In this paper, I investigate a system of quantified modal logic, due in many respects to Bressan (see [2]), from several perspectives -- both semantic and proof-theoretic. As Anderson and Belnap note in [1]: "It seems to be generally conceded that formal systems are natural or substantial if they can be looked at from several points of view. We tend to think of systems as artificial or ad hoc if most of their formal properties arise from some one notational system (...) in terms of which they are described." My efforts in this paper will be in part to lend substantiality to the system in question. Several formulation of the system are given and proved equivalent in appropriate senses. Also, some comments are made concerning possible alternative formulations. (shrink)

This paper explains and defends the idea that metaphysical necessity is the strongest kind of objective necessity. Plausible closure conditions on the family of objective modalities are shown to entail that the logic of metaphysical necessity is S5. Evidence is provided that some objective modalities are studied in the natural sciences. In particular, the modal assumptions implicit in physical applications of dynamical systems theory are made explicit by using such systems to define models of a modal temporal logic. Those assumptions (...) arguably include some necessitist principles. -/- Too often, philosophers have discussed ‘metaphysical’ modality — possibility, contingency, necessity — in isolation. Yet metaphysical modality is just a special case of a broad range of modalities, which we may call ‘objective’ by contrast with epistemic and doxastic modalities, and indeed deontic and teleological ones (compare the distinction between objective probabilities and epistemic or subjective probabilities). Thus metaphysical possibility, physical possibility and immediate practical possibility are all types of objective possibility. We should study the metaphysics and epistemology of metaphysical modality as part of a broader study of the metaphysics and epistemology of the objective modalities, on pain of radical misunderstanding. Since objective modalities are in general open to, and receive, natural scientific investigation, we should not treat the metaphysics and epistemology of metaphysical modality in isolation from the metaphysics and epistemology of the natural sciences. -/- In what follows, Section 1 gives a preliminary sketch of metaphysical modality and its place in the general category of objective modality. Section 2 reviews some familiar forms of scepticism about metaphysical modality in that light. Later sections explore a few of the many ways in which natural science deals with questions of objective modality, including questions of quantified modal logic. (shrink)

Quine ha promovido una forma de escepticismo sobre los juicios de modalidad de re que cuestiona su función y sugiere, en consecuencia, su eliminación. Wiggins, por su parte, ha argumentado que nuestros juicios ordinarios de individuación de objetos nos comprometen con un cierto esencialismo, con la aceptación de ciertos juicios de modalidad de re. Si Wiggins tuviera razón, tendríamos una respuesta potencial al escepticismo quineano de la función. Sin embargo, en este trabajo argumento que la propuesta de Wiggins es incapaz (...) de proporcionar una respuesta satisfactoria al escepticismo quineano, pues presupone aquello mismo que está en cuestión: los compromisos esencialistas se siguen sólo de la adopción previa de otros juicios de modalidad de re cuya función está precisamente en entredicho. Asimismo, argumento que esta misma conclusión puede extraerse de la consideración de algunas otras teorías contemporáneas de la individuación, similares a la de Wiggins. (shrink)

Identity is a modally inflexible relation: two objects are necessarily identical or necessarily distinct. However, identity is not alone in this respect. We will look at the relation that one object bears to some objects if and only if it is one of them. In particular, we will consider the credentials of the thesis that no matter what some objects are, an object is necessarily one of them or necessarily not one of them.

Some philosophers respond to Leibniz’s “shift” argument against absolute space by appealing to antihaecceitism about possible worlds, using David Lewis’s counterpart theory. But separated from Lewis’s distinctive system, it is difficult to understand what this doctrine amounts to or how it bears on the Leibnizian argument. In fact, the best way of making sense of the relevant kind of antihaecceitism concedes the main point of the Leibnizian argument, pressing us to consider alternative spatiotemporal metaphysics.

The counterpart theorist has a problem: there is no obvious way to understand talk about actuality in terms of counterparts. Fara and Williamson have charged that this obstacle cannot be overcome. Here I defend the counterpart theorist by offering systematic interpretations of a quantified modal language that includes an actuality operator. Centrally, I disentangle the counterpart relation from a related notion, a ‘representation relation’. The relation of possible things to the actual things they represent is variable, and an adequate account (...) of modal language must keep track of the way it is systematically shifted by modal operators. I apply my account to resolve several puzzles about counterparts and actuality. In technical appendices, I prove some important logical results about this ‘representational’ counterpart system and its relationship to other modal systems. (shrink)

I argue that propositions are contingent existents. Some propositions that in fact exist might not have existed and there might have been propositions that are distinct from every actually existing proposition. This is because some propositions are singular propositions, which are propositions containing ordinary objects as constituents, and so are ontologically dependent on the existence of those objects; had those objects not existed, then the singular propositions would not have existed. I provide both a philosophical and technical understanding of the (...) contingent status of propositions. (shrink)

It is shown that the expressive power of second-order propositional modal logic whose modalities are S4.2 or weaker is the same as that of second-order predicate logic.

