Kripke’s Modal Logic: A Historical Study

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

1 Modal logic: a semantic perspective

Narrative Semantics, Modal Logic and Fictional Possible Worlds

In the debate about Composition as Identity (CI), a recurring pattern is to ask whether a certain feature of identity is also instantiated by composition. This recurring pattern is followed when, for example, the question is asked whether a whole and its parts are indiscernible. In following this pattern, it is methodologically desirable to assume the most standard account of the philosophical problems at stake. However, when the necessity of identity and the problem whether composition is as necessary as identity are at stake, the literature about CI often violates this methodological principle, and resorts to non-standard views about modality, such as counterpart theory. In this paper, we purport to remedy this anomaly and to assess CI on the background of a standard, broadly Kripkean view of modality, and in particular of the contention that a single entity exists in more than one possible world. Given this contention, the backer of CI is forced to relativize composition and, as a consequence, identity to possible worlds, thereby introducing a non-standard kind of identity. We will discuss the charge of adhocness which might be raised against the resulting variety of CI.

Many writers have commented on the revival of philosophical theology that began in analytic circles in the 1960s. Those of us who received our training in the 50s were reacting to two things. First, our predecessors’ preoccupation with the question of religious discourse’s meaningfulness. While not necessarily disagreeing with Basil Mitchell’s, say, or John Hick’s responses to Antony Flew’s charge of ‘death by a thousand qualifications’, most of us lost interest in the debate. Second, a conviction that Hume’s and Kant’s vaunted critiques of natural theology didn’t withstand careful scrutiny. On the positive side, developments in modal logic, probability theory, and so on, offered tools for introducing a new clarity and rigour to traditional disputes. Alvin Plantinga’s work on the ontological argument, or Richard Swinburne’s use of Bayesian techniques to formulate his cumulative argument for God’s existence, are paradigmatic examples.

Previous intuitionistic justification logics included explicit justifications for all admissible rules of intuitionistic logic in order to get completeness with respect to provability semantics. We present the justification logic iJT4, which does not have these additional justification terms. We establish that iJT4 is complete with respect to modular models and that there is a realization of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit version of intuitionistic S4.

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).

This article opposes a view widely accepted in studies concerning the history of modal logic, according to which (i) the approach of C. I. Lewis towards constructing modern modal logic was purely syntactical (i.e. limited to the construction of axiomatic systems S1–S5 of propositional modal logic), and (ii) the notion of a possible world was incorporated into modern logic and philosophy mainly by authors such as Rudolf Carnap and Saul Kripke. The article presents Lewis' definition of a possible world, and his formulation of the truth-conditions of statements containing strict implication as their main connective in terms of possible worlds. The main question of the article is whether it is possible to consider Lewis' work in this area as an early stage of the development of possible world semantics, and if so, in what sense? The article concludes by answering affirmatively, due to soundness and completeness proofs with respect to S5 using Lewis' semantics.

Beliefs have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but the...

We consider the satisfiability problem for modal logic over classes of structures definable by universal first-order formulas with three variables. We exhibit a simple formula for which the problem is undecidable. This improves an earlier result in which nine variables were used. We also show that for classes defined by three-variable, universal Horn formulas the problem is decidable. This subsumes decidability results for many natural modal logics, including T, B, K4, S4, S5.

Saul Kripke's first contributions to philosophy were his papers on modal logic, which quickly made possible worlds semantics a working tool of philosophical logicians and then of philosophers more generally. The received view of the semantics of proper names around 1970 was that the correct account lay somewhere among the theories descended from classic papers of Gottlob Frege and Bertrand Russell. One of Kripke's most influential examples of a class of necessary truths which could come to be known in an a posteriori fashion were the very examples of identity sentences that Frege used to introduce the notion of the sense of a name in the first place. In the Naming and Necessity lectures, Kripke made use of the distinction between speaker's reference and semantic reference to explain some of our intuitions about the use of names.

The Semantics of Entailment

We will begin our study of modal logic with a basic system called K in honor of the famous logician Saul Kripke. K serves as the foundation for a whole family of systems. Each member of the family results from strengthening K in some way. Each of these logics uses its own symbols for the expressions it governs. For example, modal (or alethic) logics use □ for necessity, tense logics use H for what has always been, and deontic logics use O for obligation. The rules of K characterize each of these symbols and many more. Instead of rewriting K rules for each of the distinct symbols of modal logic, it is better to present K using a generic operator. Since modal logics are the oldest and best known of those in the modal family, we will adopt □ for this purpose. So □ need not mean necessarily in what follows. It stands proxy for many different operators, with different meanings. In case the reading does not matter, you may simply call □A ‘box A’.

In this paper, we present a Satisfiability Modulo Theory based (SMT-based) bounded model checking (BMC) method for Timed Real-Weighted Interpreted Systems and for the existential fragment of the Weighted Epistemic Computation Tree Logic. SMT-based bounded model checking consists in translating the existential model checking problem for a modal logic and for a model to the satisfiability problem of a quantifier-free first-order formula. We have implemented the SMT-BMC method and performed the BMC algorithm on Timed Weighted Generic Pipeline Paradigm benchmark. The preliminary experimental results demonstrate the feasibility of the method. To perform the experiments, we used the state of the art SMT-solver Z3.

Parasocial involvement, an illusion of intimacy between media consumers and media professionals, is one of the most intriguing communication phenomena ever witnessed in media Hyphen saturated societies. First discussed by Horton and Wohl in 1956, this concept has since occupied a prominent place in communication studies. While enjoying longevity, its development, much like its many counterparts in social science, has followed an interesting, inverted U, pattern. Initially, it did not draw much attention from communication researchers. This might have been due to the fact that it was overshadowed by the then-dominant ”effects” paradigm. It's being published in a psychiatry journal, also, may have delayed its debut in the communication arena. Then, its popularity started growing around early 1970s, escalating throughout the decades. The zest on the topic, however, has arrived at a plateau in early to mid- 1990s and since then has been on the decline. The primary goal of this essay is to organize, review, and interpret parasocial research published since the seminal work of Ho rton and Wohl (1956). The first part of the essay summarizes the central notions of parasocical involvement as presented by Horton and Wohl. It then traces the evolution of subsequent research-its conceptual and methodological approaches, major findings, difficulties, and development of new lines of work. Towards the end is a series of suggestions that are logically derived from the preceding review. Overall, the essay is intended to re-vitalize the paradigm through a comprehensive and systematic understanding of related issues.