Mass problems and intuitionistic higher-order logic

Basu, Sankha S.; Simpson, Stephen G.

doi:10.3233/com-150041

Cited by 2 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A special case of Example 2.3 is when we take B to be the Muchnik lattice. In that case we obtain a fragment of the first-order part of the structure studied by Basu and Simpson [2] mentioned in the introduction. Let us consider Γ = {∅} and X = ω.…”

Section: Categorical Semantics For Iqcmentioning

confidence: 99%

“…in other words a solution of the problem ∀x(B(x)) computes a solution of every B i but does so non-uniformly. If, as in [2], we take each B i to be the canonical representative of its Muchnik degree, i.e. we take B i to be upwards closed under Turing reducibility, then we have that (∀x…”

Section: Categorical Semantics For Iqcmentioning

confidence: 99%

“…In the latter case, we have that 2 ∈ ω for which t k 1 and t k 2 are incomparable. Without loss of generality, let us assume that k 1 = j.…”

Section: Furthermore If We Allow Infinite Ascending Chains Then Thimentioning

confidence: 99%

“…If, as in [2], we take each B i to be the canonical representative of its Muchnik degree, i.e. we take B i to be upwards closed under Turing reducibility, then we have that…”

Section: A ≤ P(π)(b) If and Only If (∀X)mentioning

confidence: 99%

“…Recently, Basu and Simpson [2] have independently studied an interpretation of higher-order intuitionistic logic based on the Muchnik lattice. One of the main differences between our approach and their approach is that our approach follows Kolmogorov's philosophy that the interpretation of the universal quantifier should depend uniformly on the variable.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

First-Order Logic in the Medvedev Lattice

Kuyper

2015

Stud Logica

View full text Add to dashboard Cite

Abstract.Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic, using the notion of a first-order hyperdoctrine from categorical logic, to a structure which we will call the hyperdoctrine of mass problems. We study the intermediate logic that the hyperdoctrine of mass problems gives us, and we study the theories of subintervals of the hyperdoctrine of mass problems in an attempt to obtain an analogue of Skvortsova's result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. Finally, we consider Heyting arithmetic in the hyperdoctrine of mass problems and prove an analogue of Tennenbaum's theorem on computable models of arithmetic.

show abstract

Section: Categorical Semantics For Iqcmentioning

confidence: 99%

Section: Categorical Semantics For Iqcmentioning

confidence: 99%

“…In the latter case, we have that 2 ∈ ω for which t k 1 and t k 2 are incomparable. Without loss of generality, let us assume that k 1 = j.…”

Section: Furthermore If We Allow Infinite Ascending Chains Then Thimentioning

confidence: 99%

“…If, as in [2], we take each B i to be the canonical representative of its Muchnik degree, i.e. we take B i to be upwards closed under Turing reducibility, then we have that…”

Section: A ≤ P(π)(b) If and Only If (∀X)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

First-Order Logic in the Medvedev Lattice

Kuyper

2015

Stud Logica

View full text Add to dashboard Cite

show abstract

A.N. Kolmogorov’s Calculus of Problems and Homotopy Type Theory

Rodin¹

2022

Вест. ПГУ. ФСП

View full text Add to dashboard Cite

In 1932 A.N. Kolmogorov proposed an original version of mathematical intuitionism where the distinc-tion between problems and theorems plays a central role and which differs in its content from the versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. In view of today’s historians and logicians, it remains a controversial point whether this distinction is to be treated as logical or Kol-mogorov’s «problems» should be regarded as propositions, provided the latter term is interpreted intui-tionistically. The popular BHK semantics of intuitionistic logic, so called after Brouwer, Heyting, and Kolmogorov and supposed to provide a synthesis of ideas of these authors, does not formally distinguish between problems and theorems and treats this distinction as contextual or purely linguistic. We argue that the distinction between problems and theorems is a crucial element of Kolmogorov’s approach and provide a logical interpretation of this distinction using homotopy type theory (HoTT). The notion of ho-motopy level of a given type in HoTT allows one to distinguish, after Kolmogorov, between problems re-duced to proving a sentence and problems that require realization of higher-order constructions. Thus, HoTT provides a support for Kolmogorov’s view on intuitionistic logic in his polemics with Heyting. At the same time, HoTT does not solve the problem of finding a constructive notion of negation applicable to general problems, which is raised by Kolmogorov in the same paper and still remains open.

show abstract

Mass problems and intuitionistic higher-order logic

Abstract: 2 On the contrary, logic is an application or part of mathematics (according to Brouwer).

Cited by 2 publications

References 22 publications

First-Order Logic in the Medvedev Lattice

First-Order Logic in the Medvedev Lattice

A.N. Kolmogorov’s Calculus of Problems and Homotopy Type Theory

Contact Info

Product

Resources

About