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Reference request for a modification of Bi-Intuitionistic Logic

I’ve asked this same question on math.stackexchange.com, but haven’t received any answers. I ask this question in good faith, so I hope it meets this site’s standards. I have been spending the better ...
PW_246's user avatar
  • 184
12 votes
0 answers
148 views

Limits in free cocompletion, constructively

Classically, if a locally small category $C$ has all limits of shape $K$ (for some small diagram $K$), then its free co-completion also has $K$-shapped limits. But all proof I know of that result ...
Simon Henry's user avatar
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20 votes
4 answers
2k views

Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

Last year, Andrej Bauer gave a talk showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He ...
Anon's user avatar
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6 votes
1 answer
195 views

Preserve validity between the two Kripke frames

The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame. For $n \geq 1$, let $\mathcal{C}_n$ denote the frame which is shown in Fig.1. ...
mahu's user avatar
  • 63
9 votes
0 answers
142 views

Are there good criteria for the topological models where BD-N and BD hold?

A (non-empty/inhabited) subset $S$ of $\mathbb{N}$ is said to be pseudo-bounded if for every sequence $x_n$ in $S$ we have $\lim_{n\to \infty} \frac{x_n}{n} = 0$ Clearly all bounded subsets are pseudo-...
saolof's user avatar
  • 1,793
6 votes
1 answer
143 views

Variable elimination for propositional formulas in Heyting algebras

By an (intuitionistic) propositional formula $\varphi(x_1,\ldots,x_n)$ I mean a formula built up from a (finite) number of variables $x_1,\ldots,x_n$ using connectors $\top, \bot, \land, \lor, \...
Gro-Tsen's user avatar
  • 28.2k
1 vote
0 answers
73 views

Proof of the Local Deduction Theorem, for one of many logics

I'm looking for a proof of the Local Deduction Theorem for any one of many logics. That is, a proof of the statement: $\Sigma \cup \{\phi\} \models \psi$ iff for some positive $n,$ $\Sigma \models \...
Martín S's user avatar
  • 421
2 votes
1 answer
422 views

The LNC as a mathematical theorem

One of the most intriguing things I've read about over the last few years is Diaconescu's theorem, which says that, in some forms of constructivist/intuitionistic set theory, even if the law of the ...
Kristian Berry's user avatar
7 votes
1 answer
288 views

In CZF (w/ Subset Collection removed) the Powerset axiom Implies Subset Collection

The Subset Collection axiom: $$ \forall a \forall b \exists c \forall u [\forall x \in a \exists y \in b (\psi(x,y,u)) \longrightarrow \exists d \in c (\forall x \in a \exists y \in d (\psi(x,y,u)) \...
ToucanIan's user avatar
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4 votes
0 answers
90 views

Negation-quantifier-negation blocks in nonclassical logic: reference request

I'm looking for references to discussion of a certain question in the literature on non-classical first-order logics. I suspect it must have been investigated thoroughly, but I can't seem to find ...
Beau Madison Mount's user avatar
5 votes
1 answer
161 views

Possible values of "Kripke rank" for formulae in IPL

Fix a (countable) set $\mathcal{P}$ of atomic propositional variables. Recall a Kripke model $\mathcal{K}$ for intuitionistic propositional logic (IPL) consists of: A preorder $(W,\leq)$ For each $w \...
Jordan Mitchell Barrett's user avatar
3 votes
1 answer
272 views

Subset Collection axiom

In Constructive Set Theory (CZF) the Power Set axiom is replaced with the Subset Collection axiom which I will state here: $$ \exists c \forall u [\forall x \in a \exists y \in b (\psi(x,y,u)) \...
ToucanIan's user avatar
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4 votes
0 answers
285 views

Is intuitionistic predicate logic (semantically) complete or incomplete?

According to Constructivism in Mathematics: An Introduction by Troelstra A.S. and Van Dalen (https://archive.org/details/constructivismin0002troe/page/718/mode/2up) it is proven in an intuitionisitc ...
ToucanIan's user avatar
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5 votes
1 answer
424 views

How do working constructivists get by with out the zero product property?

It is stated by Douglas Bridges in Constructive mathematics: a foundation for computable analysis that the following property, which I will call the zero product property: If $x,y \in \mathbb{R}$ and $...
ToucanIan's user avatar
  • 391
10 votes
1 answer
1k views

Why are W-types called "W"?

Why are W-types called "W"? Probably "W" means either "wellordered" or "wellfounded". (Martin-Löf uses the term "wellorder".) But these are notions ...
user347155's user avatar
2 votes
0 answers
138 views

In a constructive order/lattice theory are the arbitrary join and the weak suprema the same?

For a poset $X$ we define an upperbound $w \in X$ of a subset $S \subseteq X$ to be a weak-supremum if $(\forall a \in S (a \leq b)) \implies w \leq b$. While a supremum is defined more carefully (in ...
ToucanIan's user avatar
  • 391
7 votes
3 answers
502 views

Is there a completeness proof of intuitionistic predicate calculus using Heyting algebra semantics that is inuitionistically valid?

