Comment by seanhunter

I thought of a concrete example of why you might use intuitionist logic. Take for example the “Liar’s paradox”, which centres around a proposition such as

A: this statement (A) is false

In classical logic, statements are either true or false. So suppose A is true. If A is true, then it therefore must be false. But suppose A is false. Well if it is false then when it says it is false it is correct and therefore must be true.

Now there are various ways in classical logic [1] to resolve this paradox but in general there is a category of things for which the law of the excluded middle seems unsatisfactory. So intuitionist logic would allow you to say that A is neither true nor false, and working in that framework would allow you to derive different results from what you would get in classical logic.

It’s important to realise that when you use a different axiomatic framework the results you derive may only be valid in the alternative axiomatic system though, and not in general. Lean (to bring this back to the topic of TFA) allows you to check what axioms you are using for a given proof by doing `#print axioms`. https://lean-lang.org/doc/reference/latest/ValidatingProofs/...

[1] eg you can say that all statements include an implicit assertion of their own truth. So if I say “2 + 2 = 4” I am really saying “it is true that 2+2=4”. So the statement A resolves to “This statement is true and this statement is false”, which is therefore just false in classical logic and not any kind of paradox.