Theorems of Development.

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Theorems of Development.

(1) If x be any elective symbol, then

(2) For a function of two elective symbols we have

(3) Functions of three symbols,

From these examples the general law of development is obvious. And I desire it to be noted that this law is a mere consequence of the primary laws which have been expressed in (1), (2), (3).

This enables us to interpret any equation by a general rule.

RULE. Bring all the terms to the first side, expand this in terms of all the elective symbols involved in it, and equate to 0 every constituent whose modulus does not vanish.

For the demonstration of these and many other results, I must refer to the original work. It must be noted that on p. 66[4],z has been, through mistake, substituted for w, and that the reference on p. 80[5] should be to Prop. 2.

As an example, let us take the equation

the former implies that there are no Xs which are not-Ys, the latter that there are no Ys which are not-Xs, these together expressing the full significance of the original equation.

We can, however, often recombine the constituents with a gain of simplicity. In the present instance, subtracting (12) from (11b), we have

or

All equations are thus of equal significance which give, on expansion, the same series of constituent equations, and all are interpretable.

[4] The Mathematical Analysis of Logic

[5] Ibid.

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