According to godelian in Henkin-style completeness proofs for intuitionistic logic there are multiple intuitionstically valid proofs of the completeness of inuitionistic predicate calculus (IPC) via ...
ToucanIan's user avatar
  • 391
34 votes
3 answers
3k views

Alternatives to the law of the excluded middle

As a sentential logic, intuitionistic logic plus the law of the excluded middle gives classical logic. Is there a logical law that is consistent with intuitionistic logic but inconsistent with ...
Colin Tan's user avatar
  • 251
0 votes
0 answers
122 views

Where does intuitionistic predicate logic live in the arithmetical hierarchy?

I started reading Plisko's papers on arithmetic complexity on the arithmetic complexity of constructive logic (see for example here or here). In this context, I started wondering about the following ...
Robert Passmann's user avatar
16 votes
3 answers
1k views

Is there a semantics for intuitionistic logic that is meta-theoretically "self-hosting"?

One can study the standard semantics of classical propositional logic within classical logic set theory, so we can say that the semantics of classical logic is meta-theoretically "self-hosting&...
ttbo's user avatar
  • 163
7 votes
0 answers
278 views

Two simple cases of quantifier elimination for Heyting algebras

This extracts a simple case from a cross-post at cs.SE. Here is a fact about Intuitionistic Propositional Logic: A formula $p$ is equivalent to a formula of the form $q \lor \neg q$ if and only if $\...
მამუკა ჯიბლაძე's user avatar
1 vote
1 answer
151 views

Markov's principle from constant domain logic

I am looking for a proof of and/or a reference for the result that Markov's principle can be proved in the framework of constant domain logic. By constant domain logic, I mean intuitionistic logic ...
Erik D's user avatar
  • 338
10 votes
1 answer
418 views

Computability-theoretic results relevant to realizability

This may be a very naive question which only reflects my failure at literature search, but: Although realizability (in its original form at least) is grounded in computability, the details of ...
Noah Schweber's user avatar
10 votes
1 answer
356 views

Fibers of the morphism from the free Heyting algebra to the free Boolean algebra

For $k\in\mathbb{N}$, let $H_k$ be the free Heyting algebra on $k$ variables $p_1,\ldots,p_k$ and $B_k$ be the free Boolean algebra on the same $k$ variables. Thus, $B_k$ has $2^{2^k}$ elements (...
Gro-Tsen's user avatar
  • 28.2k
10 votes
2 answers
1k views

Henkin-style completeness proofs for intuitionistic logic

Henkin-style completeness proofs are founded on a few basic presuppositions, such as the assumptions that the language of a logical theory must be enumerable (or at least that the axiom of choice ...
Bruno Bentzen's user avatar
0 votes
1 answer
161 views

Superintuitionistic logics which are not hereditary/monotonic: impossible or possible?

An intuitionistic Kripke model is a triple $\langle W,\leq, \Vdash \rangle$, where $\langle W,\leq \rangle$ is a preordered Kripke frame, and $\Vdash$ satisfies the following condition of ...
user65526's user avatar
  • 629
0 votes
2 answers
203 views

Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic?

In posting "Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?" the answer was that they are equivalent in classical first order logic with membership "$\in$". ...
Zuhair Al-Johar's user avatar
4 votes
1 answer
752 views

Going beyond the strength of Peano arithmetic "without sets"

First-order arithmetic is fairly weak, as measured for example by its consistency strength. When a stronger theory is desired, it is common to work with (fragments of) second-order arithmetic or set ...
Robin Saunders's user avatar
3 votes
1 answer
444 views

Does cut elimination fail here?

Proving $\lnot\lnot(A\lor\lnot A)$ in intuitionistic sequent calculus with cut seems to be easy: We use cut to prove $\lnot(A\lor\lnot A)\vdash \bot$ from $\lnot(A\lor\lnot A)\vdash \lnot A \land\lnot\...
Thomas Klimpel's user avatar
3 votes
0 answers
132 views

A conservativity result of intuitionistic set theory over arithmetic

In their 1985 paper "Arithmetic Transfinite Induction and Recursive Well-Orderings", Friedman and Ščedrov prove that the theory $\mathbf{ZFI}$ is conservative over $\mathbf{HA}^*$ (see here, Theorem ...
namsap's user avatar
  • 335
6 votes
0 answers
470 views

Infinite-time Turing machines and the formal Church's thesis

Infinite-time Turing machines are known to either halt or loop in countable time. In the spirit of double-negation translation, is there a statement which is: classically equivalent to this; ...
Robin Saunders's user avatar
3 votes
1 answer
195 views

Fixed-point property and $T_0$ separation property

Each topological space $A$ with fixed-point property is $T_0$ space. Proof: suppose, two different points $a_1$ and $a_2$ belong to the same open subsets of $A$. Then the function $$f(a)= \begin{cases}...
George Cherevichenko's user avatar
2 votes
1 answer
324 views

Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle?

