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by Robert Recorde in his Whetstone of Wit, to avoid the tedious repetition of the words "is equal to;" and he chose a pair of parallel lines, because no two things can be more equal. The meaning of the sign has however been gradually extended beyond that of equality of quantities; mathematicians have themselves used it to indicate equivalence of operations. The force of analogy has been so great that writers in most other branches of science have employed the same sign. The philologist uses it to indicate the equivalence of meaning of words: chemists adopt it to signify identity in kind and equality in weight of the elements which form two different compounds. Not a few logicians, for instance Lambert, Drobitsch, George Bentham,2 Boole, have employed it as the copula of propositions. De Morgan declined to use it for this. purpose, but still further extended its meaning so as to include the equivalence of a proposition with the premises from which it can be inferred; and Herbert Spencer has applied it in a like manner.5

4

Many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience; but I hold that the common use of this sign = in so many different meanings is really founded upon a generalisation of the widest character and of the greatest importanceone indeed which it is a principal purpose of this work to explain. The employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exists, it is not only allowable, but highly desirable and even imperative, to use the symbol of equivalence with a generality of meaning corresponding to the generality of the principles involved. Accordingly De Morgan's refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations. I use the sign because I hold the contrary opinion.

1 Hallam's Literature of Europe, First Ed., vol. ii. p. 444.
2 Outline of a New System of Logic, London, 1827, pp. 133, &c.
3 An Investigation of the Laws of Thought, pp. 27, &c.

Formal Logic, pp. 82, 106. In his later work, The Syllabus of a

New System of Logic, he discontinued the use of the sign.

• Principles of Psychology, Second Ed., vol. ii. pp. 54, 55.

=

I conceive that the sign = as commonly employed, always denotes some form or degree of sameness, and the particular form is usually indicated by the nature of the terms joined by it. Thus "6,720 pounds 3 tons" is evidently an equation of quantities. The formula x = + expresses the equivalence of operations. "Exogens Dicotyledons" is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants.

=

We have great need in logic of a distinct sign for the copula, because the little verb is (or are), hitherto used both in logic and ordinary discourse, is thoroughly ambiguous. It sometimes denotes identity, as in "St. Paul's is the chef-d'oeuvre of Sir Christopher Wren;" but it more commonly indicates inclusion of class within class, or partial identity, as in "Bishops are members of the House of Lords." This latter relation involves identity, but requires careful discrimination from simple identity, as will be shown further on.

When with this sign of equality we join two nouns or logical terms, as in

=

Hydrogen The least dense clement, we signify that the object or group of objects denoted by one term is identical with that denoted by the other, in everything except the names. The general formula

A B

must be taken to mean that A and B are symbols for the same object or group of objects. This identity may sometimes arise from the mere imposition of names, but it may also arise from the deepest laws of the constitution of nature; as when we say

=

Gravitating matter Matter possessing inertia,
Exogenous plants Dicotyledonous plants,

Plagihedral quartz crystals Quartz crystals causing the plane of polarisation of light to rotate.

We shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature; but at present we are considering only the mode of expression which may be the same in either case.

Sometimes, but much less frequently, we require a symbol to indicate difference or the absence of complete

sameness. For this purpose we may generalise in like manner the symbol, which was introduced by Wallis to signify difference between quantities. The general formula

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denotes that B and C are the names of two objects or groups which are not identical with each other. Thus we may say

Acrogens Flowering plants.

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I shall also occasionally use the sign to signify in the most general manner the existence of any relation between the two terms connected by it. Thus might mean not only the relations of equality or inequality, sameness or difference, but any special relation of time, place, size, causation, &c. in which one thing may stand to another. By A B I mean, then, any two objects of though. || related to each other in any conceivable manner.

General Formula of Logical Inference.

The one supreme rule of inference consists, as I have said, in the direction to affirm of anything whatever is known of its like, equal or equivalent. The Substitution of Similars is a phrase which seems aptly to express the capacity of mutual replacement existing in any two objects which are like or equivalent to a sufficient degree. It is matter for further investigation to ascertain when and for what purposes a degree of similarity less than complete | identity is sufficient to warrant substitution. For the present we think only of the exact sameness expressed in the form

A = B.

Now if we take the letter C to denote any third conceivable object, and use the sign co in its stated meaning of indefinite relation, then the general formula of all inference may be thus exhibited :

From

we may infer

:

=

A B C
A cos C

or, in words-In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing. The identity between A

C

and B allows us indifferently to place A where B was, or B where A was; and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth. Thus if we first specify only the meaning of the sign co, we may say that if C is the weight of B, then C is also the weight of A. Similarly

If C is the father of B, C is the father of A ;
If C is a fragment of B, C is a fragment of A;
If C is a quality of B, C is a quality of A;
If C is a species of B, C is a species of A;
If C is the equal of B, C is the equal of A;

and so on ad infinitum.

We may also endow with special meanings the letterterms A, B, and C, and the process of inference will never be false. Thus let the sign mean "is height of," and let A = Snowdon,

B= Highest mountain in England or Wales,

C = 3,590 feet;

=

then it obviously follows since "3,590 feet is the height of Snowdon," and "Snowdon the highest mountain in England or Wales," that, " 3,590 feet is the height of the highest mountain in England or Wales."

One result of this general process of inference is that we may in any aggregate or complex whole replace any part by its equivalent without altering the whole. To alter is to make a difference; but if in replacing a part I make no difference, there is no alteration of the whole. Many inferences which have been very imperfectly included in logical formulas at once follow. I remember the late Prof. De Morgan remarking that all Aristotle's logic could not prove that "Because a horse is an animal, the head of a horse is the head of an animal." I conceive that this amounts merely to replacing in the complete notion head of a horse, the term "horse," by its equivalent some animal or an animal. Similarly, since

The Lord Chancellor The Speaker of the House of
Lords,

it follows that

The death of the Lord Chancellor = The death of the
Speaker of the House of Lords;

and any event, circumstance or thing, which stands in a

certain relation to the one will stand in like relation to the other. Milton reasons in this way when he says, in his Areopagitica," Who kills a man, kills a reasonable creature, God's image." If we may suppose him to mean

God's image = man some reasonable creature,

it follows that "The killer of a man is the killer of some reasonable creature," and also "The killer of God's image.'

This replacement of equivalents may be repeated over and over again to any extent. Thus if person is identical in meaning with individual, it follows that

Meeting of persons meeting of individuals;

=

=

and if assemblage meeting, we may make a new replacement and show that

=

Meeting of persons assemblage of individuals. We may in fact found upon this principle of substitution. a most general axiom in the following terms 1:—

Same parts samely related make same wholes.

If, for instance, exactly similar bricks and other materials be used to build two houses, and they be similarly placed in each house, the two houses must be similar. There are millions of cells in a human body, but if each cell of one person were represented by an exactly similar cell similarly placed in another body, the two persons would be undistinguishable, and would be only numerically different. It is upon this principle, as we shall see, that all accurate processes of measurement depend. If for a weight in a scale of a balance we substitute another weight, and the equilibrium remains entirely unchanged, then the weights must be exactly equal. The general test of equality is substitution. Objects are equally bright when on replacing one by the other the eye perceives no difference. Objects are equal in dimensions when tested by the same gauge they fit in the same manner. Generally speaking, two objects are alike so far as when substituted one for another no alteration is produced, and vice versa when alike no alteration is produced by the substitution.

1 Pure Logic, or the Logic of Quality, p. 14.

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