Then let not-X be that which any Yis when some Y is not X, and

From the conclusion of this reduction, the conclusion of Frisesomorum is justified as follows:

Another mode of effecting the short reduction of Frisesomorum is this: Let not-Y be that which any X is when no X is Y, and we have

Let some-Z be that Z which is not not-Y when some Z is not-Y,

and we have,

Any some-Z is not not-Y,

and by conversion,

Any not-y is not some-Z.

From the conclusion of this reduction, we get that of Frisesomorum thus:

In either reduction of Celantes, if we neglect the substitution of terms for their definitions, the substitutions are all of the second syllogistic figure. This of itself shows that Celantes belongs to that figure, and this is confirmed by the fact that it concludes the denial of a Case. In the same way, the reductions of Dabitis involve only substitutions in the third figure, and it concludes the denial of a Rule. Frisesomorum concludes a proposition which is at once the denial of a rule and the denial of a case: its long reduction involves one conversion in the second figure and another in the third, and its short reductions involve conversions in Frisesomorum itself. It therefore belongs to a figure which unites the characters of the second and third, and which may be termed the second-third figure in Theophrastean syllogism.

There are, then, two kinds of syllogism, the Aristotelian and Theophrastean. In the Aristotelian occur the 1st, 2d, and 3d figures, with four moods of each. In the Theophrastean occur the 2d, 3d, and 2d-3d figures, with one mood of each. The first figure is the fundamental or typical one, and Barbara is the typical mood. There is a strong analogy between the figures of syllogism and the four forms of proposition. A is the fundamental form of proposition, just as the first figure is the fundamental form of syllogism. The second and third figures are derived from the first by the contraposition of propositions, and E and I are derived from A by the contraposition of terms; thus:

Any S is P.

Any not-P is not S.

Some P is some-S.

O combines the modifications of E and 1, just as the 2d-3d figure combines the 2d and 3d. In the second-third figure, only O can be concluded, in the third only I and O, in the second only E and O, in the first either A EI O. Thus A is the first figure of proposition, E the second, I the third, O the second-third.*

§ 7. Mathematical Syllogisms.

A kind of argument very common in mathematics may be exemplified as follows:

Every part is less than that of which it is a part,

Boston is a part of the Universe;

... Boston is less than the Universe.

This may be reduced to syllogistic form thus:

Any relation of part to whole is a relation of less to greater,

The relation of Boston to the Universe is a relation of part to whole; ... The relation of Boston to the Universe is a relation of less to greater.

If logic is to take account of the peculiarities of such syllogisms, it would be necessary to consider some propositions as having three terms, subject, predicate, and object; and such propositions would be divided into active and passive. The varieties in them would be endless.

PART III. § 1. Induction and Hypothesis.

In the syllogism,

Any M is P,

'S is M;

.. ES is P;

where 'S' denotes the sum of all the classes which come under M. if the second premise and conclusion are known to be true, the first

* Hypotheticals have not been considered above, the well-known opinion having been adopted that, "If A, then B," means the same as "Every state of things in which A is true is a state of things in which B is (or will be) true."

premise is, by enumeration, true. Whence we have, as a valid demonstrative form of inference,

This is called perfect induction. It would be better to call it formal induction.

In a similar way, from the syllogism,

Any M is II' P',

where II' P' denotes the conjunction of all the characters of M, if the conclusion and first premise are true, the second premise is true by definition; so that we have the demonstrative form of argument,

Any M is II' P',

Any S is II' P';

.. Any S is M.

This is reasoning from definition, or, as it may be termed, formal hypothesis.

One half of all possible propositions are true, because every proposition has its contradictory. Moreover, for every true particular proposition there is a true universal proposition, and for every true negative proposition there is a true affirmative proposition. This follows from the fact that the universal affirmative is the type of all propositions. Hence of all possible propositions in either of the forms,

'S is M, and M is II' P',

one half are true. In an untrue proposition of either of these forms, some finite ratio of the S's or P's are not true subjects or predicates. Hence, of all propositions of either of these forms which are partly true, some finite ratio more than one half are wholly true. Hence, if in the above formulæ for formal induction or hypothesis, we substitute

S' for 'S and P' for II' P' we obtain formulæ of probable inference. This reasoning gives no determinate probability to these modes of inference, but it is necessary to consider that, however weak synthetic inference might have been at first, yet if it had the least positive tendency to produce truth, it would continually become stronger, owing to the establishment of more and more secure premises.

The rules for valid induction and hypothesis deducible from this theory are as follows:

1. The explaining syllogism, that is to say, the deductive syllogism one of whose premises is inductively or hypothetically inferred from the other and from its conclusion, must be valid.

2. The conclusion is not to be held as absolutely true, but only until it can be shown that, in the case of induction, S' was taken from some narrower class than M, or, in the case of hypothesis, that P' was taken from some higher class than M.

3. From the last rule it follows as a corollary that in the case of induction the subject of the premises must be a sum of subjects, and that in the case of hypothesis the predicate of the premises must be a conjunction of predicates.

4. Also, that this aggregate must be of different objects or qualities and not of mere names.

5. Also, that the only principle upon which the instanced subjects or predicates can be selected is that of belonging to M.*

*Positivism, apart from its theory of history and of the relations between the sciences, is distinguished from other doctrines by the manner in which it regards hypotheses. Almost all men think that metaphysical theories are valueless, because metaphysicians differ so much among themselves; but the positivists give another reason, namely, that these theories violate the sole condition of all legitimate hypothesis. This condition is that every good hypothesis must be such as is certainly capable of subsequent verification with the degree of certainty proper to the conclusions of the branch of science to which it belongs. There is, it seems to me, a confusion here between the probability of a hypothesis in itself, and its admissibility into any one of those bodies of doctrine which have received distinct names, or have been admitted into a scheme of the sciences, and which admit only conclusions which have a very high probability indeed. I have here to deal with the rule only so far as it is a general canon of the legitimacy of hypotheses, and not so far as it determines their relevancy to a particular science; and I shall, therefore, consider only another common statement of it; namely, "that no hypothesis is admissible which is not capable of verification by direct observation." The positivist regards an hypothesis, not as an inference, but as a device for stimulating and