Among the most remarkable developments in metaphysics since the 1950’s is the explosion of philosophical interest in possible worlds. This paper proposes an explanation of what possible worlds are, and argues that this proposal, the interpreted models conception, should be attractive to anyone who thinks that modal facts are primitive, and so not to be explained in terms of some non-modal notion of “possible world.” I articulate three constraints on any acceptable primitivist explanation of the nature of possible worlds, and (...) show that the interpreted models conception meets the three constraints. (shrink)

I review and reconsider some of the themes of ‘Two notions of necessity’ (Davies and Humberstone, 1980) and attempt to reach a deeper understanding and appreciation of Gareth Evans’s reflections (in ‘Reference and contingency’, 1979) on both modality and reference. My aim is to plot the relationships between the notions of necessity that Humberstone and I characterised in terms of operators in two-dimensional modal logic, the notions of superficial and deep necessity that Evans himself described, and the epistemic notion of (...) a priority. (shrink)

Lewis (The Journal of Philosophy, 65(5), 113–126, 1968) attempts to provide an account of modal talk in terms of the resources of counterpart theory, a first-order theory that eschews transworld identity. First, a regimentation of natural language modal claims into sentences of a formal first-order modal language L is assumed. Second, a translation scheme from L-sentences to sentences of the language of the theory is provided. According to Hazen (The Journal of Philosophy, 76(6), 319–338, 1979) and Fara & Williamson (Mind, (...) 114(453), 1–30, 2005), the account cannot handle certain natural language modal claims involving a notion of actuality. The challenge has two parts. First, in order to handle such claims, the initial formal modal language that natural language modal claims are regimented into must extend L with something like an actuality operator. Second, certain ways that Lewis’ translation scheme for L might be extended to accommodate an actuality operator are unacceptable. Meyer (Mind, 122(485), 27–42, 2013) attempts to defend Lewis’ approach. First, Meyer holds that in order to handle such claims, the formal modal language L$$^*$$ ∗ that we initially regiment our natural language claims into need not contain an actuality operator. Instead, we can make do with other resources. Next, Meyer provides an alternative translation scheme from L$$^*$$ ∗ -sentences to sentences of an enriched language of counterpart theory. Unfortunately, Meyer’s approach fails to provide an appropriate counterpart theoretic account of natural language modal claims. In this paper, I demonstrate that failure. (shrink)

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)

John MacFarlane explores how we might make sense of the idea that truth is relative. He provides new, satisfying accounts of parts of our thought and talk that have resisted traditional methods of analysis, including what we mean when we talk about what is tasty, what we know, what will happen, what might be the case, and what we ought to do.

It has been claimed that counterpart theory cannot support a theory of actuality without rendering obviously invalid formulas valid or obviously valid formulas invalid. We argue that these claims are not based on logical flaws of counterpart theory itself, but point to the lack of appropriate devices in first-order logic for “remembering” the values of variables. We formulate a mildly dynamic version of first-order logic with appropriate memory devices and show how to base a version of counterpart theory with actuality (...) on this. This theory is, in special cases, equivalent to modal first-order logic with actuality, and apparently does not suffer from the logical flaws that have been mentioned in the literature. (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.

I introduce and defend the semantic notion of counterfactual identity, distinguishing it from the metaphysical notion of transworld identity. After showing that Lewis's counterpart theory misconstrues counterfactual identity facts, I outline and motivate a ‘Leibnizian counterpart theory’ where the notion of counterfactual identity is adequately modelled. Finally, I show that counterfactual identity can be characterized without relying on some implausible features of Lewis's theory of conditionals.

It has been argued that a combination of game-theoretic semantics and independence-friendly (IF) languages can provide a novel approach to the conceptual foundations of mathematics and the sciences. I introduce and motivate an IF first-order modal language endowed with a game-theoretic semantics of perfect information. The resulting interpretive independence-friendly logic (IIF) allows to formulate some basic model-theoretic notions that are inexpressible in the ordinary quantified modal logic. Moreover, I argue that some key concepts of Kripke’s new theory of reference are (...) adequately modeled within IIF. Finally, I compare the logic IIF to David Lewis' counterpart theory, drawing some morals concerning the interrelation between metaphysical and semantic issues in possible-world semantics. (shrink)

The psychological orientation treats semantics as a matter of idealized computation over symbolic structures, and semantic relations like denotation as relations between linguistic expressions and these structures. I argue that results similar to Gödel's incompleteness theorems and Tarski's theorem on truth create foundational difficulties for this view of semantics.

