Originally Posted by

**nocturne**
I need to clarify. I am not trying to suggest the following:

If C(A) = C(B) then A = B

That is not any part of my argument. This is:

If C(A) = C(B) then A and B express the same hypothesis

The sets A and B may differ in some way. For example, if A = {P} and B = {~~P} then A = B is false. However, they express the same proposition and have the same logical content.

If I said 'A is equal to B' I meant only with respect to the expressing the same proposition or hypothesis, and not that both sets of statements were identical.

I thought we understood each other on the original point, but it seems like there is still some confusion left.

What does it mean to "express the same hypothesis?"

If we are *defining* it to mean that:

"C(A) = C(B) iff A and B express the same hypothesis,"

then the statement,

"if C(A) = C(B) then A and B express the same hypothesis"

simply follows from the definition, and is devoid of real content.