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3
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1answer
52 views

Intuitionistic Linear Logic

I am currently going through some papers that use the "intuitionistic version" of Girard's Linear Logic. Problem is, i seem to find very little literature on it. There is a lot done on Linear Logic ...
1
vote
1answer
56 views

If $\phi$ is $\Delta^{0}_{1}$ in the language of arithmetic, does Heyting Arithmetic prove $\forall x [\phi (x) \vee \neg \phi (x)]$?

PA is conservative over HA for $\Pi^{0}_{2}$ sentences. If $\phi$ is $\Delta^{0}_{1}$, then $\forall x [\phi (x) \vee \neg \phi (x)]$ is equivalent to a $\Pi^{0}_{2}$ sentence. Since PA trivially ...
3
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5answers
108 views

Is there a simple example of how the law of the excluded middle can be inapplicable?

Why does a logic system not use the law of the excluded middle? I studied non-classical logic (intuitionistic and modal) where double negation can't be removed and the law of excluded middle can't be ...
4
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0answers
73 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
4
votes
2answers
96 views

Non-upper bounds without excluded middle

Motivated by an earlier question, I'm curious if we can prove the following statement without the law of excluded middle: Let $E$ be a set of real numbers. A number $x$ is said to be an upper ...
2
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2answers
56 views

Double negation distributes over conjunction

In Exercise I.1.11.(ii) of Johnstone's Stone Spaces, it is claimed that in any Heyting algebra, $$\lnot\lnot (a \land b) = \lnot\lnot a \land \lnot\lnot b.$$ It is easy to see one direction: Since ...
4
votes
1answer
130 views

Model theory for intuitionistic predicate logic: a non-empty domain?

In classical logic we tend to make the assumption that the domain of quantification is non-empty. This isn't (too) problematic because classical mathematicians assume a language/mind/proof independent ...
5
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0answers
105 views

Axiom of Choice - Type Theory (Proof)

Background In Intuitionistic Type Theory (p. 27-28), Martin Löf provides a proof of the axiom of choice that is constructively valid. This version is considerably weaker than the ordinary set theory ...
7
votes
3answers
144 views

How or why does intutionistic logic proof negations from within the theory, constructively?

I'm having a little of a cognitive dissonance why, in intuitionistic logic, it's possible to show stentences like $(\neg A \land \neg B) \implies \neg(A\lor B).$ In plain text: If 'A isn't true' as ...
2
votes
1answer
84 views

Unprovable Equivalence in Type Theory

Let $\prec$ be a binary relation on a set $A$… A predicate $P(x)$ set $(x:A)$ is said to be progressive with respect to $(A,\prec)$ if \begin{equation} (\forall a:A)\Big((\forall b:A)\big(b \prec a ...
5
votes
1answer
103 views

Understanding a proof of Diaconescu's theorem

I am trying to walk through the proof of Diaconescu's theorem that the axiom of choice implies the law of excluded middle at http://plato.stanford.edu/entries/intuitionism/#ChoAxi. To paraphrase: ...
2
votes
1answer
84 views

Show formulas which are valid according to Brouwer-Heyting-Kolmogorov interpretation [closed]

How can I show, that the following formulae are valid according to Brouwer-Heyting-Kolmogorov interpretation? $(A \land B) \to (B \land A)$ $\neg (A \lor B) \to (\neg A \land \neg B)$ $A \land (B ...
8
votes
2answers
192 views

Minimal difference between classical and intuitionistic sequent calculus

Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an ...
3
votes
1answer
129 views

Axioms based on $\leftrightarrow, \lor, \bot$ for propositional intuitionistic logic?

Propositional intuitionistic logic can be axiomatized based on $\;\to, \land, \lor, \bot\;$, with modus ponens $$ \text{from }\; \phi \;\text{ and }\; \phi \to \psi \;\text{ infer }\; \psi $$ as the ...
5
votes
3answers
147 views

How can some statements be consistent with intuitionistic logic but not classical logic, when intuitionistic logic proves not not LEM?

I've heard that some axioms, such as "all functions are continuous" or "all functions are computable", are compatible with intuitionistic type theories but not their classical equivalents. But if they ...
0
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1answer
138 views

A question on intuitionistc propositional logic

Prove that: Two finite rooted frames are isomorphic iff they validate the same formulas. (This is an exercise in the book "Modal Logic" by A.Chagrov and M.Zakharyaschev)
1
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1answer
113 views

How is the double negation translation similar to CPS in functional programming languages?

On wikipedia, I found that any formula in classical logic has its double negation as its intuitionist equivalent: It is also possible to define φN by prefixing ¬¬ before every subformula of φ, ...
2
votes
1answer
127 views

Intuitionism - is it fundamentally different than “ordinary” mathematics.

I have recently had a conversation with a person who considered intuitionism to be a valid alternative for the "usual" kind of mathematics. Clearly, intuitionism differs from the type of mathematical ...
3
votes
0answers
171 views

What do ultrafinitists think about Graham's number?

I know ultrafinitists want to require not only that mathematical objects be constructible, but be constructible given finite resources (such as time). So I wonder about something like the famous ...
4
votes
4answers
156 views

Help proving $ \sim \sim (p \to q),\sim \sim p \vdash \sim \sim q $ with intuitionistic axioms

Rule modus ponens: $ p, p \to q \vdash q $ Axioms A1 $ p \to (q \to p) $ A2 $ (p \to (q \to r)) \to ((p \to q) \to (p \to r)) $ A3 $ (p \land q) \to p $ A4 $ (p \land q) \to q $ A5 $ p \to ...