Questions tagged [intuitionistic-logic]

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

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Are geometric arguments using infinitesimals valid?

This question pertains to smooth infinitesimal analysis as presented in the book A Primer of Infinitesimal Analysis by John Bell. The book uses intuitionistic logic. Let $\Delta$ denote the set of ...
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Context in Type Theory

I am reading a book on type theory. On page 105, the author says that If one views valid contexts as theories (in the sense of ordinary logics) a consistent context corresponds to a consistent ...
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Given an inhabited type in simply-typed lambda calculus (w/no basics, just variables), is there a combinator of that type that is no longer than it?

Apologies if I've got some of the terminology here wrong, typed lambda calculus is a bit new to me. Let's say we've got a type in simply-typed lambda calculus with no basic types (functions and type ...
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Universal Quantifier in Intuitionistic Logic

I have a very basic question about $\forall$ in intuitionistic first-order logic (IQL). It is well-known that in intuitionistic propositional logic (IPL), (\ref{dnlem}) and (\ref{dndne}) are both ...
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Non-existence of a dischargeable hypothesis

I meet a question about the existence/non-existence of a dischargeable hypothesis. The question is as follows: in intuitionistic logic, if for every proposition $X$, it cannot be the case that $X$ ...
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proof intuitionist logic

So here is the proposition I'd like to prove: a ∨ ¬a ⊢((a→b)→a)→a I have tried many way to find a proof, and always end up in a mess... for example: a ∨ ¬a ⊢((a→b)→a)→a a ⊢((a→b)→a)→a (...
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Proof of Angles going beyond 90 deg in Trigonometry [duplicate]

Since we’re introduced to trigonometric ratios in terms of opposite, perpendicular and hypotenuse, all of which are a part of a right angled triangle. This defines the ratios for angles greater than 0 ...
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Questions about Intuitionistic Logic

Are the following theorems of Intuitionistic Logic? $\Vdash \varphi \implies \lnot \lnot \varphi$ $\Vdash \lnot \lnot\varphi \implies \varphi$ $\Vdash (\varphi \implies \lnot\psi) \implies (\...
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Proof of Angles going beyond 90 degrees in Trigonometry

Since we’re introduced to trigonometric ratios in terms of opposite, perpendicular and hypotenuse, all of which are a part of a right angled triangle. This defines the ratios for angles greater than 0 ...
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1answer
40 views

Semantic explanation for converting intuitionistic logic into classical logic by adding LEM as an axiom

I have a question about converting intuitionistic logic (IL) into classical logic (CL) by adding LEM as an axiom. IL is usually understood as a logic without LEM. $$\textrm{LEM}:=A\vee\neg A.$$ In ...
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Intermediate logics and strong algebraic completeness

As a setup, suppose that you have a usual propositional language $\mathcal L$ over a set of propositional variables $Var$ and with symbols $\land,\lor,\rightarrow,\bot$ in the usual way. Let $L$ be an ...
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Two non-isomorphic intuitionistic Kripke frames satisfying different formulas.

I am struggling with the following problem: Being given two non-isomorphic intuitionistic Kripke frames $F_1$ and $F_2$ there is a formula $\phi$ such as: $$ F_1 \models \phi\ and\ F_2 \not \models \...
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The legitimacy of topos theory and intuitionism.

This is an exercise in critical thinking. I am not looking, therefore, for opinions on the matter; rather: I would like to know the evidence (whatever that might mean). Background: I have a ...
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Accessibility relation in non-classical logics: hereditary or not?

Recently, I am reading course materials on intuitionistic and modal logics. I have two questions about the notion of accessibility relation in Kripke semantics for intuitionistic and modal logics. ...
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Definition of proposition in a constructive setting

In a typical discrete math course, we are taught that a proposition is something that is either true or false but not both, which seems to be based on a classical interpretation. How would one go ...
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Are constant functions continuous in constructive mathematics?

