a syllogism could
|
|
be made. Whenever then the contrary of the conclusion is assumed a
|
|
syllogism will not be possible. For if A belongs to some B, and B to
|
|
all C, no syllogism is possible (as we saw) about A and C. Nor, if A
|
|
belongs to some B, and to no C, was a syllogism possible concerning
|
|
B and C. Therefore the premisses are not refuted. But when the
|
|
contradictory of the conclusion is assumed, they are refuted. For if A
|
|
belongs to all B, and B to C, A belongs to all C: but A was supposed
|
|
originally to belong to no C. Again if A belongs to all B, and to no
|
|
C, then B belongs to no C: but it was supposed to belong to all C. A
|
|
similar proof is possible if the premisses are not universal. For AC
|
|
becomes universal and negative, the other premiss particular and
|
|
affirmative. If then A belongs to all B, and B to some C, it results
|
|
that A belongs to some C: but it was supposed to belong to no C. Again
|
|
if A belongs to all B, and to no C, then B belongs to no C: but it was
|
|
assumed to belong to some C. If A belongs to some B and B to some C,
|
|
no syllogism results: nor yet if A belongs to some B, and to no C.
|
|
Thus in one way the premisses are refuted, in the other way they are
|
|
not.
|
|
|
|
From what has been said it is clear how a syllogism results in
|
|
each figure when the conclusion is converted; when a result contrary
|
|
to the premiss, and when a result contradictory to the premiss, is
|
|
obtained. It is clear that in the first figure the syllogisms are
|
|
formed through the middle and the last figures, and the premiss
|
|
which concerns the minor extreme is alway refuted through the middle
|
|
figure, the premiss which concerns the major through the last
|
|
figure. In the second figure syllogisms proceed through the first
|
|
and the last figures, and the premiss which concerns the minor extreme
|
|
is always refuted through the first figure, the premiss which concerns
|
|
the major extreme through the last. In the third figure the refutation
|
|
proceeds through the first and the middle figures; the premiss which
|
|
concerns the major is always refuted through the first figure, the
|
|
premiss which concerns the minor through the middle figure.
|
|
|
|
11
|
|
|
|
It is clear then what conversion is, how it is effected in each
|
|
figure, and what syllogism results. The syllogism per impossibile is
|
|
proved when the contradictory of the conclusion stated and another
|
|
premiss is assumed; it can be made in all the figures. For it
|
|
resembles conversion, differing only in this: conversion takes place
|
|
after a syllogism has been formed and both the premisses have been
|
|
taken, but a reduction to the impossible takes place not because the
|
|
contradictory has been agreed to already, but because it is clear that
|
|
it is true. The terms are alike in both, and the premisses of both are
|
|
taken in the same way. For example if A belongs to all B, C being
|
|
middle, then if it is supposed that A does not belong to all B or
|
|
belongs to no B, but to all C (which was admitted to be true), it
|
|
follows that C belongs to no B or not to all B. But this is
|
|
impossible: consequently the supposition is false: its contradictory
|
|
then is true. Similarly in the other figures: for whatever moods admit
|
|
of conversion admit also of the reduction per impossibile.
|
|
|
|
All the problems can be proved per impossibile in all the figures,
|
|
excepting the universal affirmative, which is proved in the middle and
|
|
third figures, but not in the first. Suppose that A belongs not to all
|
|
B, or to no B, and take besides another premiss concerning either of
|
|
the terms, viz. that C belongs to all A, or that B belongs to all D;
|
|
thus we get the first figure. If then it is supposed that A does not
|
|
belong to all B, no syllogism results whichever term the assumed
|
|
premiss concerns; but if it is supposed that A belongs to no B, when
|
|
the premiss BD is assumed as well we shall prove syllogistically
|
|
what is false, but not the problem proposed. For if A belongs to no B,
|
|
and B belongs to all D, A belongs to no D. Let this be impossible:
|
|
it is false then A belongs to no B. But the universal affirmative is
|
|
not necessarily true if the universal negative is false. But if the
|
|
premiss CA is assumed as well, no syllogism results, nor does it do so
|
|
when it is supposed that A does not belong to all B. Consequently it
|
|
is clear that the universal affirmative cannot be proved in the
|
|
first figure per impossibile.
|
|
|
|
But the particular affirmative and the universal and particular
|
|
negatives can all be proved. Suppose that A belongs to no B, and let
|
|
it have been assumed that B belongs to all or to some C. Then it is
|
|
necessary that A should belong to no C or not to all C. But this is
|
|
impossible (for let it be true and clear that A belongs to all C):
|
|
consequently if this is false, it is necessary that A should belong to
|
|
some B. But if the other premiss assumed relates to A, no syllogism
|
|
will be possible. Nor can a conclusion be drawn when the contrary of
|
|
the conclusion is supposed, e.g. that A does not belong to some B.
|
|
Clearly then we must suppose the contradictory.
|
|
|
|
Again suppose that A belongs to some B, and let it have been assumed
|
|
that C belongs to all A. It is necessary then that C should belong
|
|
to some B. But let this be impossible, so that the supposition is
|
|
false: in that case it is true that A belongs to no B. We may
|
|
proceed in the same way if the proposition CA has been taken as
|
|
negative. But if the premiss assumed concerns B, no syllogism will
|
|
be possible. If the contrary is supposed, we shall have a syllogism
|
|
and an impossible conclusion, but the problem in hand is not proved.
|
|
Suppose that A belongs to all B, and let it have been assumed that C
|
|
belongs to all A. It is necessary then that C should belong to all
|
|
B. But this is impossible, so that it is false that A belongs to all
|
|
B. But we have not yet shown it to be necessary that A belongs to no
|
|
B, if it does not belong to all B. Similarly if the other premiss
|
|
taken concerns B; we shall have a syllogism and a conclusion which
|
|
is impossible, but the hypothesis is not refuted. Therefore it is
|
|
the contradictory that we must suppose.
|
|
|
|
To prove that A does not belong to all B, we must suppose that it
|
|
belongs to all B: for if A belongs to all B, and C to all A, then C
|
|
belongs to all B; so that if this is impossible, the hypothesis is
|
|
false. Similarly if the other premiss assumed concerns B. The same
|
|
results if the original proposition CA was negative: for thus also
|
|
we get a syllogism. But if the negative proposition concerns B,
|
|
nothing is proved. If the hypothesis is that A belongs not to all
|
|
but to some B, it is not proved that A belongs not to all B, but
|
|
that it belongs to no B. For if A belongs to some B, and C to all A,
|
|
then C will belong to some B. If then this is impossible, it is
|
|
false that A belongs to some B; consequently it is true that A belongs
|
|
to no B. But if this is proved, the truth is refuted as well; for
|
|
the original conclusion was that A belongs to some B, and does not
|
|
belong to some B. Further the impossible does not result from the
|
|
hypothesis: for then the hypothesis would be false, since it is
|
|
impossible to draw a false conclusion from true premisses: but in fact
|
|
it is true: for A belongs to some B. Consequently we must not
|
|
suppose that A belongs to some B, but that it belongs to all B.