Consider Propositional Lax Logic ($PLL$) https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf The Hilbert system of $PLL$ takes as axiom ...
user65526's user avatar
  • 629
3 votes
2 answers
304 views

Modal collapse upon addition of the law of the excluded middle to an Intuitionistic modal logic

Consider Propositional Lax Logic ($PLL$) https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf The Hilbert system of $PLL$ takes as axiom ...
user65526's user avatar
  • 629
10 votes
2 answers
777 views

Adding nonconstructive disjunction to intuitionistic logic

In constructive mathematics, under realizability interpretations, we can define nonconstructive disjunction $A⅋B$ as follows: A witness for $A⅋B$ gives a candidate witness for $A$ and a candidate ...
Dmytro Taranovsky's user avatar
6 votes
1 answer
821 views

Rice's theorem in type theory

From the formula $$\forall f\colon A\to A\,\exists x\colon A\,(f(x)=x)$$ we can get the scheme $$\forall x\colon A\,(\phi(x)\vee\neg\phi(x))\Rightarrow\forall x\colon A\,\phi(x)\vee\forall x\colon A\,\...
George Cherevichenko's user avatar
8 votes
1 answer
333 views

Forcing in Constructive Set Theories

I searched on the internet, but I could not find anything useful about applications of forcing in constructive set theories. Are there any developments of forcing in CZF or IZF? Thanks in advance.
Erfan Khaniki's user avatar
5 votes
0 answers
374 views

Did Kleene constructively prove Brouwer's axioms?

Harvey Friedman's request on the FoM-forum for an overview of current intuitionistic foundations revived the following question, which I have been meaning to ask for five years. (I'm no expert on ...
Franka Waaldijk's user avatar
9 votes
2 answers
517 views

Admissibility of Harrop's rule, computationally

It is obvious that the following formula is not a theorem of intuitionistic propositional calculus (IPC). $$ (\neg A \; \to \; B \vee C) \;\; \to \;\; ((\neg A \; \to \; B) \vee (\neg A \; \to \; ...
Bob's user avatar
  • 476
21 votes
3 answers
2k views

Why would the category of sets be intuitionistic?

This question is probably really naive. And, I hope the title doesn't come off as too combative. I think that topoi of $\mathbf{Set}$-valued sheaves provide an excellent motivation for higher-order ...
goblin GONE's user avatar
  • 3,683
10 votes
2 answers
415 views

Why is the notion of algorithm a primitive one in Brouwer's intuitionism?

I've seen several times people mentioning that the notion of an algorithm / a computation is taken as a primitive notion in L. E. J. Brouwer's intuitionism. For instance, in Varieties of Constructive ...
StudentType's user avatar
7 votes
1 answer
374 views

Pure first order logic formulations of Markov's principle

Markov's principle is a statement of constructive arithmetic that allows classical reasoning on formulas of the shape $\exists x P$ when $P$ is a recursive predicate: $\neg \neg \exists x P \to \...
Matteo's user avatar
  • 71
8 votes
2 answers
908 views

Identity types: What makes Intuitionistic Type Theory *intuitionistic*?

In the opening passage of Martin-Löf's (1975) he famously says that "the theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic ...
StudentType's user avatar
13 votes
1 answer
652 views

Intutionistic Robinson Arithmetic

By Friedman translation $HA$ and $PA$ prove the same $\Pi_2$ formulas. Is it true for Intutionistic Robinson arithmetic(Robinson axioms with intutionistic logic) and classic Robinson arithmetic? ...
Erfan Khaniki's user avatar
21 votes
6 answers
3k views

How strong is Cantor-Bernstein-Schröder?

There are several questions here on MO about the Cantor-Bernstein-Schröder ((C)BS) theorem, but I could not find answers to what arose to me recently. Although I don't think I need to recall it here, ...
მამუკა ჯიბლაძე's user avatar
8 votes
2 answers
857 views

Did Bishop, Heyting or Brouwer take partial functions seriously?

The partial μ-recursive functions which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one ...
Thomas Klimpel's user avatar
9 votes
1 answer
1k views

Forcing is intuitionistic

The main idea of why it´s necessary a generic filter $G$ to extend a countable transitive $\epsilon$-interpretation (not necessarily a model) $M$ is given by the condition (for which $G$ being a ...
user40276's user avatar
  • 2,159
8 votes
3 answers
587 views

Models of intuitionistic linear logic that reflect the resource interpretation

I am interested in models of intuitionistic linear logic, that is, the logic that you get if you take classical linear logic and restrict the set of operators to $\otimes$, $1$, $\multimap$, $\times$, ...
Wolfgang Jeltsch's user avatar
24 votes
3 answers
3k views

Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic. According to the axioms of Kolmogorov, probability theory is formulated with a (normalized) probability measure $\mbox{...
Frank's user avatar
  • 457
13 votes
6 answers
2k views

Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with intuitionistic logic. It is ...
Mikhail Katz's user avatar
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