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dental form of expression. It is well known to readers of the ordinary handbooks of logic, that hypothetical propositions can always be represented in the categorical form by altering the phraseology; and the fact that the alteration required is often of the slightest possible character seems to show that there is no essential difference. Thus the proposition,

If iron contain phosphorus, it is brittle,

is hypothetical, but exactly equivalent to the categorical proposition,

Iron, containing phosphorus, is brittle;

which is of the symbolic form,

AB ABC.

=

But propositions such as,

If the barometer falls, a storm is coming,

cannot be reduced but by some such mode of expression as the following:

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The circumstances of a falling barometer are the circumstances of a storm coming.

Nevertheless, sufficient freedom in the alteration of expression being granted, they readily come under our formulæ.

36. I have as yet introduced few examples of negative propositions, because, though they may be treated in their purely negative form, it is usually more convenient to convert them into affirmative propositions. This conversion is effected by the use of negative terms, a practice not unknown to the old logic, but not nearly so much employed as it should have been. Thus the negative proposition,

A is not B

or A & B,

is much more conveniently represented by the affirmative proposition or equation,

A = Ab,

in which we denote by b the quality or fact of differing from B. The term b is in fact the name of the whole class of things, or any of them, which differ from B, so that it is a matter of indifference whether we say that A differs from B and is excluded from the class B, or that it agrees with b and is included in the class b. There are advantages, however, in employing the affirmative form.1

1 It may seem to the reader contradictory to condemn the negative proposition as sterile and incapable of affording inferences, and shortly afterwards to convert it into an affirmative proposition of fertile or inferential power. But on trial it will be found that the

37. The syllogism Celarent is now very readily brought under our single mode of inference. Take the example

(1) All metals are elements,

(2) No element can be transmuted,

(3) No metal, therefore, can be transmuted.

To represent this symbolically, let

A = metal,

B = element,

C = transmutable,

c = untransmutable.

Then the premises are

(1) A = AB
(2) B=Bc.

propositions thus obtained yield no conclusions inconsistent with my theory. Thus the negative premises,

A is not B,
B is not C,

yield the affirmative propositions or equations,

A =
== A b

B Bc.

=

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And when these premises are tested, whether on the logical slate, abacus, or logical machine referred to in a later page, they are found to give no conclusion concerning the relation of A and C. The description of A is given in the equation,

A AbC Abc, from which it appears that A may indifferently occur with or without C.

Substituting in (1) by means of (2) we get
(3) A = ABC;

or,

metals = metals, elementary, untransmutable. 38. Before proceeding to other examples of the syllogism, it will be well to point out that every affirmative proposition or equation gives rise to a corresponding equation between the negatives of the terms of the original. The general proposition of the form

A = B,

treated by the fundamental principle of reasoning, informs us that in whatever relation anything stands to A, in the same relation it stands to B, and similarly vice versa. Hence, whatever differs from A differs from B, and whatever differs from B differs from A. Now the term b denotes what differs from B, and a denotes what differs from A; so that from the single original proposition we may draw the two propositions—

a = ab
b

=

ab.

But as these propositions have an identical second member, we can make a substitution, getting

a = b.

This form of inference, though little if at all

noticed in the traditional logic, is of frequent occurrence and of great importance. It may be illustrated by such examples as

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triangle = three-sided rectilinear figure,

hence what is not a triangle = what is not a threesided rectilinear figure.

The new proposition thus obtained may be called the contrapositive of the one from which it was derived, this being a name long applied to a similar inference from the old form of proposition.

39. Though the details of this new view of logic may not yet have been perfectly worked out, much evidence of the truth of the system is to be found in the simplicity, variety, and universality of the forms of reasoning which can be evolved out of a single law of thought,-the similar treatment of similars.

The old system of the syllogism, indeed, was nominally founded on a single, or rather double, axiom or law, the dicta of Aristotle, but the mode in which these dicta led to conclusions was so far

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