|
|
Similarly if we should be proving that A does not belong to some B:
|
|
for if 'not to belong to some' and 'to belong not to all' have the
|
|
same meaning, the demonstration of both will be identical.
|
|
|
|
It is clear then that not the contrary but the contradictory ought
|
|
to be supposed in all the syllogisms. For thus we shall have necessity
|
|
of inference, and the claim we make is one that will be generally
|
|
accepted. For if of everything one or other of two contradictory
|
|
statements holds good, then if it is proved that the negation does not
|
|
hold, the affirmation must be true. Again if it is not admitted that
|
|
the affirmation is true, the claim that the negation is true will be
|
|
generally accepted. But in neither way does it suit to maintain the
|
|
contrary: for it is not necessary that if the universal negative is
|
|
false, the universal affirmative should be true, nor is it generally
|
|
accepted that if the one is false the other is true.
|
|
|
|
12
|
|
|
|
It is clear then that in the first figure all problems except the
|
|
universal affirmative are proved per impossibile. But in the middle
|
|
and the last figures this also is proved. Suppose that A does not
|
|
belong to all B, and let it have been assumed that A belongs to all C.
|
|
If then A belongs not to all B, but to all C, C will not belong to all
|
|
B. But this is impossible (for suppose it to be clear that C belongs
|
|
to all B): consequently the hypothesis is false. It is true then
|
|
that A belongs to all B. But if the contrary is supposed, we shall
|
|
have a syllogism and a result which is impossible: but the problem
|
|
in hand is not proved. For if A belongs to no B, and to all C, C
|
|
will belong to no B. This is impossible; so that it is false that A
|
|
belongs to no B. But though this is false, it does not follow that
|
|
it is true that A belongs to all B.
|
|
|
|
When A belongs to some B, suppose that A belongs to no B, and let
|
|
A belong to all C. It is necessary then that C should belong to no
|
|
B. Consequently, if this is impossible, A must belong to some B. But
|
|
if it is supposed that A does not belong to some B, we shall have
|
|
the same results as in the first figure.
|
|
|
|
Again suppose that A belongs to some B, and let A belong to no C. It
|
|
is necessary then that C should not belong to some B. But originally
|
|
it belonged to all B, consequently the hypothesis is false: A then
|
|
will belong to no B.
|
|
|
|
When A does not belong to an B, suppose it does belong to all B, and
|
|
to no C. It is necessary then that C should belong to no B. But this
|
|
is impossible: so that it is true that A does not belong to all B.
|
|
It is clear then that all the syllogisms can be formed in the middle
|
|
figure.
|
|
|
|
13
|
|
|
|
Similarly they can all be formed in the last figure. Suppose that
|
|
A does not belong to some B, but C belongs to all B: then A does not
|
|
belong to some C. If then this is impossible, it is false that A
|
|
does not belong to some B; so that it is true that A belongs to all B.
|
|
But if it is supposed that A belongs to no B, we shall have a
|
|
syllogism and a conclusion which is impossible: but the problem in
|
|
hand is not proved: for if the contrary is supposed, we shall have the
|
|
same results as before.
|
|
|
|
But to prove that A belongs to some B, this hypothesis must be made.
|
|
If A belongs to no B, and C to some B, A will belong not to all C.
|
|
If then this is false, it is true that A belongs to some B.
|
|
|
|
When A belongs to no B, suppose A belongs to some B, and let it have
|
|
been assumed that C belongs to all B. Then it is necessary that A
|
|
should belong to some C. But ex hypothesi it belongs to no C, so
|
|
that it is false that A belongs to some B. But if it is supposed
|
|
that A belongs to all B, the problem is not proved.
|
|
|
|
But this hypothesis must be made if we are prove that A belongs
|
|
not to all B. For if A belongs to all B and C to some B, then A
|
|
belongs to some C. But this we assumed not to be so, so it is false
|
|
that A belongs to all B. But in that case it is true that A belongs
|
|
not to all B. If however it is assumed that A belongs to some B, we
|
|
shall have the same result as before.
|
|
|
|
It is clear then that in all the syllogisms which proceed per
|
|
impossibile the contradictory must be assumed. And it is plain that in
|
|
the middle figure an affirmative conclusion, and in the last figure
|
|
a universal conclusion, are proved in a way.
|
|
|
|
14
|
|
|
|
Demonstration per impossibile differs from ostensive proof in that
|
|
it posits what it wishes to refute by reduction to a statement
|
|
admitted to be false; whereas ostensive proof starts from admitted
|
|
positions. Both, indeed, take two premisses that are admitted, but the
|
|
latter takes the premisses from which the syllogism starts, the former
|
|
takes one of these, along with the contradictory of the original
|
|
conclusion. Also in the ostensive proof it is not necessary that the
|
|
conclusion should be known, nor that one should suppose beforehand
|
|
that it is true or not: in the other it is necessary to suppose
|
|
beforehand that it is not true. It makes no difference whether the
|
|
conclusion is affirmative or negative; the method is the same in
|
|
both cases. Everything which is concluded ostensively can be proved
|
|
per impossibile, and that which is proved per impossibile can be
|
|
proved ostensively, through the same terms. Whenever the syllogism
|
|
is formed in the first figure, the truth will be found in the middle
|
|
or the last figure, if negative in the middle, if affirmative in the
|
|
last. Whenever the syllogism is formed in the middle figure, the truth
|
|
will be found in the first, whatever the problem may be. Whenever
|
|
the syllogism is formed in the last figure, the truth will be found in
|
|
the first and middle figures, if affirmative in first, if negative
|
|
in the middle. Suppose that A has been proved to belong to no B, or
|
|
not to all B, through the first figure. Then the hypothesis must
|
|
have been that A belongs to some B, and the original premisses that
|
|
C belongs to all A and to no B. For thus the syllogism was made and
|
|
the impossible conclusion reached. But this is the middle figure, if C
|
|
belongs to all A and to no B. And it is clear from these premisses
|
|
that A belongs to no B. Similarly if has been proved not to belong
|
|
to all B. For the hypothesis is that A belongs to all B; and the
|
|
original premisses are that C belongs to all A but not to all B.
|
|
Similarly too, if the premiss CA should be negative: for thus also
|
|
we have the middle figure. Again suppose it has been proved that A
|
|
belongs to some B. The hypothesis here is that is that A belongs to no
|
|
B; and the original premisses that B belongs to all C, and A either to
|
|
all or to some C: for in this way we shall get what is impossible. But
|
|
if A and B belong to all C, we have the last figure. And it is clear
|
|
from these premisses that A must belong to some B. Similarly if B or A
|
|
should be assumed to belong to some C.
