arXiv subjects

Sankha S. Basu

Publications and source records attributed to Sankha S. Basu.

13 recordsLinked to original sources

arXiv2026

Possibilistic Logic over a Logic of Formal Inconsistency

In this article, we have introduced a new possibilistic logic on a logic of formal inconsistency with the aim of developing a possibility theoretic framework to deal with uncertainty and inconsistency meaningfully without leading to a system collapse. We have discussed the syntax and semantics for this logic and have proved the soundness and completeness theorems. A set of new measures of consistency, contradictoriness, and triviality of a set of formulas have been defined. These have then been put to use in an example to show that this framework can provide better means of machine reasoning.

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arXiv2026

Abstract Model Structures and Compactness Theorems

The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the syntactic/semantic particularities of the corresponding logic. In this paper, using the notion of \emph{abstract model structures}, we show that one can develop a generalized notion of compactness that is independent of these. Several characterization theorems for a particular class of compact abstract model structures are also proved.

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arXiv2026

Paracomplete Probabilities

This paper presents an advance in the direction of working with probabilities in a paracomplete setting using Logics of Formal Undeterminedness (LFUs). The undeterminedness is interpreted here as missing evidence. A theorem of total paracomplete probability and a paracomplete Bayes' rule have been proved using this setup. We end with a definition of a paracomplete probability space illustrating a way to define probabilities on sets in the presence of undeterminedness.

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arXiv2026

How a computer might think

Inspired by Nuel Belnap's "How a computer should think," which gave rise to the four-valued logic FDE, we contemplate, in this article, how a computer might think if we add a fifth value for unknowable or cannot be known. We devise two new five-valued logics, UKN1 and UKN2, called the logics of the unknowable. These are different from the five-valued logic FDEe of the FDE-family. The main difference is in the number of designated truth values. While FDEe takes two designated values, UKN1 and UKN2 have three. The four-valued reducts of these logics are also different from FDE. This is due to the fact that instead of taking one of the non-designated values as neither true nor false, as in FDE, we have interpreted this as not known yet. This value although denoted by the same letter $n$, behaves differently from its namesake in FDE.

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arXiv2026

Generalized Explosion Principles: A Semantic Perspective

This article is motivated by the fact that there is a distinction between the descriptions of logical explosion from syntactic and semantic points of view. The discussion is illustrated using the concept of abstract model structures and the notions of satisfiability and finite satisfiability in these structures. Various principles of explosion have been described in terms of unsatisfiability or finite unsatisfiability. The semantic analogues of the principles of explosion introduced in [3] have also been considered among these. The article also studies the characterizations of and the interconnections between these new principles of explosion.

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arXiv2026

Suszko's Thesis and Many-valued Logical Structures

In this article, we try to formulate a definition of ''many-valued logical structure''. For this, we embark on a deeper study of Suszko's Thesis ($\mathbf{ST}$) and show that the truth or falsity of $\mathbf{ST}$ depends, at least, on the precise notion of semantics. We propose two different notions of semantics and three different notions of entailment. The first one helps us formulate a precise definition of inferentially many-valued logical structures. The second and the third help us to generalise Suszko Reduction and provide adequate bivalent semantics for monotonic and a couple of nonmonotonic logical structures. All these lead us to a closer examination of the played by language/metalanguage hierarchy vis-á-vis $\mathbf{ST}$. We conclude that many-valued logical structures can be obtained if the bivalence of all the higher-order metalogics of the logic under consideration is discarded, building formal bridges between the theory of graded consequence and the theory of many-valued logical structures, culminating in generalisations of Suszko's Thesis.

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arXiv2025

Topological semantics for a non-self-extensional LFI

In this article, we have introduced a Logic of Formal Inconsistency (LFI) that we call $\vd$. This logic is non-self-extensional, i.e., the replacement property, or the rule for substitution of equivalents, does not hold. A Hilbert-style presentation for the logic has been provided. Then, a topological semantics for $\vd$ has been described, subsequent to which we have established the Soundness and Completeness results for it with respect to this semantics.

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arXiv2024

Relational Companions of Logics

The variable inclusion companions of logics have lately been thoroughly studied by multiple authors. There are broadly two types of these companions: the left and the right variable inclusion companions. Another type of companions of logics induced by Hilbert-style presentations (Hilbert-style logics) were introduced in a recent paper. A sufficient condition for the restricted rules companion of a Hilbert-style logic to coincide with its left variable inclusion companion was proved there, while a necessary condition remained elusive. The present article has two parts. In the first part, we give a necessary and sufficient condition for the left variable inclusion and the restricted rules companions of a Hilbert-style logic to coincide. In the rest of the paper, we recognize that the variable inclusion restrictions used to define variable inclusion companions of a logic $\langle\mathcal{L},\vdash\rangle$ are relations from $\mathcal{P}(\mathcal{L})$ to $\mathcal{L}$. This leads to a more general idea of a relational companion of a logical structure, a framework that we borrow from the field of universal logic. We end by showing that even Hilbert-style logics and the restricted rules companions of these can be brought under the umbrella of the general notions of logical structures and their relational companions that are discussed here.

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arXiv2024

Generalized explosion principles

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.

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arXiv2022

Lindenbaum-type Logical Structures

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 Lindenbaum-types 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.

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arXiv2022

Negation-Free Definitions of Paraconsistency

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arXiv2021

Restricted Rules of Inference and Paraconsistency

In this paper, we study two companions to 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 (IPC) and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of Pre-Rough logic (PRL) 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.

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arXiv2014

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{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, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ 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.

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