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.
Questions tagged [intuitionistic-logic]
415 questions
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Is Smetanich's logic the second from the top in the lattice of intermediate logics?
Consider the lattice of consistent superintuitionistic logics, also known as intermediate logics. Smetanich's logic is the logic obtained from intuitionistic logic by adding the axiom $((\neg q \rightarrow p) \rightarrow (((p \rightarrow q)…
user107952
- 20,508
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Logic of the even weaker excluded middle
There exists a superintuitionistic propositional logic where $\neg p \vee \neg \neg p$ is a theorem, but $p \vee \neg p$ is not a theorem. It is called the logic of the weak excluded middle. That raises the question, is there a superintuitionistic…
user107952
- 20,508
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Is there an intuitionistic proof of "it is impossible to simulate a die with a guaranteed to terminate process involving coin flips"?
Recently, I happened to be thinking again about the question: "Is it possible to simulate a fair six-sided die using only fair coin flips?" One type of answer which tends to be given is along the lines of: flip the coin three times. On HHH, answer…
Daniel Schepler
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What are the examples for intuitionistic logic?
I have been curious about intuitionistic logic for some time and I want to know about it and I have a question, the law of the excluded middle and double negation elimination seem completely logical to me and I think they are always correct. But…
user1010242
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Is this formula provable in intuitionistic first-order logic?
I know that in classical first-order logic, the formula $(\exists x)( \exists y) x \neq y \rightarrow (\forall x)(\exists y) x \neq y$ is a theorem. Is it also a theorem of intuitionistic first-order logic? If so, what is the proof? I am asking…
user107952
- 20,508
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1 answer
Triple negation implies double negation elimination?
I have a question about intuitionistic logic regarding the relationship between the triple negation elimination rule, i.e. $\neg\neg\neg A\leftrightarrow \neg A$, and the double negation elimination. We know from Brouwer (1925) that $\neg\neg\neg…
acevik
- 98
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First-order intuitionistic logic question
I know that if $A$ is a theorem of classical propositional logic, then $\neg \neg A$ is a theorem of intuitionistic propositional logic. I read in a previous question I asked long ago, that the corresponding statement for first order logic is false.…
user107952
- 20,508
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Can intuitionistic propositional logic prove that if the negations of two statements are equivalent, so are the original statements?
I know that classical propositional logic can prove the formula $(\neg P \leftrightarrow \neg Q) \rightarrow (P \leftrightarrow Q)$. But, can intuitionistic propositional logic prove it? If not, can someone give a countermodel?
user107952
- 20,508
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Four propositions with certain properties
I am following a course in intuitionistic mathematics and I have been given an exercise about intuitionistic propositional logic.
The problem
Find four propositions $X_0, X_1, X_2, X_3$, such that the following hold:
(i) For each $i < 4$, the basic…
Tungsten
- 130
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Normal forms of proofs in intuitionistic logic?
In this post, Andrej Bauer says:
There is a theorem about normal forms of proofs in intuitionistic logic which tells us that every proof of a negation can be rearranged so that it ends with the inference rule cited above
Can someone point me to a…
user56834
- 12,925
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Examples for Kripke semantics for intuitionistic logic
In my previous question, Mees de Vries talked about Kripke semantics and
reading his links, I tried to show that $(X\wedge \neg Y)\rightarrow Z \vdash X\rightarrow (Y\vee Z)$ is valid using Kripke semantics.
Here is my answer, but I'm not sure…
Darae-Uri
- 885
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How can I show that certain inference is not intuitionistically valid?
Consider $\neg(X\wedge Y)\vdash\neg X \vee \neg Y$. The converse is valid since I can show that that inference is valid by using rules permitted. The above inference is valid classically but not intuitionistically.
But, then, How can I show that…
Darae-Uri
- 885
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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 ;)
user376326
1
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1 answer
Intuitionistic logic- proof.
$$ A \implies \neg \neg A$$
A proof of $\neg \neg A $ is a proof of $\neg A \implies \bot$. Assume $p : A $ and $q : \neg A$. Then $q(p): \bot$.
What is $q(p)$. How to understand it?
user376326