|
|
|
|
Again suppose it has been proved in the middle figure that A belongs
|
|
to all B. Then the hypothesis must have been that A belongs not to all
|
|
B, and the original premisses that A belongs to all C, and C to all B:
|
|
for thus we shall get what is impossible. But if A belongs to all C,
|
|
and C to all B, we have the first figure. Similarly if it has been
|
|
proved that A belongs to some B: for the hypothesis then must have
|
|
been that A belongs to no B, and the original premisses that A belongs
|
|
to all C, and C to some B. If the syllogism is negative, the
|
|
hypothesis must have been that A belongs to some B, and the original
|
|
premisses that A belongs to no C, and C to all B, so that the first
|
|
figure results. If the syllogism is not universal, but proof has
|
|
been given that A does not belong to some B, we may infer in the
|
|
same way. The hypothesis is that A belongs to all B, the original
|
|
premisses that A belongs to no C, and C belongs to some B: for thus we
|
|
get the first figure.
|
|
|
|
Again suppose it has been proved in the third figure that A
|
|
belongs to all B. Then the hypothesis must have been that A belongs
|
|
not to all B, and the original premisses that C belongs to all B,
|
|
and A belongs to all C; for thus we shall get what is impossible.
|
|
And the original premisses form the first figure. Similarly if the
|
|
demonstration establishes a particular proposition: the hypothesis
|
|
then must have been that A belongs to no B, and the original premisses
|
|
that C belongs to some B, and A to all C. If the syllogism is
|
|
negative, the hypothesis must have been that A belongs to some B,
|
|
and the original premisses that C belongs to no A and to all B, and
|
|
this is the middle figure. Similarly if the demonstration is not
|
|
universal. The hypothesis will then be that A belongs to all B, the
|
|
premisses that C belongs to no A and to some B: and this is the middle
|
|
figure.
|
|
|
|
It is clear then that it is possible through the same terms to prove
|
|
each of the problems ostensively as well. Similarly it will be
|
|
possible if the syllogisms are ostensive to reduce them ad impossibile
|
|
in the terms which have been taken, whenever the contradictory of
|
|
the conclusion of the ostensive syllogism is taken as a premiss. For
|
|
the syllogisms become identical with those which are obtained by means
|
|
of conversion, so that we obtain immediately the figures through which
|
|
each problem will be solved. It is clear then that every thesis can be
|
|
proved in both ways, i.e. per impossibile and ostensively, and it is
|
|
not possible to separate one method from the other.
|
|
|
|
15
|
|
|
|
In what figure it is possible to draw a conclusion from premisses
|
|
which are opposed, and in what figure this is not possible, will be
|
|
made clear in this way. Verbally four kinds of opposition are
|
|
possible, viz. universal affirmative to universal negative,
|
|
universal affirmative to particular negative, particular affirmative
|
|
to universal negative, and particular affirmative to particular
|
|
negative: but really there are only three: for the particular
|
|
affirmative is only verbally opposed to the particular negative. Of
|
|
the genuine opposites I call those which are universal contraries, the
|
|
universal affirmative and the universal negative, e.g. 'every
|
|
science is good', 'no science is good'; the others I call
|
|
contradictories.
|
|
|
|
In the first figure no syllogism whether affirmative or negative can
|
|
be made out of opposed premisses: no affirmative syllogism is possible
|
|
because both premisses must be affirmative, but opposites are, the one
|
|
affirmative, the other negative: no negative syllogism is possible
|
|
because opposites affirm and deny the same predicate of the same
|
|
subject, and the middle term in the first figure is not predicated
|
|
of both extremes, but one thing is denied of it, and it is affirmed of
|
|
something else: but such premisses are not opposed.
|
|
|
|
In the middle figure a syllogism can be made both
|
|
oLcontradictories and of contraries. Let A stand for good, let B and C
|
|
stand for science. If then one assumes that every science is good, and
|
|
no science is good, A belongs to all B and to no C, so that B
|
|
belongs to no C: no science then is a science. Similarly if after
|
|
taking 'every science is good' one took 'the science of medicine is
|
|
not good'; for A belongs to all B but to no C, so that a particular
|
|
science will not be a science. Again, a particular science will not be
|
|
a science if A belongs to all C but to no B, and B is science, C
|
|
medicine, and A supposition: for after taking 'no science is
|
|
supposition', one has assumed that a particular science is
|
|
supposition. This syllogism differs from the preceding because the
|
|
relations between the terms are reversed: before, the affirmative
|
|
statement concerned B, now it concerns C. Similarly if one premiss
|
|
is not universal: for the middle term is always that which is stated
|
|
negatively of one extreme, and affirmatively of the other.
|
|
Consequently it is possible that contradictories may lead to a
|
|
conclusion, though not always or in every mood, but only if the
|
|
terms subordinate to the middle are such that they are either
|
|
identical or related as whole to part. Otherwise it is impossible: for
|
|
the premisses cannot anyhow be either contraries or contradictories.
|
|
|
|
In the third figure an affirmative syllogism can never be made out
|
|
of opposite premisses, for the reason given in reference to the
|
|
first figure; but a negative syllogism is possible whether the terms
|
|
are universal or not. Let B and C stand for science, A for medicine.
|
|
If then one should assume that all medicine is science and that no
|
|
medicine is science, he has assumed that B belongs to all A and C to
|
|
no A, so that a particular science will not be a science. Similarly if
|
|
the premiss BA is not assumed universally. For if some medicine is
|
|
science and again no medicine is science, it results that some science
|
|
is not science, The premisses are contrary if the terms are taken
|
|
universally; if one is particular, they are contradictory.
|
|
|
|
We must recognize that it is possible to take opposites in the way
|
|
we said, viz. 'all science is good' and 'no science is good' or
|
|
'some science is not good'. This does not usually escape notice. But
|
|
it is possible to establish one part of a contradiction through
|
|
other premisses, or to assume it in the way suggested in the Topics.
|
|
Since there are three oppositions to affirmative statements, it
|
|
follows that opposite statements may be assumed as premisses in six
|
|
ways; we may have either universal affirmative and negative, or
|
|
universal affirmative and particular negative, or particular
|
|
affirmative and universal negative, and the relations between the
|
|
terms may be reversed; e.g. A may belong to all B and to no C, or to
|
|
all C and to no B, or to all of the one, not to all of the other; here
|
|
too the relation between the terms may be reversed. Similarly in the
|
|
third figure. So it is clear in how many ways and in what figures a
|
|
syllogism can be made by means of premisses which are opposed.
|
|
|
|
It is clear too that from false premisses it is possible to draw a
|
|
true conclusion, as has been said before, but it is not possible if
|
|
the premisses are opposed. For the syllogism is always contrary to the
|
|
fact, e.g. if a thing is good, it is proved that it is not good, if an
|
|
animal, that it is not an animal because the syllogism springs out
|
|
of a contradiction and the terms presupposed are either identical or
|
|
related as whole and part. It is evident also that in fallacious
|
|
reasonings nothing prevents a contradiction to the hypothesis from
|
|
resulting, e.g. if something is odd, it is not odd. For the
|
|
syllogism owed its contrariety to its contradictory premisses; if we
|
|
assume such premisses we shall get a result that contradicts our
|
|
hypothesis. But we must recognize that contraries cannot be inferred
|
|
from a single syllogism in such a way that we conclude that what is
|
|
not good is good, or anything of that sort unless a self-contradictory
|
|
premiss is at once assumed, e.g. 'every animal is white and not
|
|
white', and we proceed 'man is an animal'. Either we must introduce
|
|
the contradiction by an additional assumption, assuming, e.g., that
|
|
every science is supposition, and then assuming 'Medicine is a
|
|
science, but none of it is supposition' (which is the mode in which
|
|
refutations are made), or we must argue from two syllogisms. In no
|
|
other way than this, as was said before, is it possible that the
|
|
premisses should be really contrary.
