Four progressively ambitious systems of modal propositional logic are set forth, together with decision procedures. The simultaneous employment of parenthesis notation and parenthesis-free notation, the dual use of symbols as primitive and defined, and the introduction of a new modal operator (the truth operator) are the principal devices used to effect the development of these logics. The first two logics turn out to be "the same" as two of von Wright's systems.

Recently, it has become a custom to treat questions as a game between two subjects. Unfortunately, one rarely goesbeyond the scheme of Questioner-Scientist and Answerer-Nature, althoughthe Interlocutor so conceived displays some undesirable features. This paperargues for the idea that logic of questions can be build as a logic of the gamebetween “knowledge resources” persons or theories, rather than errant Scientist and omniscient Nature. To this end the concept of epistemically-possibleworlds is discussed, which is conceived as analogous to that of possible (...) worldsin modal logic. And, furthermore, the concepts of relation of epistemic alternativeness and of epistemically-alternative worlds are introduced. On thisbasis a version of semantics for propositional, three-valued logic of questionsis offered and semantic proofs of some theses are given. (shrink)

The aim of this paper is to present the method of Socratic proofs for seven modal propositional logics: K5, S4.2, S4.3, S4M, S4F, S4R and G. This work is an extension of [10] where the method was presented for the most common modal propositional logics: K, D, T, KB, K4, S4 and S5.

It is shown that both classical and intuitionistic propositional logics of an associative binary modality are undecidable. The proof is based on the deduction theorem for these logics.

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.

This paper is a survey of results concerning embeddings of intuitionistic propositional logic and its extensions into various classical modal systems.

Standard possible world semantics for propositional modal languages ignore truth-value gaps. However, simple considerations suggest that it should not be so. In Section 1, I identify what I take to be a correct truth-clause for necessity under the assumption that some possible worlds are incomplete (i.e., "at" which some propositions lack a truth-value). In Section 2, I build a world semantics, the semantics of TV-models, for standard modal propositional languages, which agrees with the truth-clause for necessity (...) previously identified. Sections 3–5 are devoted to systematic concerns. In particular, in Section 4, Prior’s system Q (propositional version) is given a TV-models semantics and proved adequate (i.e., sound and complete) with respect to it. (shrink)

Review of the paper mentioned in the title.

Formalizing categorical propositions of traditional logic in the language of quantifiers and propositional functions is no straightforward matter, especially when modalities get involved. Starting...

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf {IGN}\). We give (...) the bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996 ) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, (...) 2019 ) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021 ) on the Lambek Calculus. (shrink)

In recent years modal syllogistic provided by 14th century logician John Buridan has attracted increasing attention of historians of medieval logic. The widespread use of quantified modal logic with the apparatus of possible worlds semantics in current analytic philosophy has encouraged the investigation of the relation of Buridan’s theory of modality with the modern developments of symbolic modal logic. We focus on the semantics of and the inferential relations among the propositions that underlie Buridan’s theory of (...) class='Hi'>modal syllogism. First, we review all inferences between propositions of necessity, possibility, contingency, and non-contingency, with or without quod est locution, that are valid in Buridan’s semantics, and offer a comprehensive diagrammatic representation that includes them all. We then ask the question if there is a way to model those results in first order modal logic. Three ways of formalizing Buridan’s propositions in quantified modal logic are considered. Comparison of inferences between the quantified formulas and Buridan’s propositions reveals that, when supplied with a suitable formalization, Buridan’s semantics of categorical statements and immediate inferences among them can be fully captured by the quantified modal system T. (shrink)

The article deals with the interpretation of propositional attitudes in the framework of modal predicate logic. The first part discusses the classical puzzles arising from the interplay between propositional attitudes, quantifiers and the notion of identity. After comparing different reactions to these puzzles it argues in favor of an analysis in which evaluations of de re attitudes may vary relative to the ways of identifying objects used in the context of use. The second part of the article (...) gives this analysis a precise formalization from a model-and proof-theoretic perspective. (shrink)

A. Kuznetsov considered a logic which extended intuitionistic propositional logic by adding a notion of 'irreflexive modality'. We describe an extension of Kuznetsov's logic having the following properties: (a) it is the unique maximal conservative (over intuitionistic propositional logic) extension of Kuznetsov's logic; (b) it determines a new unary logical connective w.r.t. Novikov's approach, i.e., there is no explicit expression within the system for the additional connective; (c) it is axiomatizable by means of one simple additional axiom scheme.

