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This book collects the refereed proceedings of the 6th Indian Conference on Logic and Its Applications, ICLA 2015, held in Mumbai, India, in January 2015. The volume contains 13 full revised papers along with 3 invited talks presented at the conference. The papers were selected after rigorous review, from 23 submissions. They cover topics related to pure and applied formal logic, foundations and philosophy of mathematics and the sciences, set theory, model theory, proof theory, areas of theoretical computer science, artificial intelligence, systems of logic in the Indian tradition, and other disciplines which are of direct interest to mathematical and philosophical logic.



Homotopy Type Theory

Homotopy Type Theory is a new, homotopical interpretation of constructive type theory. It forms the basis of the recently proposed Univalent Foundations of Mathematics program. Combined with a computational proof assistant, and including a new foundational axiom – the Univalence Axiom – this program has the potential to shift the theoretical foundations of mathematics and computer science, and to affect the practice of working scientists. This talk will survey the field and report on some of the recent developments.
Steve Awodey

The Relevance of Relevance to Relevance Logic

I explore the question of whether the concept of relevance is relevant to the study of what Anderson and Belnap call “relevance logic.” The answer should be “Of course!” But there are some twists and turns, as is shown by the fact that it has taken over 50 years to get here. Despite protests by R. K. Meyer that the concept of relevance is not part of what he calls “relevant logic,” I suggest and defend interpreting the Routley–Meyer ternary accessibility relation using information states a, b, c, so Rabc means “in the context a, b is relevant to c.” Motivations are provided from Sperber and Wilson’s work in linguistics on relevance.
J. Michael Dunn

Logic-Automata Connections for Transformations

Pioneered by Büchi, Elgot and Trakhtenbrot, connections between automata and logics that define languages of words and trees are now well-established. During the last decade, some of these powerful connections have been extended to binary relations (transformations) of words and trees. This paper is a survey of known automata-logic connections for transformations.
Emmanuel Filiot

Truths about Simpson’s Paradox: Saving the Paradox from Falsity

There are three questions associated with Simpson’s paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP? and (iii) How to proceed when confronted with SP? An adequate analysis of the paradox starts by distinguishing these three questions. Then, by developing a formal account of SP, and substantiating it with a counter-example to causal accounts, we argue that there are no causal factors at play in answering questions (i) and (ii). Causality enters only in connection with action.
Prasanta S. Bandyopadhyay, R. Venkata Raghavan, Don Wallace Dcruz, Gordon Brittan

Some Instances of Graded Consequence in the Context of Interval-Valued Semantics

This paper proposes some instances of graded consequence relation where the object language formulae are interpreted by sub-intervals of [0, 1]. These instances represent different attitudes of decision making that may be called conservative, liberal, and moderate.
Soma Dutta, Benjamín R. C. Bedregal, Mihir Kr. Chakraborty

Neighborhood Contingency Logic

A formula is contingent, if it is possibly true and possibly false; a formula is non-contingent, if it is not contingent, i.e., if it is necessarily true or necessarily false. In this paper, we propose a neighborhood semantics for contingency logic, in which the interpretation of the non-contingency operator is consistent with its philosophical intuition. Based on this semantics, we compare the relative expressivity of contingency logic and modal logic on various classes of neighborhood models, and investigate the frame definability of contingency logic. We present a decidable axiomatization for classical contingency logic (the obvious counterpart of classical modal logic), and demonstrate that for contingency logic, neighborhood semantics can be seen as an extension of Kripke semantics.
Jie Fan, Hans van Ditmarsch

Hierarchies in Inclusion Logic with Lax Semantics

We study the expressive power of fragments of inclusion logic under the so-called lax team semantics. The fragments are defined either by restricting the number of universal quantifiers or the arity of inclusion atoms in formulae. In case of universal quantifiers, the corresponding hierarchy collapses at the first level. Arity hierarchy is shown to be strict by relating the question to the study of arity hierarchies in fixed-point logics.
Miika Hannula

A Modal Logic for Non-deterministic Information Systems

In this article, we propose a modal logic for non-deterministic information systems. A deductive system for the logic is presented and corresponding soundness and completeness theorems are proved. The logic is also shown to be decidable.
Md. Aquil Khan

Tableaux for Non-normal Public Announcement Logic

This paper presents a tableau calculus for two semantic interpretations of public announcements over monotone neighbourhood models: the intersection and the subset semantics, developed by Ma and Sano. We show that both calculi are sound and complete with respect to their corresponding semantic interpretations and, moreover, we establish that the satisfiability problem of this public announcement extensions is NP-complete in both cases. The tableau calculi has been implemented in Lotrecscheme.
Minghui Ma, Katsuhiko Sano, François Schwarzentruber, Fernando R. Velázquez-Quesada