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)

Our ordinary moral practices not only suppose that some people ought to perform some actions, and that some outcomes are morally better or worse than others, but also that there are rights, duties, goodness, and other apparently abstract moral entities. What should we make of these entities, and the talk of these entities? It is not straighforward to account for these entities in other terms. On the other hand, this paper will argue that talk of such entities is not easily (...) eliminated from good moral theorising. Nor are the prospects for paraphrasing apparent talk about these entities into less committal terms very promising. We should either find a place in our ontology for these creatures of ethics, or explain how they can be employed as a convenient fiction. (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.

Do incompatibilist arguments, like some fatalist arguments, rest on modal fallacies? If Westphal (2012) is right, then one popular argument for incompatibilism van Inwagen’s “First Formal Argument” does rest on a modal fallacy. Similarly, Warfield (2000) claims that the standard modal formulation of the master argument for incompatibilism is a modal fallacy. Here, I refute both claims. Contra Westphal, I show that the mistake in van Inwagen’s "First Formal Argument" is no modal fallacy. After that, I argue that Warfield’s charge (...) of modal fallacy can be easily avoided by using a plausible principle concerning actuality. Then, I show that this allows one to put forward a fairly simple argument for fatalism (the thesis that we aren’t able to do otherwise from what we actually do). (shrink)

In Modal Logic as Metaphysics, Timothy Williamson claims that the possibilism-actualism (P-A) distinction is badly muddled. In its place, he introduces a necessitism-contingentism (N-C) distinction that he claims is free of the confusions that purportedly plague the P-A distinction. In this paper I argue first that the P-A distinction, properly understood, is historically well-grounded and entirely coherent. I then look at the two arguments Williamson levels at the P-A distinction and find them wanting and show, moreover, that, when the N-C (...) distinction is broadened (as per Williamson himself) so as to enable necessitists to fend off contingentist objections, the P-A distinction can be faithfully reconstructed in terms of the N-C distinction. However, Williamson’s critique does point to a genuine shortcoming in the common formulation of the P-A distinction. I propose a new definition of the distinction in terms of essential properties that avoids this shortcoming. (shrink)

A person of average height would assert a truth by the conditional ‘if I were seven feet tall, I would be taller than I am,’ in which an indicative clause ‘I am’ is embedded in a subjunctive conditional. By contrast, no one would assert a truth by ‘if I were seven feet tall, I would be taller than I would be’ or ‘if I am seven feet tall, I am taller than I am’. These examples exemplify the fact that whether (...) a sentence's evaluation remains at the actual world in the scope of a modal or conditional depends on the combination of mood in the embedded and matrix clauses rather than, as is commonly thought, just on the presence of an operator ‘actually’. This essay argues that this phenomenon provides evidence that mood admits of bound and free readings along the lines of tenses and pronouns. It therefore favors the hypothesis that natural language contains variables and quantifiers for possible worlds in the object language. This, in turn, requires that the truth of a semantic value of a sentence (or whatever structure is embedded in a modal) be relativized to a sequence of worlds rather than to an individual world, and thus be distinguished from a proposition in the traditional sense. The essay also compares the framework defended with an alternative account of similar phenomena by Kai Wehmeier. (shrink)

This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.

In a recent paper, Pruss proves the validity of the rule beta-2 relative to Lewis’s semantics for counterfactuals, which is a significant step forward in the debate about the consequence argument. Yet, we believe there remain intuitive counter-examples to beta-2 formulated with the actuality operator and rigidified descriptions. We offer a novel and two-dimensional formulation of the Lewisian semantics for counterfactuals and prove the validity of a new transfer rule according to which a new version of the consequence argument can (...) be formulated. This new transfer rule is immune to the counter-examples involving the actuality operator and rigidified descriptions. However, we show that counter-examples to this new rule can also be generated, demanding that the Lewisian semantics be generalized for higher dimensions where counter-examples can always be generated. (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)

This paper recalls some applications of two-dimensional modal logic from the 1980s, including work on the logic of Actually and on a somewhat idealized version of the indicative/subjunctive distinction, as well as on absolute and relative necessity. There is some discussion of reactions this material has aroused in commentators since. We also survey related work by Leslie Tharp from roughly the same period.

