Sankha Basu - Academia.edu (original) (raw)

Drafts by Sankha Basu

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarski-and Lindenbaum-type, or of TL-type and end with a characterization and a representation theorem for these.

Papers by Sankha Basu

Research paper thumbnail of Generalized explosion principles

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent and primary. In this article, we start by asking whether a negation operator is essential for describing explosion and paraconsistency. In other words, is it possible to describe a principle of explosion and hence a notion of paraconsistency that is independent of connectives? A negation-free paraconsistency resulting from the failure of a generalized principle of explosion is presented first. We also derive a notion of quasi-negation from this and investigate its properties. Next, more general principles of explosion are considered. These are also negation-free; moreover, these principles gradually move away from the idea that an explosion requires a statement and its opposite. Thus, these principles can capture the explosion observed in logics where a statement and its negation explode only in the presence of additional information, such as in the logics of formal inconsistency.

Research paper thumbnail of Generalized explosion principles

arXiv (Cornell University), Jul 28, 2023

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent and primary. In this article, we start by asking whether a negation operator is essential for describing explosion and paraconsistency. In other words, is it possible to describe a principle of explosion and hence a notion of paraconsistency that is independent of connectives? A negation-free paraconsistency resulting from the failure of a generalized principle of explosion is presented first. We also derive a notion of quasi-negation from this and investigate its properties. Next, more general principles of explosion are considered. These are also negation-free; moreover, these principles gradually move away from the idea that an explosion requires a statement and its opposite. Thus, these principles can capture the explosion observed in logics where a statement and its negation explode only in the presence of additional information, such as in the logics of formal inconsistency.

Research paper thumbnail of Lindenbaum-Type Logical Structures

Logica Universalis

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski-and the Lindenbaum-types. The characterization theorems for the Tarski-and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaumtypes and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski-and a Lindenbaum-type, show their separations, and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski-Lindenbaum-type logical structures.

Research paper thumbnail of Lindenbaum-type Logical Structures

arXiv (Cornell University), Jul 22, 2021

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski-and the Lindenbaum-types. The characterization theorems for the Tarski-and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaumtypes and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski-and a Lindenbaum-type, show their separations, and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski-Lindenbaum-type logical structures.

Research paper thumbnail of The Field of Complex Numbers - Algebraic Closure of a Non-Archimedean Real Closed Field

Research paper thumbnail of Negation-Free Definitions of Paraconsistency

Electronic Proceedings in Theoretical Computer Science

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent. In this article, we ask whether a negation operator is essential for describing paraconsistency. In other words, is it possible to describe a notion of paraconsistency that is independent of connectives? We present two such notions of negation-free paraconsistency, one that is completely independent of connectives and another that uses a conjunction-like binary connective that we call fusion. We also derive a notion of quasi-negation from the former, and investigate its properties.

Research paper thumbnail of Negation-Free Definitions of Paraconsistency

Electronic Proceedings in Theoretical Computer Science

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent. In this article, we ask whether a negation operator is essential for describing paraconsistency. In other words, is it possible to describe a notion of paraconsistency that is independent of connectives? We present two such notions of negation-free paraconsistency, one that is completely independent of connectives and another that uses a conjunction-like binary connective that we call fusion. We also derive a notion of quasi-negation from the former, and investigate its properties.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarskiand Lindenbaum-type, or of TL-type and end with a characterization and a representation theorem for these.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarski- and Lindenbaum-type, or of TL-type, and end with a characterization and a representation theorem for these.

Research paper thumbnail of Restricted Rules of Inference and Paraconsistency

We present here two logical systems - intuitionistic paraconsistent weak Kleene logic (IPWK) and ... more We present here two logical systems - intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL) as examples of logics with some rules of inference that have variable sharing restrictions imposed on them. These systems have the same set of theorems as intuitionistic propositional logic and pre-rough logic, respectively, but are paraconsistent while the original systems are not. We discuss algebraic semantics for these logics. A contaminating element, intended to denote a state of indeterminacy, is used to extend each Heyting algebra and each pre-rough algebra. The classes of these extended Heyting algebras and the extended pre-rough algebras form models of IPWK and PPRL, respectively. We then prove the soundness and completeness results for these systems.