|
|
|
|
16
|
|
|
|
To beg and assume the original question is a species of failure to
|
|
demonstrate the problem proposed; but this happens in many ways. A man
|
|
may not reason syllogistically at all, or he may argue from
|
|
premisses which are less known or equally unknown, or he may establish
|
|
the antecedent by means of its consequents; for demonstration proceeds
|
|
from what is more certain and is prior. Now begging the question is
|
|
none of these: but since we get to know some things naturally
|
|
through themselves, and other things by means of something else (the
|
|
first principles through themselves, what is subordinate to them
|
|
through something else), whenever a man tries to prove what is not
|
|
self-evident by means of itself, then he begs the original question.
|
|
This may be done by assuming what is in question at once; it is also
|
|
possible to make a transition to other things which would naturally be
|
|
proved through the thesis proposed, and demonstrate it through them,
|
|
e.g. if A should be proved through B, and B through C, though it was
|
|
natural that C should be proved through A: for it turns out that those
|
|
who reason thus are proving A by means of itself. This is what those
|
|
persons do who suppose that they are constructing parallel straight
|
|
lines: for they fail to see that they are assuming facts which it is
|
|
impossible to demonstrate unless the parallels exist. So it turns
|
|
out that those who reason thus merely say a particular thing is, if it
|
|
is: in this way everything will be self-evident. But that is
|
|
impossible.
|
|
|
|
If then it is uncertain whether A belongs to C, and also whether A
|
|
belongs to B, and if one should assume that A does belong to B, it
|
|
is not yet clear whether he begs the original question, but it is
|
|
evident that he is not demonstrating: for what is as uncertain as
|
|
the question to be answered cannot be a principle of a
|
|
demonstration. If however B is so related to C that they are
|
|
identical, or if they are plainly convertible, or the one belongs to
|
|
the other, the original question is begged. For one might equally well
|
|
prove that A belongs to B through those terms if they are convertible.
|
|
But if they are not convertible, it is the fact that they are not that
|
|
prevents such a demonstration, not the method of demonstrating. But if
|
|
one were to make the conversion, then he would be doing what we have
|
|
described and effecting a reciprocal proof with three propositions.
|
|
|
|
Similarly if he should assume that B belongs to C, this being as
|
|
uncertain as the question whether A belongs to C, the question is
|
|
not yet begged, but no demonstration is made. If however A and B are
|
|
identical either because they are convertible or because A follows
|
|
B, then the question is begged for the same reason as before. For we
|
|
have explained the meaning of begging the question, viz. proving
|
|
that which is not self-evident by means of itself.
|
|
|
|
If then begging the question is proving what is not self-evident
|
|
by means of itself, in other words failing to prove when the failure
|
|
is due to the thesis to be proved and the premiss through which it
|
|
is proved being equally uncertain, either because predicates which are
|
|
identical belong to the same subject, or because the same predicate
|
|
belongs to subjects which are identical, the question may be begged in
|
|
the middle and third figures in both ways, though, if the syllogism is
|
|
affirmative, only in the third and first figures. If the syllogism
|
|
is negative, the question is begged when identical predicates are
|
|
denied of the same subject; and both premisses do not beg the question
|
|
indifferently (in a similar way the question may be begged in the
|
|
middle figure), because the terms in negative syllogisms are not
|
|
convertible. In scientific demonstrations the question is begged
|
|
when the terms are really related in the manner described, in
|
|
dialectical arguments when they are according to common opinion so
|
|
related.
|
|
|
|
17
|
|
|
|
The objection that 'this is not the reason why the result is false',
|
|
which we frequently make in argument, is made primarily in the case of
|
|
a reductio ad impossibile, to rebut the proposition which was being
|
|
proved by the reduction. For unless a man has contradicted this
|
|
proposition he will not say, 'False cause', but urge that something
|
|
false has been assumed in the earlier parts of the argument; nor
|
|
will he use the formula in the case of an ostensive proof; for here
|
|
what one denies is not assumed as a premiss. Further when anything
|
|
is refuted ostensively by the terms ABC, it cannot be objected that
|
|
the syllogism does not depend on the assumption laid down. For we
|
|
use the expression 'false cause', when the syllogism is concluded in
|
|
spite of the refutation of this position; but that is not possible
|
|
in ostensive proofs: since if an assumption is refuted, a syllogism
|
|
can no longer be drawn in reference to it. It is clear then that the
|
|
expression 'false cause' can only be used in the case of a reductio ad
|
|
impossibile, and when the original hypothesis is so related to the
|
|
impossible conclusion, that the conclusion results indifferently
|
|
whether the hypothesis is made or not. The most obvious case of the
|
|
irrelevance of an assumption to a conclusion which is false is when
|
|
a syllogism drawn from middle terms to an impossible conclusion is
|
|
independent of the hypothesis, as we have explained in the Topics. For
|
|
to put that which is not the cause as the cause, is just this: e.g. if
|
|
a man, wishing to prove that the diagonal of the square is
|
|
incommensurate with the side, should try to prove Zeno's theorem
|
|
that motion is impossible, and so establish a reductio ad impossibile:
|
|
for Zeno's false theorem has no connexion at all with the original
|
|
assumption. Another case is where the impossible conclusion is
|
|
connected with the hypothesis, but does not result from it. This may
|
|
happen whether one traces the connexion upwards or downwards, e.g.
|
|
if it is laid down that A belongs to B, B to C, and C to D, and it
|
|
should be false that B belongs to D: for if we eliminated A and
|
|
assumed all the same that B belongs to C and C to D, the false
|
|
conclusion would not depend on the original hypothesis. Or again trace
|
|
the connexion upwards; e.g. suppose that A belongs to B, E to A and
|
|
F to E, it being false that F belongs to A. In this way too the
|
|
impossible conclusion would result, though the original hypothesis
|
|
were eliminated. But the impossible conclusion ought to be connected
|
|
with the original terms: in this way it will depend on the hypothesis,
|
|
e.g. when one traces the connexion downwards, the impossible
|
|
conclusion must be connected with that term which is predicate in
|
|
the hypothesis: for if it is impossible that A should belong to D, the
|
|
false conclusion will no longer result after A has been eliminated. If
|
|
one traces the connexion upwards, the impossible conclusion must be
|
|
connected with that term which is subject in the hypothesis: for if it
|
|
is impossible that F should belong to B, the impossible conclusion
|
|
will disappear if B is eliminated. Similarly when the syllogisms are
|
|
negative.
