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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.

peter.petrov
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    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 06 '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 06 '17 at 18:04
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    I think this is an interesting question. Looking forward to the replies. –  Nov 06 '17 at 18:04
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    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 06 '17 at 18:06
  • interesting, ive not yet come across that line of logic yet though in all honesty it seems fairly equiviliant to a slight change of the above. statement. but this raises a fair rational for why we may find propositional and predicate logic being so prevelant.

    to a man with a hammer, everything looks like a nail.

    perhaps propositional and predicate logic is just people's first point of protocol and they just happen to work.

     – Vaas Nov 06 '17 at 18:10
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    See Intuitionistic Logic and Constructive Mathematics. – Mauro ALLEGRANZA Nov 06 '17 at 18:13
  • i'm unconvinced from what i've read so far. it seems to me (and by all means please correct me) that trying to verify mathematical uncertainities on computational output is (although very scientific) not very mathematical and adds a layer of uncertainity. Using the above arguement i dont see how

    $\exists c$ for which we can disprove $P(c)$ as the negation of $\forall x \Rightarrow P(X)$

     – Vaas Nov 06 '17 at 18:35
  • Does this question basically amount to "How do we justify the laws of logic (in particular, the law of the excluded middle)?" If not, how does it differ? (I'm not criticizing, I just want to clarify what's being asked here.) – Noah Schweber Nov 06 '17 at 22:08
  • @HagenvonEitzen I did not say it's wrong. I just wanted to understand why the proofs are based on these 2 particular logics. – peter.petrov Nov 07 '17 at 13:32
  • @NoahSchweber I didn't mean exactly how we justify it. In fact it cannot be justified, I think, if we happen to be in some other logical system e.g. a 3-valued logic. – peter.petrov Nov 07 '17 at 13:33
  • i aplogise for my ignorance. can you give an example of 3-valued logic? – Vaas Nov 07 '17 at 15:34
  • @Vaas Sure. https://en.wikipedia.org/wiki/Three-valued_logic – peter.petrov Nov 08 '17 at 16:02

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