A Pragmatistic Approach to Propositional Knowledge Based on the Successful Behavior of Belief

Every belief has a life that goes from the agent having the belief now, the transmission of the belief to other agents, and the persistence of the belief through time. In this article we propose the idea that the belief can be said to be successful in relation to any of these respects. We will call them, respectively, the first, second, and third person perspective on knowledge and investigate the requisite properties of these three perspectives.
We do not base our approach on the notion of truth as is common, or on the notion of justification, which is another basis. Our concern is not with knowledge as corresponding to truth but knowledge as corresponding to stable belief.
Aránzazu San Ginés, Rohit Parikh

Büchi Automata Optimisations Formalised in Isabelle/HOL

In applications of automata theory, one is interested in reductions in the size of automata that preserve the recognised language. For Büchi automata, two optimisations have been proposed: bisimulation reduction, which computes equivalence classes of states and collapses them, and α-balls reduction, which collapses strongly connected components (SCCs) of an automaton that only contain one single letter as edge label. In this paper, we present a formalisation of these algorithms in Isabelle/HOL, providing a formally verified implementation.
Alexander Schimpf, Jan-Georg Smaus

Nyāya’s Logical Model for Ascertaining Sound Arguments

The logical debate in India of the first millennia AD revolved around the concept of pramāna. The term pramāna was taken to mean ’the criterion of knowledge’. Current researchers of Indian philosophy are certain that Indian logicians all agreed that pramāna is the sound operation of the mental processes which produce mental knowledge episodes. Conversely, according to my research, Nyāya thinkers believed the criteria of knowledge are the rules of the use of things in everyday habitual behaviors. The issue which stood at the center of the Indian logical debate, I wish to suggest, was the following: On the one hand, there were thinkers who believed the rules of logic were prior to and independent of habitual everyday human behaviors. On the other hand, there were the Naiyāyikas who believed the rules of logic were derived from habitual everyday human behaviors and the rules of usage they provided.
Jaron Schorr

Negative Existentials and Non-denoting Terms

Logical and semantical issues surrounding non-denoting terms have been investigated since ancient times, in both the Western and Indian philosophical traditions. And in a more applied formal setting, such issues have also gained importance in constructive mathematics, as well as computer science and software engineering. The paper first presents a strategic exploration of logical treatments of reference failure in Western thought, and then goes on to provide a comparative examination of the issue in the Indian tradition, particularly with respect to the dispute between the Yogācāra-Sautrāntika school of Buddhism and the Nyāya school of Hinduism. The paper concludes by advancing a formalization of the Buddhist apoha semantical theory in terms of a dualdomain Free logic.
Paul Schweizer

Ordinals in an Algebra-Valued Model of a Paraconsistent Set Theory

This paper deals with ordinal numbers in an algebra-valued model of a paraconsistent set theory. It is proved that the collection of all ordinals is not a set in this model which is dissimilar to the other existing paraconsistent set theories. For each ordinal α of classical set theory α-like elements are defined in the mentioned algebra-valued model whose collection is not singleton. It is shown that two α-like elements (for same α) may perform conversely to validate a given formula of the corresponding paraconsistent set theory.
Sourav Tarafder

Extending Carnap’s Continuum to Binary Relations

We investigate a binary generalization of Carnap’s Continuum of Inductive Methods based on a version of Johnson’s Sufficientness Postulate for polyadic atoms and determine the probability functions that satisfy it.
Alena Vencovská

Representing Imperfect Information of Procedures with Hyper Models

When reasoning about knowledge of procedures under imperfect information, the explicit representation of epistemic possibilities blows up the S5-like models of standard epistemic logic. To overcome this drawback, in this paper, we propose a new logical framework based on compact models without epistemic accessibility relations for reasoning about knowledge of procedures. Inspired by the 3-valued abstraction method in model checking, we introduce hyper models which encode the imperfect procedural information. We give a highly non-trivial 2-valued semantics of epistemic dynamic logic on such models while validating all the usual S5 axioms. Our approach is suitable for applications where procedural information is ‘learned’ incrementally, as demonstrated by various examples.
Yanjing Wang

Erratum: Logic and Its Applications

In an earlier online version of this volume, the name of the second author was misspelled on the inner title pages and cover. This has been corrected.
Mohua Banerjee, Shankara Narayanan Krishna


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