|
|
|
|
It is clear then that when the impossibility is not related to the
|
|
original terms, the false conclusion does not result on account of the
|
|
assumption. Or perhaps even so it may sometimes be independent. For if
|
|
it were laid down that A belongs not to B but to K, and that K belongs
|
|
to C and C to D, the impossible conclusion would still stand.
|
|
Similarly if one takes the terms in an ascending series.
|
|
Consequently since the impossibility results whether the first
|
|
assumption is suppressed or not, it would appear to be independent
|
|
of that assumption. Or perhaps we ought not to understand the
|
|
statement that the false conclusion results independently of the
|
|
assumption, in the sense that if something else were supposed the
|
|
impossibility would result; but rather we mean that when the first
|
|
assumption is eliminated, the same impossibility results through the
|
|
remaining premisses; since it is not perhaps absurd that the same
|
|
false result should follow from several hypotheses, e.g. that
|
|
parallels meet, both on the assumption that the interior angle is
|
|
greater than the exterior and on the assumption that a triangle
|
|
contains more than two right angles.
|
|
|
|
18
|
|
|
|
A false argument depends on the first false statement in it. Every
|
|
syllogism is made out of two or more premisses. If then the false
|
|
conclusion is drawn from two premisses, one or both of them must be
|
|
false: for (as we proved) a false syllogism cannot be drawn from two
|
|
premisses. But if the premisses are more than two, e.g. if C is
|
|
established through A and B, and these through D, E, F, and G, one
|
|
of these higher propositions must be false, and on this the argument
|
|
depends: for A and B are inferred by means of D, E, F, and G.
|
|
Therefore the conclusion and the error results from one of them.
|
|
|
|
19
|
|
|
|
In order to avoid having a syllogism drawn against us we must take
|
|
care, whenever an opponent asks us to admit the reason without the
|
|
conclusions, not to grant him the same term twice over in his
|
|
premisses, since we know that a syllogism cannot be drawn without a
|
|
middle term, and that term which is stated more than once is the
|
|
middle. How we ought to watch the middle in reference to each
|
|
conclusion, is evident from our knowing what kind of thesis is
|
|
proved in each figure. This will not escape us since we know how we
|
|
are maintaining the argument.
|
|
|
|
That which we urge men to beware of in their admissions, they
|
|
ought in attack to try to conceal. This will be possible first, if,
|
|
instead of drawing the conclusions of preliminary syllogisms, they
|
|
take the necessary premisses and leave the conclusions in the dark;
|
|
secondly if instead of inviting assent to propositions which are
|
|
closely connected they take as far as possible those that are not
|
|
connected by middle terms. For example suppose that A is to be
|
|
inferred to be true of F, B, C, D, and E being middle terms. One ought
|
|
then to ask whether A belongs to B, and next whether D belongs to E,
|
|
instead of asking whether B belongs to C; after that he may ask
|
|
whether B belongs to C, and so on. If the syllogism is drawn through
|
|
one middle term, he ought to begin with that: in this way he will most
|
|
likely deceive his opponent.
|
|
|
|
20
|
|
|
|
Since we know when a syllogism can be formed and how its terms
|
|
must be related, it is clear when refutation will be possible and when
|
|
impossible. A refutation is possible whether everything is conceded,
|
|
or the answers alternate (one, I mean, being affirmative, the other
|
|
negative). For as has been shown a syllogism is possible whether the
|
|
terms are related in affirmative propositions or one proposition is
|
|
affirmative, the other negative: consequently, if what is laid down is
|
|
contrary to the conclusion, a refutation must take place: for a
|
|
refutation is a syllogism which establishes the contradictory. But
|
|
if nothing is conceded, a refutation is impossible: for no syllogism
|
|
is possible (as we saw) when all the terms are negative: therefore
|
|
no refutation is possible. For if a refutation were possible, a
|
|
syllogism must be possible; although if a syllogism is possible it
|
|
does not follow that a refutation is possible. Similarly refutation is
|
|
not possible if nothing is conceded universally: since the fields of
|
|
refutation and syllogism are defined in the same way.
|
|
|
|
21
|
|
|
|
It sometimes happens that just as we are deceived in the arrangement
|
|
of the terms, so error may arise in our thought about them, e.g. if it
|
|
is possible that the same predicate should belong to more than one
|
|
subject immediately, but although knowing the one, a man may forget
|
|
the other and think the opposite true. Suppose that A belongs to B and
|
|
to C in virtue of their nature, and that B and C belong to all D in
|
|
the same way. If then a man thinks that A belongs to all B, and B to
|
|
D, but A to no C, and C to all D, he will both know and not know the
|
|
same thing in respect of the same thing. Again if a man were to make a
|
|
mistake about the members of a single series; e.g. suppose A belongs
|
|
to B, B to C, and C to D, but some one thinks that A belongs to all B,
|
|
but to no C: he will both know that A belongs to D, and think that
|
|
it does not. Does he then maintain after this simply that what he
|
|
knows, he does not think? For he knows in a way that A belongs to C
|
|
through B, since the part is included in the whole; so that what he
|
|
knows in a way, this he maintains he does not think at all: but that
|
|
is impossible.
|
|
|
|
In the former case, where the middle term does not belong to the
|
|
same series, it is not possible to think both the premisses with
|
|
reference to each of the two middle terms: e.g. that A belongs to
|
|
all B, but to no C, and both B and C belong to all D. For it turns out
|
|
that the first premiss of the one syllogism is either wholly or
|
|
partially contrary to the first premiss of the other. For if he thinks
|
|
that A belongs to everything to which B belongs, and he knows that B
|
|
belongs to D, then he knows that A belongs to D. Consequently if again
|
|
he thinks that A belongs to nothing to which C belongs, he thinks that
|
|
A does not belong to some of that to which B belongs; but if he thinks
|
|
that A belongs to everything to which B belongs, and again thinks that
|
|
A does not belong to some of that to which B belongs, these beliefs
|
|
are wholly or partially contrary. In this way then it is not
|
|
possible to think; but nothing prevents a man thinking one premiss
|
|
of each syllogism of both premisses of one of the two syllogisms: e.g.
|
|
A belongs to all B, and B to D, and again A belongs to no C. An
|
|
error of this kind is similar to the error into which we fall
|
|
concerning particulars: e.g. if A belongs to all B, and B to all C,
|
|
A will belong to all C. If then a man knows that A belongs to
|
|
everything to which B belongs, he knows that A belongs to C. But
|
|
nothing prevents his being ignorant that C exists; e.g. let A stand
|
|
for two right angles, B for triangle, C for a particular diagram of
|
|
a triangle. A man might think that C did not exist, though he knew
|
|
that every triangle contains two right angles; consequently he will
|
|
know and not know the same thing at the same time. For the
|
|
expression 'to know that every triangle has its angles equal to two
|
|
right angles' is ambiguous, meaning to have the knowledge either of
|
|
the universal or of the particulars. Thus then he knows that C
|
|
contains two right angles with a knowledge of the universal, but not
|
|
with a knowledge of the particulars; consequently his knowledge will
|
|
not be contrary to his ignorance. The argument in the Meno that
|
|
learning is recollection may be criticized in a similar way. For it
|
|
never happens that a man starts with a foreknowledge of the
|
|
particular, but along with the process of being led to see the general
|
|
principle he receives a knowledge of the particulars, by an act (as it
|
|
were) of recognition. For we know some things directly; e.g. that
|
|
the angles are equal to two right angles, if we know that the figure
|
|
is a triangle. Similarly in all other cases.
