Possible Worlds Model Theory

Author(s):  
Timothy Williamson
Keyword(s):  
Author(s):  
Tim Button ◽  
Sean Walsh

Types are one of the cornerstones of contemporary model theory. Simply put, a type is the collection of formulas satisfied by an element of some elementary extension. The types can be organised in an algebraic structure known as a Lindenbaum algebra. But the contemporary study of types also treats them as the points of a certain kind of topological space. These spaces, called ‘Stone spaces’, illustrate the richness of moving back-and-forth between algebraic and topological perspectives. Further, one of the most central notions of contemporary model theory—namely stability—is simply a constraint on the cardinality of these spaces. We close the chapter by discussing a related algebra-topology ‘duality’ from metaphysics, concerning whether to treat propositions as sets of possible worlds or vice-versa. We show that suitable regimentations of these two rival metaphysical approaches are biinterpretable (in the sense of chapter 5), and discuss the philosophical significance of this rapprochement.


Author(s):  
John P. Burgess

Today there appears to be a widespread impression that W. V. Quine's notorious critique of modal logic, based on certain ideas about reference, has been successfully answered. As one writer put it some years ago: “His objections have been dead for a while, even though they have not yet been completely buried.” What is supposed to have killed off the critique? Some would cite the development of a new ‘possible-worlds’ model theory for modal logics in the 1960s; others, the development of new ‘direct’ theories of reference for names in the 1970s.These developments do suggest that Quine's unfriendliness towards any formal logics but the classical and indifference towards theories of reference for any singular terms but variables were unfortunate.


2017 ◽  
Vol 10 (2) ◽  
pp. 203-236 ◽  
Author(s):  
PETER FRITZ

AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.


2018 ◽  
Vol 82 ◽  
pp. 189-204
Author(s):  
Timothy Williamson

AbstractWe care not just how things are but how they could have been otherwise – about possibility and necessity as well as actuality. Many philosophers regard such talk as beyond the reach of respectable science, since observation tells us how things are but (allegedly) not how they could have been otherwise. I argue that, on the contrary, such criticisms are ill-founded: possibility and necessity are studied in natural science, for example through phase spaces, abstract mathematical representations of the possible states of a physical system. The possibility is objective, not merely epistemic, though it may be more restricted than pure metaphysical possibility. The elements of a phase space are very similar to Kripke's possible worlds, despite being time slices rather than total histories. The absence of explicit modal operators in the mathematical models used by scientists does not show science to be non-modal; rather, it manifests reliance on a mathematical framework for theorizing about objective possibility similar to the mathematical framework of possible worlds model theory.


2019 ◽  
pp. 151-162 ◽  
Author(s):  
Robert C. Stalnaker

This chapter was the first exposition and defense of an axiom system and model theory for a conditional logic in the possible worlds framework, a theory designed to model counterfactual propositions. It is argued, using a version of the Ramsey test, that the truth-conditions for conditionals that are provided can explain why we assess counterfactuals in the way we do. Counterexamples are given for various principles that are often taken to be valid for conditionals, but that are invalid on the semantics provided. It is argued that the theory helps to explain how propositions that are ostensibly about counterfactual possible situations can be confirmed or disconfirmed by evidence about the actual world.


1991 ◽  
Vol 36 (12) ◽  
pp. 1057-1058
Author(s):  
Marvin R. Goldfried ◽  
Douglas A. Vakoch
Keyword(s):  

Sign in / Sign up

Export Citation Format

Share Document