Refined Tableau Systems for Some Modal Logics of Confluence

Full classical S5 in natural deduction with weak normalization

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

Logical Consequence in Modal Logic: Alternative Semantic Systems for Normal Modal Logics

Beall and Restall 2000; 2001; 2006 advocate a comprehensive pluralist approach to logic, which they call Logical Pluralism, according to which there is not one true logic but many equally acceptable logical systems. They maintain that Logical Pluralism is compatible with monism about metaphysical modality, according to which there is just one correct logic of metaphysical modality. Wyatt 2004 contends that Logical Pluralism is incompatible with monism about metaphysical modality. We first suggest that if Wyatt were right, Logical Pluralism would be strongly implausible because it would get upside down a dependence relation that holds between metaphysics and logic of modality. We then argue that Logical Pluralism is prima facie compatible with monism about metaphysical modality.

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

Sören Halldén. On the logic of ‘better.’ Library of Theoria, no. 2. C. W. K. Gleerup, Lund, and Ejnar Munksgaard, Copenhagen, 1957, 112 pp. - Lennart Åqvist. Deontic logic based on a logic of ‘better.’Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23–26 August, 1962, Acta philosophica Fennica, no. 16, Helsinki1963, pp. 285–290. - Mark Fisher. A three-valued calculus for deontic logic. Theoria (Lund), vol. 27 (1961), pp. 107–118. - Lennart Åqvist. Postulate sets and decision procedures for some systems of deontic logic. Theoria (Lund), vol. 29 (1963), pp. 154–175. - Volume 31 Issue 2

Substitution Frege and extended Frege proof systems in non-classical logics

This article is a sequel to an article titled “A New Direction in AI – Toward a Computational Theory of Perceptions”, which appeared in the Spring 2001 issue of AI Magazine (volume 22, No. 1, 73-84). The concept of precisiated natural language (PNL) was briefly introduced in that article, and PNL was employed as a basis for computation with perceptions. In what follows, the conceptual structure of PNL is described in greater detail, and PNL’s role in knowledge representation, deduction, and concept definition is outlined and illustrated by examples. What should be understood is that PNL is in its initial stages of development and that the exposition that follows is an outline of the basic ideas that underlie PNL rather than a definitive theory. A natural language is basically a system for describing perceptions. Perceptions, such as perceptions of distance, likelihood, relevance, and most other attributes of physical and mental objects are intrinsically imprecise, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information. In this perspective, the imprecision of natural languages is a direct consequence of the imprecision of perceptions [44, 45]. How can a natural language be precisiated – precisiated in the sense of making it possible to treat propositions drawn from a natural language as objects of computation? This is what PNL attempts to do. In PNL, precisiation is accomplished through translation into what is termed a precisiation language. In the case of PNL, the precisiation language is the generalized-constraint language (GCL), a language whose elements are so-called generalized constraints and their combinations. What distinguishes GCL from languages such as Prolog, LISP, SQL, and, more generally, languages associated with various logical systems, for example, predicate logic, modal logic, and so on, is its so much higher expressive power. The conceptual structure of PNL mirrors two fundamental facets of human cognition, (a) partiality and (b) granularity [43]. Partiality relates to the fact that most human concepts are not bivalent, that is, are a matter of degree. Thus, we have partial understanding, partial truth, partial possibility, partial certainty, partial similarity, and partial relevance, to cite a few examples. Similarly, granularity and granulation relate to clumping of values of attributes, forming granules with words as labels, for example, young, middle-aged, and old as labels of granules of age.

In this paper I give conditions under which a matrix characterisation of validity is correct for first order logics where quantifications are restricted by statements from a theory. Unfortunately the usual definition of path closure in a matrix is unsuitable and a less pleasant definition must be used. I derive the matrix theorem from syntactic analysis of a suitable tableau system, but by choosing a tableau system for restricted quantification I generalise Wallen's earlier work on modal logics. The tableau system is only correct if a new condition I call “alphabetical monotonicity” holds. I sketch how the result can be applied to a wide range of logics such as first order variants of many standard modal logics, including non-serial modal logics.

Modal separation logics are formalisms that combine modal operators to reason locally, with separating connectives that allow to perform global updates on the models. In this work, we design Hilbert-style proof systems for the modal separation logics \(\text {MSL} ^{}(*,\langle \ne \rangle )\) and \(\text {MSL} ^{}(*,\Diamond )\), where \(*\) is the separating conjunction, \(\Diamond \) is the standard modal operator and \(\langle \ne \rangle \) is the difference modality. The calculi only use the logical languages at hand (no external features such as labels) and take advantage of new normal forms and of their axiomatisation.

Non-classical logic and ontological non-commitment, avoiding abstract objects through modal operators

In this paper I shall present three solutions to the Knower Paradox which, despite important points in common, differ in several respects. The first solution, proposed by C. A. Anderson [1] is a hierarchical solution, developed in the framework of first-order arithmetic. However I will try to show that this solution is based on an incorrect argument. The second solution, inspired by a book of R.M. Smullyan [14], is developed in the framework of modal logic and it is based on the idea of interpreting one of the basic systems of the modal logic of provability in an epistemic way. I shall give arguments in support of this solution. The third solution, proposed by P. Egree [8] attempts to connect the first and the second solutions. I will show that this attempt fails for philosophical and formal reasons.

Types are fundamental for conceptual domain modeling and knowledge representation in computer science. Frequently, monadic types used in domain models have as their instances objects (endurants, continuants), i.e., entities persisting in time that experience qualitative changes while keeping their numerical identity. In this paper, I revisit a philosophically and cognitively well-founded theory of object types and propose a system of modal logics with restricted quantification designed to formally characterize the distinctions and constraints proposed by this theory. The formal system proposed also addresses the limitations of classical (unrestricted extensional) modal logics in differentiating between types that represent mere properties (or attributions) ascribed to individual objects from types that carry a principle of identity for those individuals (the so-called sortal types). Finally, I also show here how this proposal can complement the theory of conceptual spaces by offering an account for kind-supplied principles of cross-world identity. The account addresses an important criticism posed to conceptual spaces in the literature and is in line with a number of empirical results in the literature of cognitive psychology.KeywordsCross-world identityIdentity in conceptual spacesEssential and contingent classificationSortal logicOntology

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular IΔ0+EXP. The method is adapted to obtain a similar completeness result for the Rosser logic.