|
|
|
|
By a knowledge of the universal then we see the particulars, but
|
|
we do not know them by the kind of knowledge which is proper to
|
|
them; consequently it is possible that we may make mistakes about
|
|
them, but not that we should have the knowledge and error that are
|
|
contrary to one another: rather we have the knowledge of the universal
|
|
but make a mistake in apprehending the particular. Similarly in the
|
|
cases stated above. The error in respect of the middle term is not
|
|
contrary to the knowledge obtained through the syllogism, nor is the
|
|
thought in respect of one middle term contrary to that in respect of
|
|
the other. Nothing prevents a man who knows both that A belongs to the
|
|
whole of B, and that B again belongs to C, thinking that A does not
|
|
belong to C, e.g. knowing that every mule is sterile and that this
|
|
is a mule, and thinking that this animal is with foal: for he does not
|
|
know that A belongs to C, unless he considers the two propositions
|
|
together. So it is evident that if he knows the one and does not
|
|
know the other, he will fall into error. And this is the relation of
|
|
knowledge of the universal to knowledge of the particular. For we know
|
|
no sensible thing, once it has passed beyond the range of our
|
|
senses, even if we happen to have perceived it, except by means of the
|
|
universal and the possession of the knowledge which is proper to the
|
|
particular, but without the actual exercise of that knowledge. For
|
|
to know is used in three senses: it may mean either to have
|
|
knowledge of the universal or to have knowledge proper to the matter
|
|
in hand or to exercise such knowledge: consequently three kinds of
|
|
error also are possible. Nothing then prevents a man both knowing
|
|
and being mistaken about the same thing, provided that his knowledge
|
|
and his error are not contrary. And this happens also to the man whose
|
|
knowledge is limited to each of the premisses and who has not
|
|
previously considered the particular question. For when he thinks that
|
|
the mule is with foal he has not the knowledge in the sense of its
|
|
actual exercise, nor on the other hand has his thought caused an error
|
|
contrary to his knowledge: for the error contrary to the knowledge
|
|
of the universal would be a syllogism.
|
|
|
|
But he who thinks the essence of good is the essence of bad will
|
|
think the same thing to be the essence of good and the essence of bad.
|
|
Let A stand for the essence of good and B for the essence of bad,
|
|
and again C for the essence of good. Since then he thinks B and C
|
|
identical, he will think that C is B, and similarly that B is A,
|
|
consequently that C is A. For just as we saw that if B is true of
|
|
all of which C is true, and A is true of all of which B is true, A
|
|
is true of C, similarly with the word 'think'. Similarly also with the
|
|
word 'is'; for we saw that if C is the same as B, and B as A, C is the
|
|
same as A. Similarly therefore with 'opine'. Perhaps then this is
|
|
necessary if a man will grant the first point. But presumably that
|
|
is false, that any one could suppose the essence of good to be the
|
|
essence of bad, save incidentally. For it is possible to think this in
|
|
many different ways. But we must consider this matter better.
|
|
|
|
22
|
|
|
|
Whenever the extremes are convertible it is necessary that the
|
|
middle should be convertible with both. For if A belongs to C
|
|
through B, then if A and C are convertible and C belongs everything to
|
|
which A belongs, B is convertible with A, and B belongs to
|
|
everything to which A belongs, through C as middle, and C is
|
|
convertible with B through A as middle. Similarly if the conclusion is
|
|
negative, e.g. if B belongs to C, but A does not belong to B,
|
|
neither will A belong to C. If then B is convertible with A, C will be
|
|
convertible with A. Suppose B does not belong to A; neither then
|
|
will C: for ex hypothesi B belonged to all C. And if C is
|
|
convertible with B, B is convertible also with A, for C is said of
|
|
that of all of which B is said. And if C is convertible in relation to
|
|
A and to B, B also is convertible in relation to A. For C belongs to
|
|
that to which B belongs: but C does not belong to that to which A
|
|
belongs. And this alone starts from the conclusion; the preceding
|
|
moods do not do so as in the affirmative syllogism. Again if A and B
|
|
are convertible, and similarly C and D, and if A or C must belong to
|
|
anything whatever, then B and D will be such that one or other belongs
|
|
to anything whatever. For since B belongs to that to which A
|
|
belongs, and D belongs to that to which C belongs, and since A or C
|
|
belongs to everything, but not together, it is clear that B or D
|
|
belongs to everything, but not together. For example if that which
|
|
is uncreated is incorruptible and that which is incorruptible is
|
|
uncreated, it is necessary that what is created should be
|
|
corruptible and what is corruptible should have been created. For
|
|
two syllogisms have been put together. Again if A or B belongs to
|
|
everything and if C or D belongs to everything, but they cannot belong
|
|
together, then when A and C are convertible B and D are convertible.
|
|
For if B does not belong to something to which D belongs, it is
|
|
clear that A belongs to it. But if A then C: for they are convertible.
|
|
Therefore C and D belong together. But this is impossible. When A
|
|
belongs to the whole of B and to C and is affirmed of nothing else,
|
|
and B also belongs to all C, it is necessary that A and B should be
|
|
convertible: for since A is said of B and C only, and B is affirmed
|
|
both of itself and of C, it is clear that B will be said of everything
|
|
of which A is said, except A itself. Again when A and B belong to
|
|
the whole of C, and C is convertible with B, it is necessary that A
|
|
should belong to all B: for since A belongs to all C, and C to B by
|
|
conversion, A will belong to all B.