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)

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality (...) operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5. (shrink)

The addition of "actually" operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing "actually" operators has concentrated entirely upon extensions of KT5 and has employed a particular modeltheoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing "actually" operators, the weakest of which are conservative extensions of K, (...) using a novel generalisation of the standard semantics. (shrink)

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).

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.

John MacFarlane has recently presented a novel argument in support of truth- relativism. According to this, contextualists fail to accommodate retrospective reassessments of propositional contents, when it comes to languages which are rich enough to express actuality. The aim of this note is twofold. First, it is to argue that the argument can be effectively rejected, since it rests on an inadequate conception of actuality. Second, it is to offer a more plausible account of actuality in branching time, along the (...) line of Lewis (1970/1983). (shrink)

In this paper, I explore Rosen’s ‘transcendental’ objection to constructive empiricism—the argument that in order to be a constructive empiricist, one must be ontologically committed to just the sort of abstract, mathematical objects constructive empiricism seems committed to denying. In particular, I assess Bueno’s ‘partial structures’ response to Rosen, and argue that such a strategy cannot succeed, on the grounds that it cannot provide an adequate metalogic for our scientific discourse. I conclude by arguing that this result provides some interesting (...) consequences in general for anti-realist programmes in the philosophy of science.Keywords: Constructive empiricism; Mathematical fictionalism; Partial structures; Modality. (shrink)

While many are impressed with the utility of possible worlds in linguistics and philosophy, few can accept the modal realism of David Lewis, who regards possible worlds as sui generis entities of a kind with the concrete world we inhabit.1 Not all uses of possible worlds require exotic ontology. Consider, for instance, the use of Kripke models to establish formal results in modal logic. These models contain sets often regarded for heuristic reasons as sets of “possible worlds”. But the “worlds” (...) in these sets can be anything at all; they can be numbers, or people, or sh. The set of worlds, together with the accessibility relation and the rest of the model, is used as a purely formal structure.2 One can even go beyond establishing results about formal systems and apply Kripke models to English, as Charles Chihara has recently argued.3 Chihara shows, for instance, how to use Kripke models (plus primitive modal notions) to give an account of validity for English modal sentences. In other cases worlds are not really needed at all. It is often vivid to give a counterexample thus: “There is a possible world in which P. Since your theory implies that in all worlds, not-P, your theory is wrong.” But the counterexample could just as easily be given using modal operators: “Possibly, P. Since your theory implies that it is necessary that not-P, your theory is wrong.”. (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.

Among modal claims, claims that involve the notions of broadly logical possibility and necessity, one that seems almost trivial is this: that if some proposition is possible, then it is possibly true. However, there is an argument, due in its essentials to the medieval philosopher and logician Jean Buridan, to the effect that this seemingly trivial claim is, in fact, untrue. ;Briefly put, the argument is this. Let Q be the proposition that Quine does not exist. Since Quine's existence is (...) contingent, Q is possible. But, propositions are ontologically dependent upon their constituents, and Quine is a constituent of Q. So, necessarily, were Quine not to exist, neither would Q exist. And, a necessary condition of a proposition's being true is that it exist. Hence, there are propositions, such as Q, that are possible, but not possibly true. I call this conclusion, together with certain other similar claims, 'Buridanism'. ;My main topic is the evaluation of the argument for Buridanism, and the exploration of the consequences of Buridanism for modalist theories of possible worlds and for modal logic. I begin by examining the two crucial premises in the argument for Buridanism: N-dependency, according to which propositions expressed by sentences containing proper names are singular propositions that depend ontologically upon the referents of those names, and Buridan's Thesis, according to which a proposition can be neither necessary, nor possible, nor true, unless it exists. ;Robert Adams has argued that accepting Buridanism forces us to revise the modalist conception of possible worlds as maximal and consistent propositions or sets of propositions. I show that this is not the case, and provide an analysis of a proposition's being true relative to a world that allows us to maintain the modalist account while embracing Buridanism. ;Standard systems of quantified modal logic are prone to several related problems that I refer to as the problems of contingency. These systems often include as theorems formulas that seem to presuppose that everything that exists, exists necessarily. I develop a system of modal logic, BML, based upon Buridanism, that is able to overcome these difficulties. (shrink)