# Why are proofs based on the 2 classical logics?

Mathematics recognizes that there are logics different from the propositional logic and the predicate logic. I have studied a few of them myself.

Then... why are all proofs in math textbooks based on propositional and predicate logics only?

I mean, why do these proofs assume that it's either $A$ or not $A$? Or... why do they assume that the negation of "for all $X$, we have that $P(x)$ holds true" is "there exists an $X$ such that not $P(x)$"?

Isn't this a flaw in the credibility of math itself (or say of all these texts)?

I guess it's not but I am not sure why.

• If you think that it is wrong to use classical logics for all proofs, do you mean that there exist some proofs that better do not use classical logic? – Hagen von Eitzen Nov 6 '17 at 18:03
• i'm also curious about the alternative. ie what is the alternative to $¬(\forall x \Rightarrow p(x)) = \exists x \nRightarrow p(x)$ – Vaas Nov 6 '17 at 18:04
• I think this is an interesting question. Looking forward to the replies. – user370967 Nov 6 '17 at 18:04
• In constructive logic, the negation of “for all $x$, we have that $P(x)$ holds true” is not “there exists an $x$ such that not $P(x)$”. Instead, it is the stronger statement “Here is a particular $c$ for which we can disprove $P(c)$”. (This is of course sufficient in classical logic as well.) Constructive logic is widely used in some contexts. – MJD Nov 6 '17 at 18:06
• – Mauro ALLEGRANZA Nov 6 '17 at 18:13