Is the world internally connected by a web of necessary connections or is everything loose and independent? Followers of David Hume accept the latter by upholding Hume’s Dictum, according to which there are no necessary connections between distinct existences. Roughly put, anything can coexist with anything else, and anything can fail to coexist with anything else. Hume put it like this: “There is no object which implies the existence of any other if we consider these objects in themselves.” Since Hume’s (...) day, Hume’s Dictum has played a major role in philosophy, especially in contemporary metaphysics. In ruling out necessary connections, Hume’s Dictum implies that causal relations and the laws of nature are contingent. It makes other demands on our metaphysics of science as well; for example, it places restrictions on our metaphysics of counterfactuals, dispositions, and explanation. The principle has applications in other areas too. Hume’s Dictum places constraints on our theories of abstract objects and moral properties. For example, Hume’s Dictum rules out the view that moral properties are distinct from, but necessarily connected to, non-moral properties. While Hume’s Dictum is a prominent principle in metaphysics, few Humeans have defended the principle at length. It is assumed more often than it is argued for. As a result, Hume’s Dictum no longer enjoys the reputation it once did. It’s common to hear complaints about the lack of arguments in support of Hume’s Dictum. Moreover, philosophers are increasingly willing to defend views that are at odds with Hume’s Dictum. A striking example of this is the recent explosion of interest in views that take causal relations and the laws of nature to be necessary. Humeans can no longer afford to be complacent. An extended defense of Hume’s Dictum is called for, and that’s what this dissertation aims to offer. This dissertation has three parts. In the first part, I lay out some preliminaries and offer a precise formulation of Hume’s Dictum. In the second part, I develop three arguments in favor of Hume’s Dictum. In the final part, I defend Hume’s Dictum against objections. (shrink)

In this paper the language of first-order modal logic is enriched with an operator @ ('actually') such that, in any model, the evaluation of a formula @A at a possible world depends on the evaluation of A at the actual world. The models have world-variable domains. All the logics that are discussed extend the classical predicate calculus, with or without identity, and conform to the philosophical principle known as serious actualism. The basic logic relies on the system K, whereas others (...) correspond to various properties that the actual world may have. All the logics are axiomatized. (shrink)

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

We investigate and compare two major approaches to enhancing the expressive capacities of modal languages, namely the addition of subjunctive markers on the one hand, and the addition of scope-bearing actuality operators, on the other. It turns out that the subjunctive marker approach is not only every bit as versatile as the actuality operator approach, but that it in fact outperforms its rival in the context of cross-world predication.

The purpose of the present paper is to challenge some received assumptions about the logical analysis of modal English, and to show that these assumptions are crucial to certain debates in current philosophy of language. Specifically, I will argue that the standard analysis in terms of quantified modal logic mistakenly fudges important grammatical distinctions, and that the validity of Kripke's modal argument against description theories of proper names crucially depends on ensuing equivocations.

Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures. Both modal and two-sorted predicate logic are natural languages for speaking about such models. In this paper we compare their expressivity. We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models. Decidable criteria for modal definability are presented.

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)

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments. -/- There (...) are two main indispensability arguments in the literature, though one has received nearly all of the attention. They correspond to two ways in which we use mathematics in science and in everyday life. We use mathematical language to help us describe non-mathematical reality; and we use mathematical reasoning to help us perform inferences concerning non-mathematical reality using only a feasible amount of cognitive power. The former use is the starting point of the Quine-Putnam indispensability argument ([Quine, 1980a], [Quine, 1980b], [Quine, 1981a], [Quine, 1981b], [Putnam, 1979a], [Putnam, 1979b]); the latter provides the basis for Ketland’s more recent argument ([Ketland, 2005]). I begin by considering the Quine-Putnam argument and introduce instrumental nominalism to defuse it. Then I show that Ketland’s argument can be defused in a similar way. (shrink)

Thought experiments are widely used in the informal explanation of Relativity Theories; however, they are not present explicitly in formalized versions of Relativity Theory. In this paper, we present an axiom system of Special Relativity which is able to grasp thought experiments formally and explicitly. Moreover, using these thought experiments, we can provide an explicit definition of relativistic mass based only on kinematical concepts and we can geometrically prove the Mass Increase Formula in a natural way, without postulates of conservation (...) of mass and momentum. (shrink)

This paper explains how to obtain quantification over times in a tense logic in which all temporal distinctions are ultimately spelled out in terms of the two simple tense operators “it was the case that” and “it will be the case that.” The account of times defended here is similar to what is known as “linguistic ersatzism” about possible worlds, but there are noteworthy differences between these two cases. For example, while linguistic ersatzism would support actualism, the view of times (...) defended here does not support presentism. (shrink)

