A System of Logic, Ratiocinative and Inductive - John Stuart Mill

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therefore therefore All A is C No A is C Some A is C Some A is not C

      Second Figure.

No C is B All C is B No C is B All C is B
All A is B No A is B Some A is B Some A is not B
therefore therefore therefore therefore
No A is C No A is C Some A is not C Some A is not C

      Third Figure.

All B is C No B is C Some B is C All B is C Some B is not C No B is C
All B is A All B is A All B is A Some B is A All B is A Some B is A
therefore therefore therefore therefore therefore therefore
Some A is C Some A is not C Some A is C Some A is C Some A is not C Some A is not C

      Fourth Figure.

All C is B All C is B Some C is B No C is B No C is B
All B is A No B is A All B is A All B is A Some B is A
therefore therefore therefore therefore therefore
Some A is C Some A is not C Some A is C Some A is not C Some A is not C

      In these exemplars, or blank forms for making syllogisms, no place is assigned to singular propositions; not, of course, because such propositions are not used in ratiocination, but because, their predicate being affirmed or denied of the whole of the subject, they are ranked, for the purposes of the syllogism, with universal propositions. Thus, these two syllogisms—

All men are mortal, All men are mortal,
All kings are men, Socrates is a man,
therefore therefore
All kings are mortal, Socrates is mortal,

      are arguments precisely similar, and are both ranked in the first mood of the first figure.50

      [pg 128]

      The reasons why syllogisms in any of the above forms are legitimate, that is, why, if the premises are true, the conclusion must inevitably be so, and why this is not the case in any other possible mood (that is, in any other combination of universal and particular, affirmative and negative propositions), any person taking interest in these inquiries may be presumed to have either learned from the common-school books of the syllogistic logic, or to be capable of discovering for himself. The reader may, however, be referred, for every needful explanation, to Archbishop Whately's Elements of Logic, where he will find stated with philosophical precision, and explained with remarkable perspicuity, the whole of the common doctrine of the syllogism.

      All valid ratiocination; all reasoning by which, from general propositions [pg 129] previously admitted, other propositions equally or less general are inferred; may be exhibited in some of the above forms. The whole of Euclid, for example, might be thrown without difficulty into a series of syllogisms, regular in mood and figure.

      Though a syllogism framed according to any of these formulæ is a valid argument, all correct ratiocination admits of being stated in syllogisms of the first figure alone. The rules for throwing an argument in any of the other figures into the first figure, are called rules for the reduction of syllogisms. It is done by the conversion of one or other, or both, of the premises. Thus an argument in the first mood of the second figure, as—

      No C is B

       All A is B

       therefore

       No A is C,

      may be reduced as follows. The proposition, No C is B, being a universal negative, admits of simple conversion, and may be changed into No B is C, which, as we showed, is the very same assertion in other words—the same fact differently expressed. This transformation having been effected, the argument assumes the following form:

      No B is C

       All A is B

       therefore

       No A is C,

      which is a good syllogism in the second mood of the first figure. Again, an argument in the first mood of the third figure must resemble the following:

      All B is C

       All B is A

       therefore

       Some A is C,

      where the minor premise, All B is A, conformably to what was laid down in the last chapter respecting universal affirmatives, does not admit of simple conversion, but may be converted per accidens, thus, Some A is B; which, though it does not express the whole of what is asserted in the proposition All B is A, expresses, as was formerly shown, part of it, and must therefore be true if the whole is true. We have, then, as the result of the reduction, the following syllogism in the third mood of the first figure:

      All B is C

       Some A is B,

      from which it obviously follows, that

      Some A is C.

      In the same manner, or in a manner on which after these examples it is not necessary to enlarge, every mood of the second, third, and fourth figures may be reduced to some one of the four moods of the first. In other words, every conclusion which can be proved in any of the last three figures, may be proved in the first figure from the same premises, with a slight alteration in the mere manner of expressing them. Every valid ratiocination, therefore, may be stated in the first figure, that is, in one of the following forms:

      [pg 130]

Every B is C No B is C

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