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]
447 questions
72
votes
14 answers
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
- 735
35
votes
4 answers
Difference between proof of negation and proof by contradiction
I stumbled across Andrej Bauer's Proof of negation and proof by contradiction in which Andrej 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…
Erwan Aaron
- 927
31
votes
5 answers
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
- 2,311
28
votes
0 answers
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…
Arrow
- 14,390
20
votes
2 answers
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,…
Taroccoesbrocco
- 18,243
15
votes
1 answer
Is it known whether there are atoms in the complete lattice of intermediate logics?
By "intermediate logic" I mean a (non-trivial) propositional logic at least as strong as intuitionistic logic whose set of theorems is closed under modus ponens and closed under substitution of propositional letters with formulas. The complete…
RichardBerry
- 151
15
votes
1 answer
15
votes
4 answers
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
- 60,745
14
votes
2 answers
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
- 3,246
14
votes
3 answers
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
- 7,462
13
votes
3 answers
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
- 20,307
13
votes
2 answers
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…
Bunny
- 3,384
12
votes
1 answer
Is Hilbert's epsilon calculus a conservative extension of intuitionist logic?
Epsilon calculus is an extension to First Order Logic, by including the epsilon operator. The epsilon operator works as a sort of choice operator, which selects a witness for each true existential statement. I'll notate the epsilon operator as…
Jade Vanadium
- 5,046
- 10
- 25
12
votes
1 answer
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…
ferdinand
- 652
- 3
- 9
11
votes
1 answer
How Do Heyting Algebras Relate To Logic?
My question is, broadly speaking, how are Heyting algebras related to logic ? It would be great if someone could answer this question without being too technical (or point to easy to read literature). I know about category theory, but not much about…
Richard Southwell
- 1,903