According to Hans Kamp and Frank Vlach, the two-dimensional tense operators “now” and “then” are ineliminable in quantified tense logic. This is often adduced as an argument against tense logic, and in favor of an extensional account that makes use of explicit quantification over times. The aim of this paper is to defend tense logic against this attack. It shows that “now” and “then” are eliminable in quantified tense logic, provided we endow it with enough quantificational structure. The operators might (...) not be redundant in some other systems of tense logic, but this merely indicates a lack of quantificational resources and does not show any deep-seated inability of tense logic to express claims about time. The paper closes with a brief discussion of the modal analogue of this issue, which concerns the role of the actuality operator in quantified modal logic. (shrink)

Fara and Williamson (Mind, 2005) argue that counterpart theory is unable to account for modal claims that use an actuality operator. This paper argues otherwise. Rather than provide a different counterpart translation of the actuality operator itself, the solution presented here starts out with a quantified modal logic in which the actuality operator is redundant, and then translates the sentences of this logic into claims of counterpart theory.

I argue that ‘actually’ does not have a reading according to which it is synonymous with the actuality operator of modal logic, and propose an alternative account of ‘actually’. The cases that have been thought to show that ‘actually’ is synonymous with the actuality operator are modal and counterfactual sentences in which an embedded clause's evaluation is held fixed at the world of the context. In these cases, though, this embedded clause's evaluation is not due to the presence of ‘actually’. (...) As an alternative, I propose that ‘actually’ is a presupposition trigger along the lines of ‘even’ or ‘too’, and it signals that presupposed information states in the discourse are evolving in a non-standard way. (shrink)

ABSTRACT Backwards-looking operators Saarinen, E. [1979. “Backwards-Looking Operators in Tense Logic and in Natural Language.” In Essays on Mathematical and Philosophical Logic, edited by J. Hintikka, I. Niiniluoto, and E. Saarinen, 341–367. Dordrecht: Reidel] that have the material in their scope depend on higher intensional operators, are known to increase the expressivity of some intensional languages and have thus played a central role in debates about approaches to intensionality in terms of implicit parameters vs. variables explicitly quantifying over them. The (...) current contribution takes a look at these operators from a type-logical perspective. It is shown that extending Gallin's translation from intensional type logic to two-sorted type theory so as to include a version of Yanovich's backwards-looking operators, does not increase the expressive power of formulae with exclusively intensional parameters. The result, which makes use of a theorem by Zimmermann : 65–77]), is illustrated by pertinent examples from the literature. The paper closes by indicating alternative strategies of incorporating backwards-looking operators. (shrink)

While standard first-order modal logic is quite powerful, it cannot express even very simple sentences like “I could have been taller than I actually am” or “Everyone could have been smarter than they actually are”. These are examples of cross-world predication, whereby objects in one world are related to objects in another world. Extending first-order modal logic to allow for cross-world predication in a motivated way has proven to be notoriously difficult. In this paper, I argue that the standard accounts (...) of cross-world predication all leave something to be desired. I then propose an account of cross-world predication based on quantified hybrid logic and show how it overcomes the limitations of these previous accounts. I will conclude by discussing various philosophical consequences and applications of such an account. (shrink)

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified (...) only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages. (shrink)

Jean-Yves Béziau (‘Classical Negation can be Expressed by One of its Halves’, Logic Journal of the IGPL 7 (1999), 145–151) has given an especially clear example of a phenomenon he considers a sufficiently puzzling to call the ‘paradox of translation’: the existence of pairs of logics, one logic being strictly weaker than another and yet such that the stronger logic can be embedded within it under a faithful translation. We elaborate on Béziau’s example, which concerns classical negation, as well as (...) giving some additional background (especially from intuitionistic logic) to the example. Our interest is more on the logical exploration of the phenomenon Béziau’s case exemplifies than on the question of whether that phenomenon is (even prima facie ) paradoxical, though in Section 5 we do approach the latter question – somewhat obliquely – by considering an analogous phenomenon which it is hard to find puzzling. (shrink)

This article studies seriously actualistic quantified modal logics. A key component of the language is an abstraction operator by means of which predicates can be created out of complex formulas. This facilitates proof of a uniform substitution theorem: if a sentence is logically true, then any sentence that results from substituting a predicate abstract for each occurrence of a simple predicate abstract is also logically true. This solves a problem identified by Kripke early in the modern semantic study of quantified (...) modal logic. A tableau proof system is presented and proved sound and complete with respect to logical truth. The main focus is on seriously actualistic T, an extension of T, but the results established hold also for systems based on other propositional modal logics. Following Menzel it is shown that the formal language studied also supports an actualistic account of truth simpliciter. (shrink)

We present two-dimensional tableau systems for the actuality, fixedly, and up-arrow operators. All systems are proved sound and complete with respect to a two-dimensional semantics. In addition, a decision procedure for the actuality logics is discussed.