The standard proof that a constant function $c: X \to Y$, $x \mapsto y_0$ is continuous proceeds as follows: if $U \subseteq Y$ is open, then either $c^{-1}(U)=X$ if $y_0 \in U$, or $c^{-1}(U)=\...
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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 ...
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Possible worlds with different formal semantics

I want to ask a basic question regarding modal logic: is it formally possible that a statement such as $$\forall x(P(x)\vee\neg P(x))$$ is true at a world $w_1$, but not in another world $w_2$? For ...
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Watered down axiom of choice

I'm wondering what the following statement is equivalent to in an intuitionistic framework. For a union of sets $C = \bigcup\limits_{i \in I} X_i,$ if $S \in C,$ there exists $s \in I$ such that $...
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intuitionistic probability theory

I wonder whether the law of excluded middle not being available in intuitionistic logic might provide an obstacle when working with an axiomatization of probability theory using the Kolmogorov axioms. ...
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Provide a Kripke model to prove that $\neg(\neg P\lor \neg Q\lor R)\to (P\land Q \land R)$ is false in intuitionistic logic

Could anyone check my working please? Provide a Kripke model to prove that $\neg(\neg P\lor \neg Q\lor \neg R)\to (P\land Q \land R)$ is false in intuitionistic logic. Here $k_0$ forces $\neg(\neg ...
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Resource suggestion for practicing modal logic and intuitionistic logic

any suggestion ( Book , Pdf , online resource , ... ) for solving problems about modal logic and intuitionistic logic and it would be really great if there was a solution for it .
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Proving $\Gamma \vdash \phi$ Implies $\Gamma \vDash \phi$ (for Institutionistic propositional logic and Heying algebras)

I'm trying to prove that $\Gamma \vdash \phi$ implies $\Gamma \vDash \phi$ (for Institutionistic propositional logic and Heying algebras), by induction with respect to natural deduction proofs of ...
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For which logic output of funcntion $\tau$ will be K logic?

We have $ L = \{\land , \lor , \to , \bot , \lnot \} $ and $L_\Box = \{\land , \lor , \to , \bot , \lnot ,\Box \}$ and then we define function $\tau : L \to L_\Box$ like below $\tau(p) = \...
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Satisfaction relation in Intuitionistic Logic

I would like to clarify what concerns me in satisfaction relation in Kripke frames for intuitionistic logic (INT). Firstly, is it a true statement that given a Kripke Model $$M = \langle W, R, \models ...
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Relation between Heyting algebras and Intuitionistic logic

We can associate a logic to an algebra in the following way: Let $(A, \leq)$ be an algebra and $D \subseteq A$ a set of designated elements. Then we define a valuation as $v: Prop \to A$. Then we ...
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What does intuitionism/constructivism say about this proof?

The Wikipedia article on Euler's totient function lists, in the Growth rate section, the following: $$\varphi(n)<\frac{n}{e^\gamma\log\log n}\quad\text{ for infinitely many }n$$ and says: "The ...
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A Question About the Order of Learning from the Book “Lectures on the Curry-Howard Isomorphism” (1998)

I'm learning from this book: https://disi.unitn.it/~bernardi/RSISE11/Papers/curry-howard.pdf (Lectures on Curry-Howard Isomorphism - 1998 version) for some project. And due to time constraints, I ...
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In Intuitionistic Kripke semantics, is proving a formula not forced at one node the same as proving it not derivable/invalid?

I am having trouble understanding whether proving a formula not forced at one node in Kripke semantics is the same as proving that it is not derivable/invalid in Intuitionistic logic. For example, ...
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Can we prove using intuitionist logic that $\neg \exists r: \forall x: [P(x,r) \iff \neg P(x,x)]$

Can we prove using intuitionist logic that $\neg \exists r: \forall x: [P(x,r) \iff \neg P(x,x)]$ where $P$ is a binary logical predicate?
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What are the restrictions on substitution of terms in Hilbert-style calculus vis-à-vis intuitionistic logic?

I apologize if the title of this question can be better formulated, but I recently encountered a situation where what I thought was valid substitution led to an incorrect thing—at least I think. I ...
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Introductions to intuitionistic logic?

Short version: What is a good shortish introduction to intuitionistic logic, accessible to a relative beginner in logic? Long version: I'm revising the frequently used Teach Yourself Logic Study ...
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How HSAT (Heyting satisfiability) can be proven to be PSPACE-COMPLETE if not every intuitionistic formula has disjunctive or alternative normal form?

