C. I. Lewis on Possible Worlds

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

In a very short time Saul Kripke provided a suitable and rigorous semantics for different axiomatic modal systems and established a series of related results. Many key ideas were already in the air in the late Fifties, but it was Kripkean articles’ merit to system atically introduce comprehensive devices and solutions. Later on, the spreading of possible-worlds semantics massively changed the approach to modal logic, which enormously increased in popularity after that. Since Kripke’s work in modal logic is central to the development of the discipline, the aim of this essay is to present the fundamental results published between 1959 and 1965. Indeed, it was in such a brief and early phase of his career that Kripke was able to conceive the main novelties that would become central to the subsequent academic debates about modality. Here, their presentation will follow the original historical progressive introduction. Particular attention will be given to the interconnection between articles, their similarities in structure and the unified analysis produced by means of them. It actually appears quite impressive that, already in 1959, Kripke seemed to have planned all the developments he would present, one after the other, in the following years. First, an overview of the background where Kripke’s ideas start to rise is given. Then, each text’s results are individually briefly analysed.

1 Modal logic: a semantic perspective

Bisimulation is a central concept in the model theory of modal logic with extensive computational applications. It is a relation between two models in which related states have identical atomic properties and matching transition possibilities. Bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula. In recent years, fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities. In this paper, we define the notion of fuzzy bisimulation and prove that, in a fuzzified version, it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic.

Clarence Irving Lewis (b. 1883–d. 1964) is arguably the most important philosopher bridging the pragmatism of the golden age of William James and Charles Sanders Peirce and the analytic quasi-pragmatism of philosophers like W. V. Quine, Nelson Goodman, Wilfrid Sellars, and Hilary Putnam (the first three of whom were taught by him). Lewis’s philosophy as a whole reveals a unified systematic development from his dissertation in 1910, his early work in logic, the development of his epistemology in the 1920s and 1930s, his account of value theory in the 1940s and 1950s, culminating in his work in ethics, which occupied him until his death. Along the way he offered a devastating critique of American absolute idealism and offered a rich epistemology grounded in a Peircean kind of pragmatism. Early in his career Lewis wrote the first the history of logic in English, and, critical of the paradoxes of material implication, he developed an account of strict implication and a set of successively stronger modal logics, the S systems becoming the father of modern modal logic. Lewis was the most influential American philosopher from the mid-1930s until after his retirement in the 1950s. His work helped shape American philosophy as an academic endeavor and contributor to the growing acceptance of rigorous philosophical analysis and European logical empiricism. Lewis spent practically his entire career at Harvard University, bridging the Harvard of James and Royce and the modern department of Quine and Goodman. During his career he wrote six books and a hundred or so papers and reviews. A student of Josiah Royce, William James, and Ralph Barton Perry, a contemporary of Hans Reichenbach, Rudolf Carnap, and the logical empiricists of the 1930s and 1940s, and the teacher of Quine, William Frankena, Goodman, Roderick Chisholm, Roderick Firth, Sellars, and others, he played a pivotal role in shaping the marriage between pragmatism and empiricism that has come to dominate much of current analytic philosophy. Despite his significant contributions, his work soon became neglected and misinterpreted, lost in the influx of interest in Wittgenstein and the philosophy of language. Fortunately, this neglect has begun to wane.

We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

Stig Kanger — born of Swedish parents in China in 1924 — was professor of Theoretical Philosophy at Uppsala University from 1968 until his death in 1988. He received his Ph. D. from Stockholm University in 1957 under the supervision of Anders Wedberg. Kanger’s dissertation, Provability in Logic, was remarkably short, only 47 pages, but also very rich in new ideas and results. By combining Gentzen-style techniques with a model theory a la Tarski, Kanger obtained new and simplified proofs of central metalogical results of classical predicate logic: Godel’s completeness theorem, Lowenheim-Skolem’s theorem and Gentzen’s Hauptsatz. The part that had the greatest impact, however, was the 15 pages devoted to modal logic. There Kanger developed a new semantic interpretation for quantified modal logic which had a close family resemblance to semantic theories that were developed around the same time by Jaakko Hintikka, Richard Montague and Saul Kripke (independently of each other and independently of Kanger).

Ultraproducts and possible worlds semantics in institutions

Even if Leibniz didn’t have the opportunity to actually conceive an explicit modal logic system, remains the fact that he had worked out a modal metaphysics, of which the inaugural act, in his Elementa juris naturalis (c. 1671) was an obvious reference to the Apulean square of opposition. Later, scholars acknowledged in this passage probably one of the first sketch of deontic logic of norms. His modal metaphysics rather deals with the so-called alethic modalities, sometimes expounded through a language such as R.M. Adams wondered whether Leibniz could be “a sort of grandfather of possible worlds semantics for modal logic”. In the following study, the Apulean square is used as a hermeneutic tool: however, is the square really well-fitted to express some basic implications of the modal metaphysics? The most remarkable point is that Leibniz was well aware of the K-distributive axiom □(p⊃q)⊃(□p⊃□q) common to the main modal systems today. This awareness dissuaded him to trust the too easy solution of the necessitarianism issue grounded on the well-known distinction coming from Boethius between a “necessitas consequentiae” and a “necessitas consequentis”.

The classical analytic tableau method has been extended successfully to modal logics (see e.g. Fitting 1983; Fitting 1993; Gore 1992) and also to relevant and paraconsistent logics Bloesch 1993a; Bloesch 1993b. The classical connection method has been extended to modal and intuitionistic logics Wallen 1990, and the purpose of this paper is to investigate whether a similar adaptation to relevant logic is possible. A hybrid method is developed for B+, with a specific solution to the “multiplicity problem”, as in the technique of modal semantic diagrams introduced in Hughes and Cresswell 1968. Proofs of soundness and completeness are also given.

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators - `like it is necessary' in philosophy, `it is believed' in cognitive science, `it is provable' in mathematics and `it is true after executing a program' in computer science. It is an advanced text which starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantical and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist can use the book as a source of references. For the first time results and methods of many directions in propositional modal logic - from completeness and duality to algorithmic problems - are collected and systematically presented in one volume. Unlike other books, modal logic is treated here as a uniform theory rather than a collection of a few particular systems. It is the only book presenting the theory of superintuitionistic logics.

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.

This paper introduces a two-dimensional modal logic to represent agents’ concurrent common knowledge in distributed systems. Unlike common knowledge, concurrent common knowledge is a kind of agreement reachable in asynchronous environments. As a proper semantics to concurrent common knowledge we present the closed sub-product of modal logics. We axiomatize the presented logic issuing an idea of the soundness and completeness proofs.

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.