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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.

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    $\begingroup$ 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? $\endgroup$ – Hagen von Eitzen Nov 6 '17 at 18:03
  • $\begingroup$ i'm also curious about the alternative. ie what is the alternative to $¬(\forall x \Rightarrow p(x)) = \exists x \nRightarrow p(x)$ $\endgroup$ – Vaas Nov 6 '17 at 18:04
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    $\begingroup$ I think this is an interesting question. Looking forward to the replies. $\endgroup$ – user370967 Nov 6 '17 at 18:04
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    $\begingroup$ 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. $\endgroup$ – MJD Nov 6 '17 at 18:06
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    $\begingroup$ See Intuitionistic Logic and Constructive Mathematics. $\endgroup$ – Mauro ALLEGRANZA Nov 6 '17 at 18:13

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