Possible Worlds Semantics and Fiction

The canonical version of possible worlds semantics for story prefixes is due to David Lewis. This paper reassesses Lewis's theory and draws attention to some novel problems for his account.     

From Modal Discourse to Possible Worlds

The possible-worlds semantics for modality says that a sentence is possibly true if it is true in some possible world. Given classical prepositional logic, one can easily prove that every consistent set of propositions can be embedded in a ‘maximal consistent set’, which in a sense represents a possible world. However the construction depends on the fact that standard modal logics are finitary, and it seems false that an infinite collection of sets of sentences each finite subset of which is intuitively ‘possible’ in natural language has the property that the whole set is possible. The argument of the paper is that the principles needed to shew that natural language possibility sentences involve quantification over worlds are analogous to those used in infinitary modal logic.     

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

If is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate , we tackle both problems. Given a frame <W,R> consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret at every world in such a way that A holds at a world wW if and only if A holds at every world vW such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several paradoxes (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the -inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.     

Semantics for the Logic of Essence

This paper provides a possible worlds semantics for the system of the author's previous paper The Logic of Essence. The basic idea behind the semantics is that a statement should be taken to be true in virtue of the nature of certain objects just in case it is true in any possible world compatible with the nature of those objects. It is shown that a slight variant of the original system is sound and complete under the proposed semantics.     

Kripke Frame with Graded Accessibility and Fuzzy Possible World Semantics

A possible world structure consist of a set W of possible worlds and an accessibility relation R. We take a partial function r(·,·) to the unit interval [0, 1] instead of R and obtain a Kripke frame with graded accessibility r Intuitively, r(x, y) can be regarded as the reliability factor of y from x We deal with multimodal logics corresponding to Kripke frames with graded accessibility in a fairly general setting. This setting provides us with a framework for fuzzy possible world semantics. The basic propositional multimodal logic gK (grated K) is defined syntactically. We prove that gK is sound and complete with respect to this semantics. We discuss some extensions of gK including logics of similarity relations and of fuzzy orderings. We present a modified filtration method and prove that gK and its extensions introduced here are decidable.     

Everyday Life Reasoning, Possible Worlds and Cultural Processes

Discussing Faiciuc’s paper, I first tackle the problem of fallacies in deductive reasoning showing how, in a possible world theory, non correct forms of reasoning can be useful strategies for discovery, providing these strategies remain at a hypothesis level. Secondly, everyday reasoning and its specificity in comparison to logical-normative one are analyzed. This topic stresses the notion of interpretation and, in this context, the role of the community and of cultural canons shared by the subject. From this point of view, reasoning does not occur, only, in the brain of a person but in everyday exchanges occurring between individuals and the history of their community.

Modal Logic for Other-World Agnostics: Neutrality and Halldén Incompleteness

The logic of ‘elsewhere,’ i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions (explaining the use of the word ‘elsewhere’), as well as in the case in which they represent moments of time. This logic is applied here to the alethic modal case, in which the points are thought of as possible worlds, with the suggestion that its deployment clarifies aspects of a position explored by John Divers un-der the name ‘modal agnosticism.’ In particular, it makes available a logic whose Halldén incompleteness explicitly registers the agnostic element of the position – its neutrality as between modal realism and modal anti-realism.     

Algebraic Semantics for Deductive Systems

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics.     

Refutations,Proofs, and Models in the Modal Logic K4

In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk's logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.     