We provide a Hilbert-style axiomatization of the logic of , as well as a two-dimensional semantics with respect to which our logics are sound and complete. Our completeness results are quite general, pertaining to all such actuality logics that extend a normal and canonical modal basis. We also show that our logics have the strong finite model property and permit straightforward first-order extensions.

A new puzzle of modal recombination is presented which relies purely on resources of first-order modal logic. It shows that naive recombinatorial reasoning, which has previously been shown to be inconsistent with various assumptions concerning propositions, sets and classes, leads to inconsistency by itself. The context sensitivity of modal expressions is suggested as the source of the puzzle, and it is argued that it gives us reason to reconsider the assumption that the notion of metaphysical necessity is in good standing.

Many philosophers, following David Lewis, believe that we should look to counterpart theory, not quantified modal logic, as a means of understanding modal discourse. We argue that this is a mistake. Significant parts of modal discourse involve either implicit or explicit reference to what is actually the case, raising the question of how talk about actuality is to be represented counterpart-theoretically. By considering possible modifications of Lewis's counterpart theory, including actual modifications due to Graeme Forbes and Murali Ramachandran, we argue (...) that no coherent version of counterpart theory can provide a plausible representation of talk about actuality, and so, we conclude, counterpart theory should be rejected. (shrink)

We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal.

An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.

§1. Introduction. The problem raised by the liar paradox has long been an intriguing challenge for all those interested in the concept of truth. Many “solutions” have been proposed to solve or avoid the paradox, either prescribing some linguistical restriction, or giving up the classical true-false bivalence or assuming some kind of contextual dependence of truth, among other possibilities. We shall not discuss these different approaches to the subject in this paper, but we shall concentrate on a kind of formal (...) construction which was originated by Kripke's paper “Outline of a theory of truth” [11] and which, in different forms, reappears in later papers by various authors.The main idea can be presented as follows: assume a first order language ℒ containing, among other unspecified symbols, a predicate symbolTintended to represent the truth predicate for ℒ. Assume, also, a fixed modelM= 〈D, I〉 whereDcontains all sentences of ℒ andIinterprets all non-logical symbols of ℒ exceptTin the usual way. In general,Dmight contain many objects other than sentences of ℒ but as that would raise the problem of the meaning of sentences in whichTis applied to one of these objects, we shall assume that this is not the case. (shrink)

This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I argue that the interaction between the multi-dimensional intensional framework and intensional plural quantification enables epistemic possibilities to target the haecceitistic properties of individuals. I outline the elements of plural logic, and I specify, then, a multi-dimensional intensional formula encoding the relation between the epistemic possibility of haecceity comprehension and its metaphysical possibility. I conclude by addressing objections from the indeterminacy (...) of ontological principles relative to the space of epistemic possibilities, and from the consistency of epistemic modal space. (shrink)

§1. Introduction. By means of what semantic features is a proper name tied to its bearer? This is a puzzling question indeed: proper names — like “Aristotle” or “Paris” — are syntactically simple, and it therefore does not seem possible to reduce their meanings, by means of a principle of compositionality, to the meanings of more basic, and hence perhaps more tractable, linguistic elements.

A glass couldn't contain water unless it contained H2O-molecules. Likewise, a man couldn't be a bachelor unless he was unmarried. Now, the latter is what we would call a conceptual or analytical truth. It's also what we would call a priori. But it's hardly a conceptual or analytical truth that if a glass contains water, then it contains H2O-molecules. Neither is it a priori. The fact that water is composed of H2O-molecules was an empirical discovery made in the eighteenth century. (...) The fact that all bachelors are unmarried was not. But neither is a logical truth, so how do we explain the difference? Two-dimensional semantics is a framework that promises to shed light on these issues. The main purpose of this thesis is to understand and evaluate this framework in relation to various alternatives, to see whether some version of it can be defended. I argue that it fares better than the alternatives. However, much criticism of two-dimensionalism has focused on its alleged inability to provide a proper semantics for certain epistemic operators, in particular the belief operator and the a priori operator. In response to this criticism, a two-dimensional semantics for belief ascriptions is developed using structured propositions. In connection with this, a number of other issues in the semantics of belief ascriptions are addressed, concerning indexicals, beliefs de se, beliefs de re, and the problem of logical omniscience. (shrink)