Articles of intuitionistic logic

List of equivalents to Law of the Excluded Middle under intuitionistic logic

What are commonly-known equivalents of LEM among logicians, assuming intuitionistic axioms? I’ll list a few to begin with. Below, $\varphi$ and $\psi$ denote arbitrary sentences. (DNE) Double negation elimination: $\neg \neg \varphi \vdash \varphi$ (RAA) Reductio ad absurdum: $\neg \varphi \to \psi, \neg \varphi \to \neg \psi \vdash \varphi$ (P) Peirce’s Law: $\vdash ((\varphi \to […]

Intuitionistic proof of $\neg\neg(\neg\neg P \rightarrow P)$

How do you prove $\neg\neg(\neg\neg P \rightarrow P)$ in intuitionistic logic? I know this statement to be intuitionistically provable because of Glivenko’s theorem. However, I wish to prove it intuitionistically. The relevant axioms I happen to be using are: (1) $\neg P = P\rightarrow\bot$ and (2) $\bot \rightarrow P$.

Does the law of the excluded middle imply the existence of “intangibles”?

First off, I’m not sure if “intangible” is standard terminology, Wikipedia defines an intangible object to be: “objects that are proved to exist, but which cannot be explicitly constructed”. So if someone could point me towards better terminology, I’d appreciate it. The linked article from Wikipedia claims that the axiom of choice implies the existence […]

Prove that formula is not tautology.

Show Kripke’s model proving that the following formula isn’t tautology in intuitionistic logic. $ \neg ( p \wedge q ) \implies (\neg p \vee \neg q) $ Please help/hint me 😉

What's an example of a number that is neither rational nor irrational?

Of course in regular logic, the answer is there aren’t any. But in intuitionistic logic, there might be, as seen by this answer: https://math.stackexchange.com/a/1437130/49592. My question is, as per that answer, what is a specific number that is neither rational nor irrational (note that the link above uses a different definition of irrational than the […]

Difference between proof of negation and proof by contradiction

I stumbled across article titled “Proof of negation and proof by contradiction” in which the author differentiates proof by contradiction and proof by negation and denounces an abuse of language that is “bad for mental hygiene”. I get that it is probably a hyperbole but I am genuinely curious about what’s so horrible in using […]

Are proofs by contradiction really logical?

Let’s say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction. My question is this: How does one go from “so there’s a contradiction if we don’t have $A$” to concluding that “we have $A$”? That, to me, seems the exact opposite of logical. It sounds like we […]

All real functions are continuous

I’ve heard that within the field of intuitionistic mathematics, all real functions are continuous (i.e. there are no discontinuous functions). Is there a good book where I can find a proof of this theorem?