Research paper thumbnail of Mass problems and

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik top... more In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x ∃y A(x, y)) ⇒ ∃w ∀x A(x, wx) and a bounding principle (∀x ∃y A(x, y)) ⇒ ∃z ∀x ∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.

Research paper thumbnail of Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik top... more In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass prob-lems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x∃y A(x, y)) ⇒ ∃w ∀xA(x,wx) and a bound-ing principle (∀x∃y A(x, y)) ⇒ ∃z ∀x∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not cont...

Research paper thumbnail of Restricted Rules of Inference and Paraconsistency

Logic Journal of the IGPL

In this paper, we study two companions of a logic, viz., the left variable inclusion companion an... more In this paper, we study two companions of a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems—intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL)—are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of intuitionistic propositional logic and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of pre-rough logic but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of Płonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.

Research paper thumbnail of A model of intuitionism based on Turing degrees

Intuitionism is a constructive approach to mathematics introduced in the early part of the twetie... more Intuitionism is a constructive approach to mathematics introduced in the early part of the twetieth century by L. E. J. Brouwer and formalized by his student A. Heyting. A. N. Kolmogorov, in 1932, gave a natural but non-rigorous interpretation of intuitionism as a calculus of problems. In this document, we present a rigorous implementation of Kolmogorov’s ideas to higher-order intuitionistic logic using sheaves over the poset of Turing degrees with the topology of upward closed sets. This model is aptly named as the Muchnik topos, since the lattice of upward closed subsets of Turing degrees is isomorphic to the lattice of Muchnik degrees which were introduced in 1963 by A. A. Muchnik in an attempt to formalize the notion of a problem in Kolmogorov’s calculus of problems.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarskiand Lindenbaum-type, or of TL-type and end with a characterization and a representation theorem for these.

Research paper thumbnail of Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik top... more In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x ∃y A(x, y)) ⇒ ∃w ∀x A(x, wx) and a bounding principle (∀x ∃y A(x, y)) ⇒ ∃z ∀x ∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarski-and Lindenbaum-type, or of TL-type and end with a characterization and a representation theorem for these.

Research paper thumbnail of Generalized explosion principles

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent and primary. In this article, we start by asking whether a negation operator is essential for describing explosion and paraconsistency. In other words, is it possible to describe a principle of explosion and hence a notion of paraconsistency that is independent of connectives? A negation-free paraconsistency resulting from the failure of a generalized principle of explosion is presented first. We also derive a notion of quasi-negation from this and investigate its properties. Next, more general principles of explosion are considered. These are also negation-free; moreover, these principles gradually move away from the idea that an explosion requires a statement and its opposite. Thus, these principles can capture the explosion observed in logics where a statement and its negation explode only in the presence of additional information, such as in the logics of formal inconsistency.

Research paper thumbnail of Generalized explosion principles

arXiv (Cornell University), Jul 28, 2023

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent and primary. In this article, we start by asking whether a negation operator is essential for describing explosion and paraconsistency. In other words, is it possible to describe a principle of explosion and hence a notion of paraconsistency that is independent of connectives? A negation-free paraconsistency resulting from the failure of a generalized principle of explosion is presented first. We also derive a notion of quasi-negation from this and investigate its properties. Next, more general principles of explosion are considered. These are also negation-free; moreover, these principles gradually move away from the idea that an explosion requires a statement and its opposite. Thus, these principles can capture the explosion observed in logics where a statement and its negation explode only in the presence of additional information, such as in the logics of formal inconsistency.

Research paper thumbnail of Lindenbaum-Type Logical Structures

Logica Universalis

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski-and the Lindenbaum-types. The characterization theorems for the Tarski-and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaumtypes and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski-and a Lindenbaum-type, show their separations, and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski-Lindenbaum-type logical structures.

Research paper thumbnail of Lindenbaum-type Logical Structures

arXiv (Cornell University), Jul 22, 2021

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski-and the Lindenbaum-types. The characterization theorems for the Tarski-and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaumtypes and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski-and a Lindenbaum-type, show their separations, and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski-Lindenbaum-type logical structures.