|
|
|
|
When, of two opposites A and B, A is preferable to B, and
|
|
similarly D is preferable to C, then if A and C together are
|
|
preferable to B and D together, A must be preferable to D. For A is an
|
|
object of desire to the same extent as B is an object of aversion,
|
|
since they are opposites: and C is similarly related to D, since
|
|
they also are opposites. If then A is an object of desire to the
|
|
same extent as D, B is an object of aversion to the same extent as C
|
|
(since each is to the same extent as each-the one an object of
|
|
aversion, the other an object of desire). Therefore both A and C
|
|
together, and B and D together, will be equally objects of desire or
|
|
aversion. But since A and C are preferable to B and D, A cannot be
|
|
equally desirable with D; for then B along with D would be equally
|
|
desirable with A along with C. But if D is preferable to A, then B
|
|
must be less an object of aversion than C: for the less is opposed
|
|
to the less. But the greater good and lesser evil are preferable to
|
|
the lesser good and greater evil: the whole BD then is preferable to
|
|
the whole AC. But ex hypothesi this is not so. A then is preferable to
|
|
D, and C consequently is less an object of aversion than B. If then
|
|
every lover in virtue of his love would prefer A, viz. that the
|
|
beloved should be such as to grant a favour, and yet should not
|
|
grant it (for which C stands), to the beloved's granting the favour
|
|
(represented by D) without being such as to grant it (represented by
|
|
B), it is clear that A (being of such a nature) is preferable to
|
|
granting the favour. To receive affection then is preferable in love
|
|
to sexual intercourse. Love then is more dependent on friendship
|
|
than on intercourse. And if it is most dependent on receiving
|
|
affection, then this is its end. Intercourse then either is not an end
|
|
at all or is an end relative to the further end, the receiving of
|
|
affection. And indeed the same is true of the other desires and arts.
|
|
|
|
23
|
|
|
|
It is clear then how the terms are related in conversion, and in
|
|
respect of being in a higher degree objects of aversion or of
|
|
desire. We must now state that not only dialectical and
|
|
demonstrative syllogisms are formed by means of the aforesaid figures,
|
|
but also rhetorical syllogisms and in general any form of
|
|
persuasion, however it may be presented. For every belief comes either
|
|
through syllogism or from induction.
|
|
|
|
Now induction, or rather the syllogism which springs out of
|
|
induction, consists in establishing syllogistically a relation between
|
|
one extreme and the middle by means of the other extreme, e.g. if B is
|
|
the middle term between A and C, it consists in proving through C that
|
|
A belongs to B. For this is the manner in which we make inductions.
|
|
For example let A stand for long-lived, B for bileless, and C for
|
|
the particular long-lived animals, e.g. man, horse, mule. A then
|
|
belongs to the whole of C: for whatever is bileless is long-lived. But
|
|
B also ('not possessing bile') belongs to all C. If then C is
|
|
convertible with B, and the middle term is not wider in extension,
|
|
it is necessary that A should belong to B. For it has already been
|
|
proved that if two things belong to the same thing, and the extreme is
|
|
convertible with one of them, then the other predicate will belong
|
|
to the predicate that is converted. But we must apprehend C as made up
|
|
of all the particulars. For induction proceeds through an
|
|
enumeration of all the cases.
|
|
|
|
Such is the syllogism which establishes the first and immediate
|
|
premiss: for where there is a middle term the syllogism proceeds
|
|
through the middle term; when there is no middle term, through
|
|
induction. And in a way induction is opposed to syllogism: for the
|
|
latter proves the major term to belong to the third term by means of
|
|
the middle, the former proves the major to belong to the middle by
|
|
means of the third. In the order of nature, syllogism through the
|
|
middle term is prior and better known, but syllogism through induction
|
|
is clearer to us.
|
|
|
|
24
|
|
|
|
We have an 'example' when the major term is proved to belong to
|
|
the middle by means of a term which resembles the third. It ought to
|
|
be known both that the middle belongs to the third term, and that
|
|
the first belongs to that which resembles the third. For example let A
|
|
be evil, B making war against neighbours, C Athenians against Thebans,
|
|
D Thebans against Phocians. If then we wish to prove that to fight
|
|
with the Thebans is an evil, we must assume that to fight against
|
|
neighbours is an evil. Evidence of this is obtained from similar
|
|
cases, e.g. that the war against the Phocians was an evil to the
|
|
Thebans. Since then to fight against neighbours is an evil, and to
|
|
fight against the Thebans is to fight against neighbours, it is
|
|
clear that to fight against the Thebans is an evil. Now it is clear
|
|
that B belongs to C and to D (for both are cases of making war upon
|
|
one's neighbours) and that A belongs to D (for the war against the
|
|
Phocians did not turn out well for the Thebans): but that A belongs to
|
|
B will be proved through D. Similarly if the belief in the relation of
|
|
the middle term to the extreme should be produced by several similar
|
|
cases. Clearly then to argue by example is neither like reasoning from
|
|
part to whole, nor like reasoning from whole to part, but rather
|
|
reasoning from part to part, when both particulars are subordinate
|
|
to the same term, and one of them is known. It differs from induction,
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|
because induction starting from all the particular cases proves (as we
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|
saw) that the major term belongs to the middle, and does not apply the
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|
syllogistic conclusion to the minor term, whereas argument by
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|
example does make this application and does not draw its proof from
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|
all the particular cases.
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25
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|
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By reduction we mean an argument in which the first term clearly
|
|
belongs to the middle, but the relation of the middle to the last term
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|
is uncertain though equally or more probable than the conclusion; or
|
|
again an argument in which the terms intermediate between the last
|
|
term and the middle are few. For in any of these cases it turns out
|
|
that we approach more nearly to knowledge. For example let A stand for
|
|
what can be taught, B for knowledge, C for justice. Now it is clear
|
|
that knowledge can be taught: but it is uncertain whether virtue is
|
|
knowledge. If now the statement BC is equally or more probable than
|
|
AC, we have a reduction: for we are nearer to knowledge, since we have
|
|
taken a new term, being so far without knowledge that A belongs to
|
|
C. Or again suppose that the terms intermediate between B and C are
|
|
few: for thus too we are nearer knowledge. For example let D stand for
|
|
squaring, E for rectilinear figure, F for circle. If there were only
|
|
one term intermediate between E and F (viz. that the circle is made
|
|
equal to a rectilinear figure by the help of lunules), we should be
|
|
near to knowledge. But when BC is not more probable than AC, and the
|
|
intermediate terms are not few, I do not call this reduction: nor
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|
again when the statement BC is immediate: for such a statement is
|
|
knowledge.
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|
|
|
26
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|
|
|
An objection is a premiss contrary to a premiss. It differs from a
|
|
premiss, because it may be particular, but a premiss either cannot
|
|
be particular at all or not in universal syllogisms. An objection is
|
|
brought in two ways and through two figures; in two ways because every
|
|
objection is either universal or particular, by two figures because
|
|
objections are brought in opposition to the premiss, and opposites can
|
|
be proved only in the first and third figures. If a man maintains a
|
|
universal affirmative, we reply with a universal or a particular
|
|
negative; the former is proved from the first figure, the latter
|
|
from the third. For example let stand for there being a single
|
|
science, B for contraries. If a man premises that contraries are
|
|
subjects of a single science, the objection may be either that
|
|
opposites are never subjects of a single science, and contraries are
|
|
opposites, so that we get the first figure, or that the knowable and
|
|
the unknowable are not subjects of a single science: this proof is
|
|
in the third figure: for it is true of C (the knowable and the
|
|
unknowable) that they are contraries, and it is false that they are
|
|
the subjects of a single science.
|
|
|
|
Similarly if the premiss objected to is negative. For if a man
|
|
maintains that contraries are not subjects of a single science, we
|
|
reply either that all opposites or that certain contraries, e.g.
|
|
what is healthy and what is sickly, are subjects of the same
|
|
science: the former argument issues from the first, the latter from
|
|
the third figure.
|
|
|
|
In general if a man urges a universal objection he must frame his
|
|
contradiction with reference to the universal of the terms taken by
|
|
his opponent, e.g. if a man maintains that contraries are not subjects
|
|
of the same science, his opponent must reply that there is a single
|
|
science of all opposites. Thus we must have the first figure: for
|
|
the term which embraces the original subject becomes the middle term.
