Cantor Paradoxes, Possible Worlds and Set Theory (original) (raw)

The Russell-Kaplan paradox and other modal paradoxes: A new solution

Nordic Journal of Philosophical Logic, 1999

The most successful semantics for modal logic in the narrow sense, the logic of possibility, necessity, impossibility and contingency, has been possible worlds semantics. And though various kinds of algebraic semantics (as in Bealer 1982) are emerging as noteworthy rivals to ...

Logically possible worlds and counterpart semantics for modal logic

Philosophy of Logic, Handbook of the Philosophy …

The paper reviews the technical results from modal logic as well as their philosophical significance. It focuses on possible worlds semantics in general and on the notions of a possible world, of accessibility, and of an object.

Another Problem in Possible World Semantics

Advances in Modal Logic, 2020

In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics—the atomicity of the algebra of propositions—but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic.

Worlds and Propositions Set Free

2014

The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory to define worlds. Next the authors show that an object-theoretic analysis of the Kaplan paradox reveals that there is no genuine paradox at all, as the central premise of the paradox is simply a logical falsehood and hence can be rejected on the strongest possible grounds --- not only in object theory but for the very framework of propositional modal logic in which Kaplan frames his argument. The authors close by fending off a possible objection that object theory avoids the Russell paradox only by refusing to incorporate set theory and, hence, that the object theoretic solution is only a consequence of the theory's weakness.

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

Journal of Philosophical Logic, 2003

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 w∈W if and only if A holds at every world v∈W 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 ω-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.

Impossible Worlds and Propositions - The Philosophical Quarterly

The Philosophical Quarterly, 2010

Accounts of propositions as sets of possible worlds have been criticized for conflating distinct impossible propositions. In response to this problem, some have proposed to introduce impossible worlds to represent distinct impossibilities, endorsing the thesis that impossible worlds must be of the same kind; this has been called the parity thesis. I show that this thesis faces problems, and propose a hybrid account which rejects it: possible worlds are taken as concrete Lewisian worlds, and impossibilities are represented as set-theoretic constructions out of them. This hybrid account (1) distinguishes many intuitively distinct impossible propositions; (2) identifies impossible propositions with extensional constructions; (3) avoids resorting to primitive modality, at least so far as Lewisian modal realism does.

Compatibility, compossibility, and epistemic modality

Proceedings of the 23rd Amsterdam Colloquium, 2022

We give a theory of epistemic modals in the framework of possibility semantics and axiomatize the corresponding logic, arguing that it aptly characterizes the ways in which reasoning with epistemic modals does, and does not, diverge from classical modal logic.

On Propositions as Sets of Possible Worlds

Hoffmann has recently argued that propositions cannot be sets of possible worlds. I show that this is not the case if 'proposition' is understood in the correct way and that Hoffmann's arguments rest on premises that should be rejected.

On Modal Set Theory. Three Routes to the Iterative Conception ​

2022

The thesis presents an assessment of the three main theories available in the contemporary debate over the modal profile intrinsic to the Iterative Conception of Set. The outcome of the research offers a prospect of what is, according to my analysis, the best route to the potential hierarchy of sets. The work starts with a preliminary historical overview of the path the Iterative Conception (Ch.0), before turning to the contemporary discussion . Each theory is then presented according to three criteria: the informal profile, the formal regimentation, and the cost-benefit analysis. The first proposal considered is Øystein Linnebo’s modal set theory, the very first contemporary view that combines set theory and modal logic in a satisfactory way, via a framework that resorts to plural resources (Ch. 1). The second account is due to James Studd and it basically amounts to an improvement over Linnebo’s theory by capturing the potential hierarchy of sets via a bimodal approach (Ch. 2). The third and most recent account is due to Tim Button, who champions a view, originally stated by Hilary Putnam, according to which the modal and the non-modal accounts of set theory are, in fact, equivalent (Ch. 3). Finally, I offer my own general assessment, taking a stand, contra Button, in favor of the ultimate irreducibility of modal set theory to its non-modal counterpart, and, contra Studd, in favor of Linnebo’s approach (Ch. 4). However, some space for an improvement towards a more viable bimodal enhancement is left over among the possible extensions of the debate. ​