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]
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 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:
| Every B is C | No B is C |
| All A is B, | All A is B, |
| Some A is B, | Some A is B, |
| therefore | therefore |
| All A is C. | No A is C. |
| Some A is C. | Some A is not C. |
Or, if more significant symbols are preferred:
To prove an affirmative, the argument must admit of being stated in this form:
All animals are mortal;
All men/Some men/Socrates are animals;
therefore
All men/Some men/Socrates are mortal.
To prove a negative, the argument must be capable of being expressed in this form:
No one who is capable of self-control is necessarily vicious;
No one who is capable of self-control is necessarily vicious;
All negroes/Some negroes/Mr. A’s negro are capable of self-control;
therefore
No negroes are/Some negroes are not/Mr. A’s negro is not necessarily vicious.
Though all ratiocination admits of being thrown into one or the other of these forms, and sometimes gains considerably by the transformation, both in clearness and in the obviousness of its consequence; there are, no doubt, cases in which the argument falls more naturally into one of the other three figures, and in which its conclusiveness is more apparent at the first glance in those figures, than when reduced to the first. Thus, if the proposition were that pagans may be virtuous, and the evidence to prove it were the example of Aristides; a syllogism in the third figure,
50
Professor Bain denies the claim of Singular Propositions to be classed, for the purposes of ratiocination, with Universal; though they come within the designation which he himself proposes as an equivalent for Universal, that of Total. He would even, to use his own expression, banish them entirely from the syllogism. He takes as an example,
Socrates is wise,
Socrates is poor, therefore
Some poor men are wise,
or more properly (as he observes) “one poor man is wise.” “Now, if wise, poor, and a man, are attributes belonging to the meaning of the word Socrates, there is then no march of reasoning at all. We have given in Socrates,
“But the example in this form does not do justice to the syllogism of singulars. We must suppose both propositions to be real, the predicates being in no way involved in the subject. Thus
Socrates was the master of Plato,
Socrates fought at Delium,
The master of Plato fought at Delium.
“It may fairly be doubted whether the transitions, in this instance, are any thing more than equivalent forms. For the proposition ‘Socrates was the master of Plato and fought at Delium,’ compounded out of the two premises, is obviously nothing more than a grammatical abbreviation. No one can say that there is here any change of meaning, or any thing beyond a verbal modification of the original form. The next step is, ‘The master of Plato fought at Delium,’ which is the previous statement cut down by the omission of Socrates. It contents itself with reproducing a part of the meaning, or saying less than had been previously said. The full equivalent of the affirmation is, ‘The master of Plato fought at Delium, and the master of Plato was Socrates:’ the new form omits the last piece of information, and gives only the first. Now, we never consider that we have made a real inference, a step in advance, when we repeat
The first argument, as will have been seen, rests upon the supposition that the name Socrates has a meaning; that man, wise, and poor, are parts of this meaning; and that by predicating them of Socrates we convey no information; a view of the signification of names which, for reasons already given (Note to § 4 of the chapter on Definition,
The second part of Mr. Bain’s argument, in which he contends that even when the premises convey real information, the conclusion is merely the premises with a part left out, is applicable, if at all, as much to universal propositions as to singular. In every syllogism the conclusion contains less than is asserted in the two premises taken together. Suppose the syllogism to be
All bees are intelligent,
All bees are insects, therefore
Some insects are intelligent:
one might use the same liberty taken by Mr. Bain, of joining together the two premises as if they were one—“All bees are insects and intelligent”—and might say that in omitting the middle term