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|
|
|
If the objection is particular, the objector must frame his
|
|
contradiction with reference to a term relatively to which the subject
|
|
of his opponent's premiss is universal, e.g. he will point out that
|
|
the knowable and the unknowable are not subjects of the same
|
|
science: 'contraries' is universal relatively to these. And we have
|
|
the third figure: for the particular term assumed is middle, e.g.
|
|
the knowable and the unknowable. Premisses from which it is possible
|
|
to draw the contrary conclusion are what we start from when we try
|
|
to make objections. Consequently we bring objections in these
|
|
figures only: for in them only are opposite syllogisms possible, since
|
|
the second figure cannot produce an affirmative conclusion.
|
|
|
|
Besides, an objection in the middle figure would require a fuller
|
|
argument, e.g. if it should not be granted that A belongs to B,
|
|
because C does not follow B. This can be made clear only by other
|
|
premisses. But an objection ought not to turn off into other things,
|
|
but have its new premiss quite clear immediately. For this reason also
|
|
this is the only figure from which proof by signs cannot be obtained.
|
|
|
|
We must consider later the other kinds of objection, namely the
|
|
objection from contraries, from similars, and from common opinion, and
|
|
inquire whether a particular objection cannot be elicited from the
|
|
first figure or a negative objection from the second.
|
|
|
|
27
|
|
|
|
A probability and a sign are not identical, but a probability is a
|
|
generally approved proposition: what men know to happen or not to
|
|
happen, to be or not to be, for the most part thus and thus, is a
|
|
probability, e.g. 'the envious hate', 'the beloved show affection'.
|
|
A sign means a demonstrative proposition necessary or generally
|
|
approved: for anything such that when it is another thing is, or
|
|
when it has come into being the other has come into being before or
|
|
after, is a sign of the other's being or having come into being. Now
|
|
an enthymeme is a syllogism starting from probabilities or signs,
|
|
and a sign may be taken in three ways, corresponding to the position
|
|
of the middle term in the figures. For it may be taken as in the first
|
|
figure or the second or the third. For example the proof that a
|
|
woman is with child because she has milk is in the first figure: for
|
|
to have milk is the middle term. Let A represent to be with child, B
|
|
to have milk, C woman. The proof that wise men are good, since
|
|
Pittacus is good, comes through the last figure. Let A stand for good,
|
|
B for wise men, C for Pittacus. It is true then to affirm both A and B
|
|
of C: only men do not say the latter, because they know it, though
|
|
they state the former. The proof that a woman is with child because
|
|
she is pale is meant to come through the middle figure: for since
|
|
paleness follows women with child and is a concomitant of this
|
|
woman, people suppose it has been proved that she is with child. Let A
|
|
stand for paleness, B for being with child, C for woman. Now if the
|
|
one proposition is stated, we have only a sign, but if the other is
|
|
stated as well, a syllogism, e.g. 'Pittacus is generous, since
|
|
ambitious men are generous and Pittacus is ambitious.' Or again
|
|
'Wise men are good, since Pittacus is not only good but wise.' In this
|
|
way then syllogisms are formed, only that which proceeds through the
|
|
first figure is irrefutable if it is true (for it is universal),
|
|
that which proceeds through the last figure is refutable even if the
|
|
conclusion is true, since the syllogism is not universal nor
|
|
correlative to the matter in question: for though Pittacus is good, it
|
|
is not therefore necessary that all other wise men should be good. But
|
|
the syllogism which proceeds through the middle figure is always
|
|
refutable in any case: for a syllogism can never be formed when the
|
|
terms are related in this way: for though a woman with child is
|
|
pale, and this woman also is pale, it is not necessary that she should
|
|
be with child. Truth then may be found in signs whatever their kind,
|
|
but they have the differences we have stated.
|
|
|
|
We must either divide signs in the way stated, and among them
|
|
designate the middle term as the index (for people call that the index
|
|
which makes us know, and the middle term above all has this
|
|
character), or else we must call the arguments derived from the
|
|
extremes signs, that derived from the middle term the index: for
|
|
that which is proved through the first figure is most generally
|
|
accepted and most true.
|
|
|
|
It is possible to infer character from features, if it is granted
|
|
that the body and the soul are changed together by the natural
|
|
affections: I say 'natural', for though perhaps by learning music a
|
|
man has made some change in his soul, this is not one of those
|
|
affections which are natural to us; rather I refer to passions and
|
|
desires when I speak of natural emotions. If then this were granted
|
|
and also that for each change there is a corresponding sign, and we
|
|
could state the affection and sign proper to each kind of animal, we
|
|
shall be able to infer character from features. For if there is an
|
|
affection which belongs properly to an individual kind, e.g. courage
|
|
to lions, it is necessary that there should be a sign of it: for ex
|
|
hypothesi body and soul are affected together. Suppose this sign is
|
|
the possession of large extremities: this may belong to other kinds
|
|
also though not universally. For the sign is proper in the sense
|
|
stated, because the affection is proper to the whole kind, though
|
|
not proper to it alone, according to our usual manner of speaking. The
|
|
same thing then will be found in another kind, and man may be brave,
|
|
and some other kinds of animal as well. They will then have the
|
|
sign: for ex hypothesi there is one sign corresponding to each
|
|
affection. If then this is so, and we can collect signs of this sort
|
|
in these animals which have only one affection proper to them-but each
|
|
affection has its sign, since it is necessary that it should have a
|
|
single sign-we shall then be able to infer character from features.
|
|
But if the kind as a whole has two properties, e.g. if the lion is
|
|
both brave and generous, how shall we know which of the signs which
|
|
are its proper concomitants is the sign of a particular affection?
|
|
Perhaps if both belong to some other kind though not to the whole of
|
|
it, and if, in those kinds in which each is found though not in the
|
|
whole of their members, some members possess one of the affections and
|
|
not the other: e.g. if a man is brave but not generous, but possesses,
|
|
of the two signs, large extremities, it is clear that this is the sign
|
|
of courage in the lion also. To judge character from features, then,
|
|
is possible in the first figure if the middle term is convertible with
|
|
the first extreme, but is wider than the third term and not
|
|
convertible with it: e.g. let A stand for courage, B for large
|
|
extremities, and C for lion. B then belongs to everything to which C
|
|
belongs, but also to others. But A belongs to everything to which B
|
|
belongs, and to nothing besides, but is convertible with B: otherwise,
|
|
there would not be a single sign correlative with each affection.
|
|
|
|
-THE END-
|
|
.
|