Questions tagged [intuitionistic-logic]

Intuitionistic logic refer constructive logic, a logical system avoiding deduction rules like *Reductio ad absurdum*.

Intuitionistic logic refers to constructive logic, a logical system avoiding deduction rules like Reductio ad absurdum.

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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…
Simp
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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?
gifty
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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…
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How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the entities in the definition of an elementary topos come…
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Semantics for minimal logic

Minimal logic is a fragment of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet). Essentially, from proof-theoretical viewpoint,…
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Excluded middle, double negation, contraposition and Peirce's law in minimal logic

Minimal logic does not assume any falsity $\bot$ or negation $\neg$, so the above mentioned laws can (apart from Peirce's) not be stated as usual. However, if we fix some propositional variable $F$, we can use it to define a kind of negation by…
Léreau
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Is there a decision procedure for intuitionistic propositional logic?

Is intutionistic propositional logic decidable? If so, what is a decision procedure for it, like tableaux for classical propositional logic? EDIT: In the first revision I mistook "predicate" for "propositional".
Pteromys
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Can we make intuitionistic logic "intuitive"?

This is something I am wondering about, because all the formulations I've seen of the logic seem fairly difficult to grasp, e.g. lists of abstract axioms that have a few missing versus classical logic. Yet, I've had the following idea floating…
The_Sympathizer
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Intuitionistic logic plus $A → B \lor C \vdash ( A → B ) \lor ( A → C )$

The following is a classically valid deduction for any propositions $A,B,C$. $\def\imp{\rightarrow}$ $A \imp B \lor C \vdash ( A \imp B ) \lor ( A \imp C )$. But I'm quite sure it isn't intuitionistically valid, although I don't know how to prove…
user21820
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Why is it called linear logic?

Why is it called "Linear" Logic? What's linear about it?
psquid
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Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not accept it in contrasts to the formalists. I'm curious…
Red
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About Gödel-Gentzen negative translation

I have a question about Gödel-Gentzen negative translation. According to the Wikipedia article for negative translations, "a sentence $\phi$ may not imply its negative translation $\phi^{\rm N}$". I am not sure if I understand correctly this…
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Does double negation distribute over disjunction intuitionistically?

Does the following equivalence $$\lnot \lnot (A \lor B) \leftrightarrow (\lnot \lnot A \lor \lnot \lnot B)$$ hold in propositional intuitionistic logic? And in propositional minimal logic? (In propositional classical logic this is obvious since $A…
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What is the difference between intuitionistic, classical, modal and linear logic?

I am currently going through Philip Wadler's "Proposition as Types" and a passage of the introduction has struck me: Propositions as Types is a notion with breadth. It applies to a range of logics including propositional, predicate,…
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Defined negation in intuitionistic linear logic

Is it possible to define a negation in intuitionistic linear logic, the way one does in intuitionistic logic, i.e. $A^{\bot} \equiv A \multimap \mathbf{0}$ (or, as it would be written in intuitionistic logic, $\neg A \equiv A \to \bot$)? While I can…
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