night, and which we continue to do even in our dreams.'MACAULAY, Essay on Bacon.

[E.]

123. The Claimant has undoubtedly many peculiarities of gait and manner which were characteristic of the missing baronet. Are not these therefore proofs of identity equivalent to the evidence of imposture afforded by the absence of tattoo-marks which the genuine man is proved to have possessed?

124. Even if it could be shown that animals perform certain actions which men could only perform by the aid of reason, it would by no means necessarily follow that animals perform them by its aid.

125. If there's neither Mind nor Matter,
Mill's existence, too, we shatter :

If you still believe in Mill,

Believe as well in Mind and Matter.

[c.]

[E.]

126. If we accept Aristotle's testimony, we may infer that Anaximander was not one of the Ionian philosophers that accepted as the One material principle a mean term between Water and Air; for, in the Physics, we read that he held the substances in nature to have been produced from the primordial principle by a process of secretion and not by a process of condensation and rarefaction; while in the De Coelo it is stated that other mode of production than the last-named was not put forward by any who adopted such a mean term for their principle. What syllogistic form (figure and mood) does this inference most naturally assume? [R.]

CHAPTER XXI

ELEMENTS OF EQUATIONAL LOGIC

I. THE Symbols employed in this system are the following:

A, B, C, or other capital letters, signify qualities, or groups of qualities, forming the. common part, or intensive meaning, of terms, or names of objects and classes of objects.

=

a, b, c, or other small italic letters, are the corresponding negative terms; thus a signifies the absence of one or more of the qualities signified by A. This notation for negatives was proposed by De Morgan (Formal Logic, p. 38). The mark is the sign of Identity of Meaning of the terms between which it stands; thus A B indicates that the qualities signified by A are identical with the qualities signified by B. The sign

signifies unexclusive alternation, including the ordinary meanings of both the conjunctions or and and. Thus A B means the qualities of A or those of B, or those of both A and B, if they happen to coincide.

Juxtaposition of two letters forms a term whose meaning is the sum of the qualities signified by the two letters: thus AB means a union of the qualities of A and B.

2. The Laws of Combination of these symbols are as

follow :

The Law of Commutation.

ABBA: that is to say, the sum of qualities of A and B is evidently the same as the sum of qualities of B and A. The way of arriving at the sum may be different, but the result is identical.

The Law of Simplicity. AA = A: if we have the same qualities twice over we get the same as if we named them

once.

and

The Law of Unity.

AAA the qualities of A

the qualities of A are simply the qualities of A.

or

The Law of Distribution.

A(B. C)
and

The qualities of A with those of B those of C are the

or

3. The Laws of Thought are the foundation of all reasoning, and may thus be symbolically stated :

of }

AAB Ab.

Excluded Middle .

The Law of Contradiction. Aa = o.

The successive application of the Law of Duality to two, three, four, five or more terms, gives rise to the development of all possible logical combinations, called the Logical Alphabet, the first few columns of which are given below. The combinations for six terms are given in the Principles of Science, p. 94 (first ed. vol. i. p. 109).

4. The one sole and all sufficient rule of inference is the

following RULE OF SUBSTITUTION.

FOR ANY TERM SUBSTITUTE WHAT IS STATED IN ANY PREMISE TO BE IDENTICAL IN MEANING WITH THAT TERM.

The term may consist of any single letter, any juxtaposed letters, or any group of alternatives connected by the sign, the sign of unexclusive alternation.

5. It is assumed as a necessary law that every term must have its negative. This was called the Law of Infinity in my first logical essay (Pure Logic, p. 65 ; see also p. 45); but as pointed out by Mr. A. J. Ellis, it is assumed by De Morgan, in his Syllabus, article 16. Thence arises what I propose to call the CRITERIOn of Consistency, stated as follows:

Any two or more propositions are contradictory when, and only when, after all possible substitutions are made, they occasion the total disappearance of any term, positive or negative, from the Logical Alphabet.

The principle of this criterion was explained in p. 65 of the Essay on Pure Logic referred to, but subsequent inquiry, and the writings of Mr. A. J. Ellis, have tended to show the supreme importance of the criterion.

The processes of this equational system of Logic are fully treated in the first seven chapters of the Principles of Science, and they are now amply illustrated by the problems which follow.

6. How do you express in the new logic the four Aristotelian forms of proposition indicated by the vowels A, E, I, and O?

part of the class B, namely A B, which is the equational mode of asserting that the A's form part of the B's. The second expresses similarly that the A's are found among the not-B's. In the third form some is expressed by the symbol C, and the proposition asserts that some A's (CA) are identical with a part of the class B. Some difficulties may arise about this form, owing to the ambiguity of the Aristotelian some, as elsewhere discussed (pp. 151--158). The fourth proposition is evidently the negative form of the third.

7. How shall we express equationally the assertion of Hobbes (De Corpore Politico, I. i. 13),

that Irresistible might in the state of nature is right'?

Might' is the principal part of the subject, but it is qualified or restricted in this proposition by the adjective