The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Propositions, Sets, and Worlds

If we agree with Michael Jubien that propositions do not exist, while accepting the existence of abstract sets in a realist mathematical ontology, then the combined effect of these ontological commitments has surprising implications for the metaphysics of modal logic, the ontology of logically possible worlds, and the controversy over modal realism versus actualism. Logically possible worlds as maximally consistent proposition sets exist if sets generally exist, but are equivalently expressed as maximally consistent conjunctions of the same propositions in corresponding sets. A conjunction of propositions, even if infinite in extent, is nevertheless itself a proposition. If sets and hence proposition sets exist but propositions do not exist, then whether or not modal realism is true depends on which of two apparently equivalent methods of identifying, representing, or characterizing logically possible worlds we choose to adopt. I consider a number of reactions to the problem, concluding that the best solution may be to reject the conventional model set theoretical concept of logically possible worlds as maximally consistent proposition sets, and distinguishing between the actual world alone as maximally consistent and interpreting all nonactual merely logically possible worlds as submaximal. I am grateful to the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS), Royal Netherlands Academy of Arts and Sciences (KNAW), for supporting this among related research projects in philosophical logic and philosophy of mathematics during my Resident Research Fellowship in 2005-2006.     

Validity and Necessity

In this paper I argue against the commonly received view that Kripke’s formal Possible World Semantics (PWS) reflects the adoption of a metaphysical interpretation of the modal operators. I consider in detail Kripke’s three main innovations vis-à-vis Carnap’s PWS: a new view of the worlds, variable domains of quantification, and the adoption of a notion of universal validity. I argue that all these changes are driven by the natural technical development of the model theory and its related notion of validity: they are dictated by merely formal considerations, not interpretive concerns. I conclude that Kripke’s model theoretic semantics does not induce a metaphysical reading of necessity, and is formally adequate independently of the specific interpretation of the modal operators.     

Quantification over Sets of Possible Worlds in Branching-Time Semantics

Temporal logic is one of the many areas in which a possible world semantics is adopted. Prior's Ockhamist and Peircean semantics for branching-time, though, depart from the genuine Kripke semantics in that they involve a quantification over histories, which is a second-order quantification over sets of possible worlds. In the paper, variants of the original Prior's semantics will be considered and it will be shown that all of them can be viewed as first-order counterparts of the original semantics.     

Ideal Worlds — Wishful Thinking in Deontic Logic

The ideal world semantics of standard deontic logic identifies our obligations with how we would act in an ideal world. However, to act as if one lived in an ideal world is bad moral advice, associated with wishful thinking rather than well-considered moral deliberation. Ideal world semantics gives rise to implausible logical principles, and the metaphysical arguments that have been put forward in its favour turn out to be based on a too limited view of truth-functional representation. It is argued that ideal world semantics should be given up in favour of other, more plausible uses of possible worlds for modelling normative subject-matter.     

Probability Semantics for Quantifier Logic

By supplying propositional calculus with a probability semantics we showed, in our 1996, that finite stochastic problems can be treated by logic-theoretic means equally as well as by the usual set-theoretic ones. In the present paper we continue the investigation to further the use of logical notions in probability theory. It is shown that quantifier logic, when supplied with a probability semantics, is capable of treating stochastic problems involving countably many trials.     

On Permutation in Simplified Semantics

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) (there called c6) and permutation (A → (B → C)) → (B → (A → C)) (there called c7). We show that the modelling conditions for assertion and permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections.     

The Genesis of Hi-Worlds: Towards a Principle-Based Possible World Semantics

A Leibnizian semantics proposed by Becker in 1952 for the modal operators has recently been reviewed in Copeland’s paper The Genesis of Possible World Semantics (Copeland in J Philos Logic 31:99–137, 2002), with a remark that “neither the binary relation nor the idea of proving completeness was present in Becker’s work”. In light of Frege’s celebrated Sense-Determines-Reference principle, we find, however, that it is Becker’s semantics, rather than Kripke’s semantics, that has captured the true spirit of Frege’s semantic program. Furthermore, for Kripke’s possible world semantics to fit in Frege’s framework of senses, worlds and referents, it will have to be thoroughly reformulated. By introducing the notion of a hi-world into the picture, we manage to keep the key ingredients of Becker’s semantics intact, while at the same time solve a fatal problem that used to shadow Becker’s original semantics—it had not been able to make sense of inhomogeneous modality. The resulting generalized Beckerian semantics provides, in effect, a Beckerian analysis of the Kripkean possible worlds. It reveals the subtle hierarchical internal structure of a Kripkean world that has not been discovered before.