Bob Hale’s distinguished record of research places him among the most important and influential contemporary analytic metaphysicians. In his deep, wide ranging, yet highly readable book Necessary Beings, Hale draws upon, but substantially integrates and extends, a good deal his past research to produce a sustained and richly textured essay on — as promised in the subtitle — ontology, modality, and the relations between them. I’ve set myself two tasks in this review: first, to provide a reasonably thorough (if not (...) exactly comprehensive) overview of the structure and content of Hale’s book and, second, to a limited extent, to engage Hale’s book philosophically. I approach these tasks more or less sequentially: Parts I and 2 of the review are primarily expository; in Part 3 I adopt a somewhat more critical stance and raise several issues concerning one of the central elements of Hale’s account, his essentialist theory of modality. (shrink)

This paper develops some modal metatheory for quantified modal logic. In such a theory, the logic of a first-order modal object-language is made sensitive to the modal facts, stated in the metalanguage. This is radically different from possible worlds semantics, which reduces questions of validity to questions of nonmodal set theory. We consider theories which characterize a notion of truth under a second-order interpretation, where an operator for metaphysical necessity is treated homophonically. The form they take is crucially influenced by (...) whether the Barcan formulas are part of the metalogic. The main result is that, with the Barcan formulas, formulating a modal metatheory is easy; principles of classical semantics transfer over smoothly with natural adaptations to the modal setting. Without the Barcan formulas, it seems to be much harder. Several options are considered, but each involves commitments plausibly in tension with the underlying philosophical motivations for rejecting them. (shrink)

How does vagueness interact with metaphysical modality and with restrictions of it, such as nomological modality? In particular, how do definiteness, necessity (understood as restricted in some way or not), and actuality interact? This paper proposes a model-theoretic framework for investigating the logic and semantics of that interaction. The framework is put forward in an ecumenical spirit: it is intended to be applicable to all theories of vagueness that express vagueness using a definiteness (or: determinacy) operator. We will show how (...) epistemicists, supervaluationists, and theorists of metaphysical vagueness like Barnes and Williams (2010) can interpret the framework.  We will also present a complete axiomatization of the logic we recommend to both epistemicists and local supervaluationists. . (shrink)

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme □∃xφ ⊃ ∃x□φ.

The delicate point in the formalistic position is to explain how the non-intuitionistic classical mathematics is significant, after having initially agreed with the intuitionists that its theorems lack a real meaning in terms of which they are true (S. C. Kleene, 1952).

In a recent article in Analysis, Graeme Hunter and William Seager (1981) attempt to rescue counterpart theory (CT) from some objections of Hazen 1979. They see these objections as arising from ‘uncritical use of the translation scheme originally proposed by Lewis’, and intend to meet them by refraining from use of that scheme. But they do not offer a new scheme; they say ‘…it is no more necessary to have one to capture the sense of modal idiom than it is (...) to capture the sense of quantificational idiom…Appeal to truth value is the single most important criterion of correct translation’ (Hunter and Seager 1981:72). Thus, where the scheme of Lewis (1968) translates a truth by a falsehood or conversely, Hunter and Seager simply produce a sentence which they claim to be a better translation, without articulating any structural principles they employ to arrive at their candidate. A friend of CT should not be happy with this procedure. Let Lm be first-order modal language and Lc be the language of counterpart theory. Sentences of Lc are intended to interpret or elucidate the meanings of Lm-sentences in the strong sense that they should be the outputs of an adequate theory of meaning, which we can formulate as a model theory, for Lm, written in Lc plus set theory. The interest of CT resides in its potential to provide an alternative approach to the model theory of modal logic, and from the clauses of any such alternative model theory we should be able to read off a translation scheme for Lm into Lc, just as we can in the standard case. If CT’s motivation is sound, which Hunter and Seager do not dispute, yet there is no translation scheme and so no counterpart-theoretic model theory, one might suspect some defect in the original language Lm. And in fact, Hunter and Seager find certain Lm-sentences multiply ambiguous with respect to Lc; but if a counterpart-theoretic model theory is possible, to say.. (shrink)

This paper is part of a general programme of developing and investigating particular first- order modal theories. In the paper, a modal theory of propositions is constructed under the assumption that there are genuinely singular propositions, ie. ones that contain individuals as constituents. Various results on decidability, axiomatizability and definability are established.

I identify a notion of logical grounding, clarify it, and show how it can be used (i) to characterise various consequence relations, and (ii) to give a precise syntactic account of the notion of “groundedness” at work in the literature on the paradoxes of truth.

Dummett's logic LC quantified, Q-LC, is shown to be characterized by the extended frame Q+, ,D, where Q+ is the set of non-negative rational numbers, is the numerical relation less or equal then and D is the domain function such that for all v, w Q+, Dv and if v w, then D v . D v D w . Moreover, simple completeness proofs of extensions of Q-LC are given.

It is shown that the modally first-degree formulas of quantificational S5 constitute a reduction class. This is done by defining prenex normal forms for quantificational S5, and then showing that for any formula A there is a formula B in prenex normal form, such that B is modally first-degree and is provable if and only if A is provable.

A system of modal logic with the operator is proposed, and proved complete. In contrast with a previous one by Stalnaker and Thomason, this system does not require two categories of singular terms.