Research paper thumbnail of The Field of Complex Numbers - Algebraic Closure of a Non-Archimedean Real Closed Field

Research paper thumbnail of Negation-Free Definitions of Paraconsistency

Electronic Proceedings in Theoretical Computer Science

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent. In this article, we ask whether a negation operator is essential for describing paraconsistency. In other words, is it possible to describe a notion of paraconsistency that is independent of connectives? We present two such notions of negation-free paraconsistency, one that is completely independent of connectives and another that uses a conjunction-like binary connective that we call fusion. We also derive a notion of quasi-negation from the former, and investigate its properties.

Research paper thumbnail of Negation-Free Definitions of Paraconsistency

Electronic Proceedings in Theoretical Computer Science

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosi... more Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent. In this article, we ask whether a negation operator is essential for describing paraconsistency. In other words, is it possible to describe a notion of paraconsistency that is independent of connectives? We present two such notions of negation-free paraconsistency, one that is completely independent of connectives and another that uses a conjunction-like binary connective that we call fusion. We also derive a notion of quasi-negation from the former, and investigate its properties.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarskiand Lindenbaum-type, or of TL-type and end with a characterization and a representation theorem for these.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarski- and Lindenbaum-type, or of TL-type, and end with a characterization and a representation theorem for these.

Research paper thumbnail of Restricted Rules of Inference and Paraconsistency

We present here two logical systems - intuitionistic paraconsistent weak Kleene logic (IPWK) and ... more We present here two logical systems - intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL) as examples of logics with some rules of inference that have variable sharing restrictions imposed on them. These systems have the same set of theorems as intuitionistic propositional logic and pre-rough logic, respectively, but are paraconsistent while the original systems are not. We discuss algebraic semantics for these logics. A contaminating element, intended to denote a state of indeterminacy, is used to extend each Heyting algebra and each pre-rough algebra. The classes of these extended Heyting algebras and the extended pre-rough algebras form models of IPWK and PPRL, respectively. We then prove the soundness and completeness results for these systems.

Research paper thumbnail of Mass problems and

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik top... more In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x ∃y A(x, y)) ⇒ ∃w ∀x A(x, wx) and a bounding principle (∀x ∃y A(x, y)) ⇒ ∃z ∀x ∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.

Research paper thumbnail of Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik top... more In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass prob-lems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x∃y A(x, y)) ⇒ ∃w ∀xA(x,wx) and a bound-ing principle (∀x∃y A(x, y)) ⇒ ∃z ∀x∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not cont...

Research paper thumbnail of Restricted Rules of Inference and Paraconsistency

Logic Journal of the IGPL

In this paper, we study two companions of a logic, viz., the left variable inclusion companion an... more In this paper, we study two companions of a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems—intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL)—are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of intuitionistic propositional logic and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of pre-rough logic but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of Płonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.

Research paper thumbnail of A model of intuitionism based on Turing degrees

Intuitionism is a constructive approach to mathematics introduced in the early part of the twetie... more Intuitionism is a constructive approach to mathematics introduced in the early part of the twetieth century by L. E. J. Brouwer and formalized by his student A. Heyting. A. N. Kolmogorov, in 1932, gave a natural but non-rigorous interpretation of intuitionism as a calculus of problems. In this document, we present a rigorous implementation of Kolmogorov’s ideas to higher-order intuitionistic logic using sheaves over the poset of Turing degrees with the topology of upward closed sets. This model is aptly named as the Muchnik topos, since the lattice of upward closed subsets of Turing degrees is isomorphic to the lattice of Muchnik degrees which were introduced in 1963 by A. A. Muchnik in an attempt to formalize the notion of a problem in Kolmogorov’s calculus of problems.

Research paper thumbnail of Lindenbaum-type Logical Structures: Introduction and Characterization

In this paper, we study some classes of logical structures from the universal logic standpoint, v... more In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the previously known Tarski-type, and the new Lindenbaum-type logical structures defined here. The characterization theorems for these logical structures have been proved as well. The following natural questions have been dealt with next. Is every Tarski-type logical structure also of Lindenbaum-type and/or vice-versa? Finally, we study the logical structures that are both of Tarskiand Lindenbaum-type, or of TL-type and end with a characterization and a representation theorem for these.

Research paper thumbnail of Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik top... more In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x ∃y A(x, y)) ⇒ ∃w ∀x A(x, wx) and a bounding principle (∀x ∃y A(x, y)) ⇒ ∃z ∀x ∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.