Namelesss wrote: ↑April 24th, 2018, 10:14 pm

Ultimately, there is no difference between the 'consistent' and the 'inconsistent'!

That's a good way of describing the relationship between the two.

Mosesquine wrote: ↑April 25th, 2018, 8:11 am

The sentence "some contradiction exists in a theory" means the same as the sentence "the theory is wrong".

The proposition "some contradiction exists in a theory" is logically equivalent to the proposition "the theory is right and wrong."

*Proof*. Assume some contradiction exists in a theory. Name the theory

*T*. Then, just as in my original post, the following two propositions are true in

*T*, by

*ex contradictione quodlibet*.

9.

*p*
10. It is not true that

*p*.

So, in a metatheory of

*T*, the following two propositions are true.

11.

*p* is true in

*T*.

12. It is not true that "

*p* is true in

*T*."

Since (12) is the negation of (11), some contradiction exists. Therefore, by

*ex contradictione quodlibet*,

*T* is right and wrong. Discharge the assumption.

To prove the converse, assume a theory is right and wrong. Name the theory

*T*. Since

*T* is right and wrong, by definition of wrong,

*T* is right and not right. So, some contradiction exists. By

*ex contradictione quodlibet*, some contradiction exists in

*T*. Discharge the assumption. This concludes the proof.

Mosesquine wrote: ↑April 25th, 2018, 8:11 am

We shift from the inconsistent theory P to the consistent theory P-2.

I'm not quite sure what the purpose of making theories P and P-2 was.