Define “ is identical with ” as meaning “whatever is true of is true of ,”* i.e.*

“ always implies .” (1)

Dem.

If is identical with and “ is identical with , then is identical with a.

Although the conclusion is sound, the above reasoning embodies a vicous-circle fallacy. The following is the above demonstration written out in full:

“ is identical with ” is, by definition, “.” (2)

. [(1). ] (3)

.implies. . (4)

is identical to

(1) is a function in the form of

(2) is a function of and can be written as in which is an apparent variable instead of a real variable.

In step (3), (2) is taken as a possible value of , but (2) involves the totality (all possible values) of , thus a value of is determined by its totality and violates vicious-circle principle.

Note: *A value of is obtained by assigning a , namely , etc. A value of is obtained by assigning an , namely , etc. The latter determination is NOT used in this post.*