Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete (...) analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)

We discuss the origin and development of the universal logic project. We describe in particular the structure of UNILOG, a series of events created for promoting the universal logic project, with a school, a congress, a secret speaker and a contest. We explain how the contest has evolved into a session of logic prizes.

After explaining the interdisciplinary aspect of the series of events organized around the square of opposition since 2007, we discuss papers related to the 4th World Congress on the Square of Opposition which was organized in the Vatican at the Pontifical Lateran University in 2014. We distinguish three categories of work: those dealing with the evolution and development of the theory of opposition, those using the square as a metalogical tool to give a better understanding of various systems of logic (...) and those related with applications of the theory of opposition to conceptual analysis and pedagogy. (shrink)

Logics that have many truth values—more than just True and False—have been argued to be useful in the analysis of very many philosophical and linguistic puzzles. In this paper, which is a followup to, we will start with a particularly well-motivated four-valued logic that has been studied mainly in its propositional and first-order versions. And we will then investigate its second-order version. This four-valued logic has two natural three-valued extensions: what is called a “gap logic”, and what is called a (...) “glut logic”. We mention various results about the second-order version of these logics as well. And we then follow our earlier papers, where we had added a specific conditional connective to the three valued logics, and now add that connective to the four-valued logic under consideration. We then show that, although this addition is “conservative” in the sense that no new theorems are generated in the four-valued logic unless they employ this new conditional in their statement, nevertheless the resulting second-order versions of these logics with and without the conditional are quite different in important ways. We close with a moral for logical investigations in this realm. (shrink)

In this work, we propose a meaningful extension of description logics for non-monotonic reasoning. We introduce \, a logic allowing for the representation of and reasoning about both typical class-membership and typical instances of a relation. We propose a preferential semantics for \ in terms of partially-ordered DL interpretations which intuitively captures the notions of typicality we are interested in. We define a tableau-based algorithm for checking \ knowledge-base consistency that always terminates and we show that it is sound and (...) complete w.r.t. our preferential semantics. The general framework we here propose can serve as the foundation for further exploration of non-monotonic reasoning in description logics and similarly structured logics. (shrink)

Płonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities. Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Płonka sum representation in terms of dualisable algebras.

According to a very widespread interpretation of the metaphysical nature of quantum entities—the so-called Received View on quantum non-individuality—, quantum entities are non-individuals. Still according to this understanding, non-individuals are entities for which identity is restricted or else does not apply at all. As a consequence, it is said, such approach to quantum mechanics would require that classical logic be revised, given that it is somehow committed with the unrestricted validity of identity. In this paper we examine the arguments to (...) the inadequacy of classical logic to deal with non-individuals, as previously defined, and argue that they fail to make a good case for logical revision. In fact, classical logic may accommodate non-individuals in that specific sense too. What is more pressing for the Received View, it seems, is not a revision of logic, but rather a more adequate metaphysical characterization of non-individuals. (shrink)

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames, the semantic approach presented in this article is suitable for modeling applied logics, as it respects the intended interpretation of the logical operators. This is made possible by restricting admissible interpretations.

A possibility of defining logical constants within abstract logical frameworks is discussed, in relation to abstract definition of logical consequence. We propose using duals as a general method of applying the idea of invariance under replacement as a criterion for logicality.

We continue the investigations initiated in the recent papers where Bayes logics have been introduced to study the general laws of Bayesian belief revision. In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using the Bayes rule. In this paper we take the more general Jeffrey formula as a conditioning device and study the corresponding modal logics that we call Jeffrey logics, focusing mainly on the countable case. The containment relations among (...) these modal logics are determined and it is shown that the logic of Bayes and Jeffrey updating are very close. It is shown that the modal logic of belief revision determined by probabilities on a finite or countably infinite set of elementary propositions is not finitely axiomatizable. The significance of this result is that it clearly indicates that axiomatic approaches to belief revision might be severely limited. (shrink)

A general meta-logical theory is developed by considering ontological disputes in the systems of metaphysics. The usefulness of this general meta-logical theory is demonstrated by considering the case of the ontological dispute between the metaphysical systems of Lewis’ Modal Realism and Terence Parsons’ Meinongianism. Using Quine’s criterion of ontological commitments and his views on ontological disagreement, three principles of metalogic is formulated. Based on the three principles of metalogic, the notions of independent variable and dependent variable are introduced. Then, the (...) ontological dispute between Lewis’ Modal Realism and Terence Parsons’ Meinongianism are restated in the light of the principles of metalogic. After the restatement, Independent variable and dependent variables are fixed in both Lewis’ Modal Realism and Terence Parsons’ Meinongianism to resolve the dispute. Subsequently, a new variety of quantifiers are introduced which is known as functionally isomorphic quantifiers to provide a formal representation of the resolution of the dispute. The specific functionally isomorphic quantifier which is developed in this work is known as st-quantifier. It is indicated that how st-quantifier which is one of the functionally isomorphic quantifiers can function like existential quantifier. It is also shown that there is some kind of inconsistency which is unavoidable in stating the ontological disagreement and therefore, paraconsistent logic is a requirement in stating the ontological disputes. (shrink)

This paper begins the study of first-order functions, which are a generalization of truth-functions. The concepts of truth-table and systems of truth-functions, both introduced in propositional logic by Post, are also generalized and studied in the quantificational setting. The general facts about these concepts are given in the first five sections, and constitute a “general theory” of first-order functions. The central theme of this paper is the relation of definition among notions expressed by formulas of first-order logic. We emphasize that (...) logic is not concerned only with the consequence relation among notions expressed by formulas. It also attends to the relation of definition among notions, where a notion is defined from other notions. Sections 5 and 6 deal exclusively with the relation of definition among notions expressed by formulas of first-order logic. In these sections, we study the systems of first-order functions, which are the sets of first-order functions closed under definitions. Sections 7 and 8 are concerned with the relativization of first-order functions to a class of structures. The relativization to a class of structures is a fundamental operation which is used in order to relate the theory of first-order functions with set theory and first-order model theory, a subject which we have barely scratched the surface. The apparatus developed in this paper enables us to define what is a vehicle for the foundation of classical mathematics in set theory, and, in Sect. 8, we prove that first-order logic with one binary predicate variable is not a minimal vehicle for the foundation of classical mathematics in set theory. Sections 9 and 10 introduce further operations and ideals of first-order functions. Besides some results on the influence of the arguments of a first-order function, a result about definability is proved in Sect. 10.1. It is this theorem that provides necessary and sufficient conditions for a first-order function to be in a finitely generated ideal. In Sect. 11, this result is applied to the problem of predicate definability in classes of structures, the problem with which Beth’s theorem dealt in the case of elementary classes. (shrink)