There's a proof in R. Statman, Intuitionistic propositional logic is polynomial-space complete. Is it constructive? Would that be possible to formulate HSAT without reffering to second order classic ...
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Does every intuitionistic formula have disjunctive or conjuctive normal form?

As in title - does every intuitionistic formula have disjunctive and conjuctive normal form? I guess that this is correct but I couldn't find any information on that.
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Does constructive dilemma hold in intuitionistic logic?

I mean this law: $ (( p \Rightarrow r) \wedge (q\Rightarrow r) \wedge (p \vee q)) \Rightarrow r$ It seems to me that it does hold for INT. Is there any non-esoteric logic that excludes that law?
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$A \rightarrow \neg \neg A$ in constructive logic with and without explosion

I have been reading about constructive logic and I also recently became aware of minimal logic. My understanding of minimal logic is not clear, and I don't get how it is defined, or what suffices as a ...
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Proof methods w/o Contradiction (Intuitionistic Frameworks of Math)

Regarding the standard formulation of the division algorithm: If $a,b$ are integers with $b > 0$, then there exists unique $q,r$ such that $a = bq + r$ with $0 \leq r < b$. The standard ...
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Does the BHK interpretation induce a category?

I was recently introduced to the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic, and since I am just starting out with basic category theory and have heard that topos logic is ...
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Relation between classical implication and intuitionistic implication

Recently, I have read an article on combining classical and intuitionistic implications. On page 9, in their Proposition 6, the authors say that $$A\Rightarrow((A\Rightarrow B)\rightarrow (A\...
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What is the definition of an embedding of one logic in another?

An oft-cited result about intuitionistic logic is that Proposition 1. Classical logic can be embedded into intuitionistic logic. This is the Godel-Gentzen double-negation translation: given a ...
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Does the law of the excluded middle hold for finite sets in intuitionist set theory?

There are local laws of the excluded middle in intuitionist logic, for instance I believe you can prove in Heyting Arithmetic that for all $\Sigma_1$ propositions, that $p \vee \neg p$. Is there ...
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Can you define non-effectively calculable function in constructive mathematics?

Pretty simple just what the title says. Can you define non-effectively calculable functions in constructive mathematics? Does this answer differ in any way if it were in an intuitionistic logic?
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Defining sets in intuitionistic logic

I'm somewhat familiar with the school of intuitionistic logic. I know that an intuitionistic logician thinks of infinity as constructive as apposed to complete. Thus a intuitionistic logician cannot ...
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Doesn't cut elimination make intuitionistic logic equivalent to classical logic?

Suppose we have a proof by contradiction of $A$, meaning we've proven $(A \to \bot) \to \bot$. If we eliminate all cuts in the proof, then the last step of the proof will be an implication ...
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How exactly does the currry-howard formalization of logic capture the semantics of LEM not holding?

Let $p$ be a proposition and $P$ the collection of propositions. In classical logic, the law of excluded middle holds, and we can model the semantics of this as saying that there is a function $\text{...
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Proof of the Disjunction Property for propositional Intuitionistic Logic, using Normalization for natural deduction

I'm doing a Proof Theory course and this is an exercise. I've tried a few things but nothing works. Any help would be much appreciated. (Note it's regular Normalization, not strong).
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Equivalence Between Law of Excluded Middle and Self-Implication

We know that $P \to Q$ is equivalent to $\neg P \lor Q$, as can be verified easily in truth table. Now suppose we have proof for self-implication below [the axiom system is Lukasiewicz's, with L1: $P ...
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Noetherian sets without LEM

A noetherian ring can be defined as a ring in which any nonempty set of ideals has a maximal element. They're pretty nice objects. One can obviously generalize this to a bunch of different algebraic ...
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Difference between $\lambda$-$\mu$-calculus and intuitionistic type theory + LEM for classical proofs?

I have some experience with using type theory to do proofs in intuitionistic logic. If I want to prove theorems that require classical logic, I simply pose the law of excluded middle (LEM) as an axiom....
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Is intuitionistic first-order logic with no function or relation symbols decidable?

Classical first-order logic with no function or relation symbols is decidable. If I'm not mistaken, this is essentially because any formula (with possible free variables) has truth value uniquely ...

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