Now, in all judgment we think “is,” but in few judgments predicate
“equal to.” In quantitative judgments we may think x = y, or,
as Boole proposes, x = vy = 00y or, as Jevons proposes, x = xy, or, as
Venn proposes, x which is not y = 0; and equational symbolic logic
is useful whenever we think in this quantitative way. But it is a
byway of thought. In most judgments all we believe is that x is
(or is not) y, that a thing is (or is not) determined, and that the
thing signified by the subject is a thing signified by the predicate,
but not that it is the only thing, or equal to everything signified
by the predicate. The symbolic logic, which confuses “is” with
“is equal to,” having introduced a particular kind of predicate
into the copula, falls into the mistake of reducing all predication
to the one category of the quantitative; whereas it is more often
in the substantial, e.g. “I am a man,” not “I am equal to a man,”
or in the qualitative, e.g. “I am white,” not “I am equal to white,”
or in the relative, e.g. “I am born in sin,” not “I am equal to born
in sin.” Predication, as Aristotle saw, is as various as the categories
of being. Finally, the great difficulty of the logic of judgment is
to find the mental act behind the linguistic expression, to ascribe
to it exactly what is thought, neither more nor less, and to apply
the judgment thought to the logical proposition, without expecting
to find it in ordinary propositions. Beneath Hamilton’s postulate
there is a deeper principle of logic—A rational being thinks only to
the point, and speaks only to understand and be understood.
Inference
The nature and analysis of inference have been so fully treated in the Introduction that here we may content ourselves with some points of detail.
1. False Views of Syllogism arising from False Views of Judgment.—The false views of judgment, which we have been examining, have led to false views of inference. On the one hand, having reduced categorical judgments to an existential form, Brentano proposes to reform the syllogism, with the results that it must contain four terms, of which two are opposed and two appear twice; that, when it is negative, both premises are negative; and that, when it is affirmative, one premise, at least, is negative. In order to infer the universal affirmative that every professor is mortal because he is a man, Brentano’s existential syllogism would run as follows:—
| There is not a not-mortal man. |
On the other hand, if on the plan of Sigwart categorical universals were reducible to hypotheticals, the same inference would be a pure hypothetical syllogism, thus:—
| If anything is a man it is mortal. |
But both these unnatural forms, which are certainly not analyses of any conscious process of categorical reasoning, break down at once, because they cannot explain those moods in the third figure, e.g. Darapti, which reason from universal premises to a particular conclusion. Thus, in order to infer that some wise men are good from the example of professors, Brentano’s syllogism would be the following non-sequitur:—
| There is not a not-good professor. |
So Sigwart’s syllogism would be the following non-sequitur:—
| If anything is a professor, it is good. |
But as by the admission of both logicians these reconstructions of Darapti are illogical, it follows that their respective reductions of categorical universals to existentials and hypotheticals are false, because they do not explain an actual inference. Sigwart does not indeed shrink from this and greater absurdities; he reduces the first figure to the modus ponens and the second to the modus tollens of the hypothetical syllogism, and then, finding no place for the third figure, denies that it can infer necessity; whereas it really infers the necessary consequence of particular conclusions. But the crowning absurdity is that, if all universals were hypothetical, Barbara in the first figure would become a purely hypothetical syllogism—a consequence which seems innocent enough until we remember that all universal affirmative conclusions in all sciences would with their premises dissolve into mere hypothesis. No logic can be sound which leads to the following analysis:—
| If anything is a body it is extended. |
Sigwart, indeed, has missed the essential difference between the categorical and the hypothetical construction of syllogisms. In a categorical syllogism of the first figure, the major premise, “Every M whatever is P,” is a universal, which we believe on account of previous evidence without any condition about the thing signified by the subject M, which we simply believe sometimes to be existent (e.g. “Every man existent”), and sometimes not (e.g., “Every centaur conceivable”); and the minor premise, “S is M,” establishes no part of the major, but adds the evidence of a particular not thought of in the major at all. But in a hypothetical syllogism of the ordinary mixed type, the first or hypothetical premise is a conditional belief, e.g. “If anything is M it is P,” containing a hypothetical antecedent, “If anything is M,” which is sometimes a hypothesis of existence (e.g. “If anything is an angel”), and sometimes a hypothesis of fact (e.g. “If an existing man is wise”); and the second premise or assumption, “Something is M,” establishes part of the first, namely, the hypothetical antecedent, whether as regards existence (e.g. “Something is an angel”), or as regards fact (e.g. “This existing man is wise”). These very different relations of premises are obliterated by Sigwart’s false reduction of categorical universals to hypotheticals. But even Sigwart’s errors are outdone by Lotze, who not only reduces “Every M is P” so “If S is M, S is P,” but proceeds to reduce this hypothetical to the disjunctive, “If S is M, S is P1 or P2 or P3,” and finds fault with the Aristotelian syllogism because it contents itself with inferring “S is P” without showing what P. Now there are occasions when we want to reason in this disjunctive manner, to consider whether S is P1 or P2 or P3, and to conclude that “S is a particular P”; but ordinarily all we want to know is that “S is P”; e.g. in arithmetic, that 2 + 2 are 4, not any particular 4, and in life that all our contemporaries must die, without enumerating all their particular sorts of deaths. Lotze’s mistake is the same as that of Hamilton about the quantification of the predicate, and that of those symbolists who held that reasoning ought always to exhaust all alternatives by equations. It is the mistake of exaggerating exceptional into normal forms of thought, and ignoring the principle that a rational being thinks only to the point.
2. Quasi-syllogisms.—Besides reconstructions of the syllogistic fabric, we find in recent logic attempts to extend the figures of the syllogism beyond the syllogistic rules. An old error that we may have a valid syllogism from merely negative premises (ex omnibus negativis), long ago answered by Alexander and Boethius, is now revived by Lotze, Jevons and Bradley, who do not perceive that the supposed second negative is really an affirmative containing a “not” which can only be carried through the syllogism by separating it from the copula and attaching it to one of the extremes, thus:—
| The just are not unhappy (negative). |
Here the minor being the infinite term “not-recognized” in the conclusion, must be the same term also in the minor premise. Schuppe, however, who is a fertile creator of quasi-syllogisms, has managed to invent some examples from two negative premises of a different kind:—
| (1) | (2) | (3) |
| No M is P. | No M is P. | No P is M. |
| S is not P. | S is not M. | S is not M. |
| ∴Neither S nor M is P. | ∴S may be P. | ∴S may be P. |
But (1) concludes with a mere repetition, (2) and (3) with a contingent “may be,” which, as Aristotle says, also “may not be,” and therefore nihil certo colligitur. The same answer
