A Theorem in Logic

There are three Propositions, A, B, and C.

It is given that
“If A is true, B is true. (i)
If C is true, then if A is true B is not true. (ii)

Number (ii) amounts to this:—

If C is true, then (i) is not true.

But, ex hypothesi, (i) is true.

∴ C cannot be true; for the assumption of C involves an absurdity.

This Theorem in Hypotheticals—that the Propositions, numbered (i) and (ii), together prove that C cannot be true—may be illustrated by the following algebraical example:—

Let $a x + (a - b) y + z = 5$ ; (1)
$b x + z = 6$ (2)

Equation (1) may be stated as a Hypothetical, thus:—

“If $a x$ , $(a - b) y$ , and z be added together, the number ‘5’ is obtained”.

Let ‘A’ mean “ $a x$ , $(a - b) y$ , and z are added together”;
‘B’ ” “the number ‘5’ is obtained”;
‘C’ ” “ $a = b$ ”.

Then we have
“If A is true B is true”.

Assume that C is true; i. e. that $a = b$ .

Then $(a x + (a - b) y + z)$ becomes $(b x + z)$ , which, by Equation (2), must always $= 6$ .

Hence
“If C is true, then if A is true B is not true”.

Therefore C cannot be true;
i. e. ‘a’ cannot = ‘b’.