Classical propositional logic plays a prominent role in industrial applications, and yet the complexity of this logic is presumed to be non-feasible. Tractable systems such as depth-bounded boolean logics approximate classical logic and can be seen as a model for resource-bounded agents whose reasoning style is nonetheless classical. In this paper we first study a hierarchy of tractable logics that is not defined by depth. Then we extend it into a modal logic where modalities make explicit (...) the assumptions discharged in propositional proofs, thereby expressing blueprints for proofs. A natural deduction system is provided that permits to reason about and manage such proof blueprints. (shrink)

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p 1, . . . , p n , = (p 1, . . . , p n-1, p n ) intuitively states that the value of p n is determined by those of p 1, . . . , p n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic (...) is complete for nondeterministic exponential time.In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness.We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction.In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster. (shrink)

BERTRAND RUSSELL ON MODALITY AND LOGICAL RELEVANCE - SECOND EDITION of 2015. Praise for the first edition of 1999: "In the twenty-nine years since Russell's death, much of the major scholarship has drawn heavily on his manuscripts and unpublished correspondence. The author shows that the published Russell is capable of new interpretations; in particular, that modal notions such as possibility have a greater place in various aspects of his logical and philosophical thought than has been previously imagined." -Ivor Grattan-Guinness, (...) Foreword to the first edition. EXCERPT FROM PUBLISHED REVIEW: "Dejnožka's book is the first full-length study of modality in Russell. It is useful for its very full survey of passages in which Russell makes use of or alludes to modal notions. Dejnožka's command of Russell's huge output is indeed impressive and his utilization of it thorough...." - Nicholas Griffin, Studia Logica. EXCERPT FROM PUBLISHED REVIEW: "Dejnožka's book raises a very important point in the history of formal logic. Until now the major studies on this topic have drawn heavily on the development of classical logic as standardized by Gottlob Frege and Bertrand Russell. Dejnožka challenges the reader to open his mind for a new interpretation of Russell's work, in particular that modal and relevance notions have a greater place in his philosophy of logic than has been stressed before.... "Dejnožka rightly observes that many of Russell's insights on modality are a result of his discussions with Hugh MacColl, who was indeed the first to seriously attempt to develop formal modal logic. This particularly applies to Russell's conception of a modal logic without modal operators....That is, classical logic can be used to simulate modal expressions. Thus, the notions of (logical) necessity and possibility are not 'fundamental notions'....On this basis, Dejnožka develops a higher level of modality, where the quantification scope extends to the predicates yielding what Russell calls 'fully general propositions'.... "The best studied translation method is known as the standard translation, and it is quite compatible with Dejnožka's suggestions.... "Dejnožka's book is full of material which stimulates [one] to rethink Russell's philosophy of logic and...it is greatly to the author's credit that he brings to light such a wealth of crucial issues in the history and philosophy of logic. - Shahid Rahman, History and Philosophy of Logic. BOOK DESCRIPTION: This book is the only exhaustive study of Russell on modality and logical relevance ever written. This is the second edition, revised over a period of over sixteen years, and over twice as long. Russell initially rejects the possibility of a modal logic in an unpublished paper written ca. 1903-05 and read to an audience in 1905. But the very next year, he adopts the theory of modality he had rejected, and he repeats that theory in published works for many years to come. That theory, together with other texts scattered among Russell's writings, implies eight modal logics which Russell himself never expressly accepted. And Russell's express acceptance of the early Wittgenstein's theory of entailment ("following from") as truth-ground containment implies a deductive relevance logic in Russell's writings which Russell himself never expressly accepted. The book ends with a look at John Maynard Keynes and Anglo-American evidence law as the origins of Russell's theory of probability as degree of logical relevance. Thus the book is written toward the reunion of modern classical logicians with modal logicians and relevantists. On the reunion of modern classical logicians with relevantists, please see also The Concept of Relevance and the Logic Diagram Tradition (2012). Thus I hope to have created two new fields of Russell studies. I may have created a new legal field as well: the history of logical relevance in evidence law: 132 pages (368-500) in chapter 10. (shrink)

This paper treats a kind of a modal logic based on the intuitionistic propositional logic which arose from the provability predicate in the first order arithmetic. The semantics of this calculus is presented in both a relational and an algebraic way.Completeness theorems, existence of a characteristic model and of a characteristic frame, properties of FMP and FFP and decidability are proved.

We show how mereotopological notions can be expressed by extending intuitionistic propositional logic with propositional quantification and a strong modal operator. We first prove completeness for the logics wrt Kripke models; then we trace the correspondence between Kripke models and topological spaces that have been enhanced with an explicit notion of expressible region. We show how some qualitative spatial notions can be expressed in topological terms. We use the semantical and topological results in order to show (...) how in some extensions of the logics it is possible to express connectedness, non-emptiness and a set of jointly exhaustive, pairwise disjoint, binary relations that play a significant role in qualitative spatial reasoning. (shrink)

We introduce modal propositional substructural logics with strong negation, and prove the completeness theorems (with respect to Kripke models) for these logics.

We introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation (...) of quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here backwards from the cube (...) to the line and the square; and then forwards from the cube to the four-dimensional polyhedron. (shrink)