“Any X is not Y,” becomes, by conversion, “Any Y is not X.” The term “not-X” is then introduced, being defined as that which Y is when it is not X. Then “Z is X” becomes “Any not-X is not Z”; and, the premises being transposed, the reduction is effected.
Dabitis and its long reduction are as follows:—
“Some Y is Z” becomes, by conversion, “Some Z is Y.” Then the term “some-Z” is introduced, being defined as that Z which is Y if “some Z is Y.” Then “Any Z is X” becomes “Some X is some-Z,” and, the premises being transposed, the reduction is effected.
Frisesomorum is,
Let some-Y be that Y which is Z when some Y is Z; and then we have,
Then let not-X be that which any Y is when some Y is not X, and we have,
which yields by conversion,
and we thus obtain the reduction,
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,
and by conversion,
Thus we obtain as the reduced form,
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:—
O combines the modifications of E and I, 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 E I O. Thus A is the first figure of proposition, E the second, I the third, O the second-third.7
§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,
where Σ’ 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 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,
where Π′ 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,
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,
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