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*.

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*.

346 questions

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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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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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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…

Erwan Aaron

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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…

Arrow

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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,…

Taroccoesbrocco

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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 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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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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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? What's linear about it?

psquid

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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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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…

ferdinand

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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…

Taroccoesbrocco

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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,…

Erwan Aaron

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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…

wen

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