Quantified Modal Logic

Can we do science without numbers? How much contingency is there? These seemingly unrelated questions--one in the philosophy of math and science and the other in metaphysics--share an unexpectedly close connection. For as it turns out, a radical answer to the second leads to a breakthrough on the first. The radical answer is new view about modality called compossible immutabilism. The breakthrough is a new strategy for doing science without numbers. One of the chief benefits of the new strategy is (...) that, unlike the existing substantialism approach from Field (1980), the new strategy naturally generalizes to theories formulated in terms of state space. (shrink)

It is often assumed that pluralities are rigid, in the sense of having all and only their actual members necessarily. This assumption is operative in standard approaches to modal plural logic. I argue that a sceptical approach towards the assumption is warranted.

Here I first raise an argument purporting to show that Lewis’ Modal Realism ends up being entirely trivial. But although I reject this line, the argument reveals how difficult it is to interpret Lewis’ thesis that possibilia “exist.” Five natural interpretations are considered, yet upon reflection, none appear entirely adequate. On the three different “concretist” interpretations of ‘exist’, Modal Realism looks insufficient for genuine ontological commitment. Whereas, on the “multiverse” interpretation, Modal Realism acknowledges physical possibilities only--and worse, (assuming either axiom (...) S5 or axiom B) each possibilium ends up as a necessary physical existent. Finally, on the “broadly Actualist” of ‘exist’, Modal Realism is either inconsistent or it mistakenly identifies the unrestricted quantifier with the unrestricted Actualist quantifier. The upshot is that it remains obscure in what non-trivial sense Lewisian possibilia “exist.”. (shrink)

It is commonplace to formalize propositions involving essential properties of objects in a language containing modal operators and quantifiers. Assuming David Lewis’s counterpart theory as a semantic framework for quantified modal logic, I will show that certain statements discussed in the metaphysics of modality de re, such as the sufficiency condition for essential properties, cannot be faithfully formalized. A natural modification of Lewis’s translation scheme seems to be an obvious solution but is not acceptable for various reasons. Consequently, the only (...) safe way to express some intuitions regarding essential properties is to use directly the language of counterpart theory without modal operators. (shrink)

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world either (...) makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages. (shrink)

This essay considers Kant’s theory of modality in light of a debate in contemporary modal metaphysics and modal logic concerning the Barcan formulas. The comparison provides a new and fruitful perspective on Kant’s complex and sometimes confusing claims about possibility and necessity. Two central Kantian principles provide the starting point for the comparison: that the possible must be grounded in the actual and that existence is not a real predicate. Both are shown to be intimately connected to the Barcan formulas, (...) and Kant’s views on what he distinguishes as three different kinds of modality are then considered in light of this connection. (shrink)

Actualism is a widely-held view in the metaphysics of modality that arises in response to the thesis of possibilism, the doctrine that, in addition to the things that actually exist alongside us in the causal order, there are merely possible things as well, things that, in fact, fail to be actual but which could have been. The central motivation for possibilism is to explain what it is about reality that grounds such intuitively true propositions as that Wittgenstein (who was childless) (...) could have had children. In answer, possibilists argue that we must simply broaden our understanding of reality, of what there is in the broadest sense, beyond the actual, beyond what actually exists, so that it also includes the merely possible. In particular, says the possibilist, there are merely possible people, things that are not, in fact, people but which could have been. So, for the possibilist, the proposition that Wittgenstein could have had children is grounded in the fact that, among the possibilia, there are those that could have been his children. Actualism is (at the least) the denial of possibilism; to be an actualist is to deny that there are any possibilia. Put another way, for the actualist, there is no realm of reality, or being, beyond actual existence; to be is to exist, and to exist is to be actual. This article investigates the origins and nature of the debate between possibilists and actualists, with a particular focus on the implications of the debate for quantified modal logic. -/- . (shrink)

Lewis (1968) claims that his language of Counterpart Theory (CT) interprets modal discourse and he adverts to a translation scheme from the language of Quantifed Modal Logic (QML) to CT. However, everybody now agrees that his original translation scheme does not always work, since it does not always preserve the ‘intuitive’ meaning of the translated QML-formulas. Lewis discusses this problem with regard to the Necessitist Thesis, and I will extend his discourse to the analysis of the Converse Barcan Formula. Everyone (...) also agrees that there are CT-formulas that can express the QMLcontent that gets lost through the translation. The problem is how we arrive to them. In this paper, I propose new translation rules from QML to CT, based on a suggestion by Kaplan. However, I will claim that we cannot have ‘the’ translation scheme from QML to CT. The reason being that de re modal language is ambiguous. Accordingly, there are diferent sorts of QML, depending on how we resolve such ambiguity. Therefore, depending on what sort of QML we intend to translate into CT, we need to use the corresponding translation scheme. This suggests that all the translation problems might just disappear if we do what Lewis did not: begin with a fully worked out QML that tells us how to understand de re modal discourse. (shrink)

This paper develops an account of pluralities based on the following simple claim: some things are nothing over and above the individual things they comprise. For some, this may seem like a mysterious statement, perhaps even meaningless; for others, like a truism, trivial and inferentially inert. I show that neither reaction is correct: the claim is both tractable and has important consequences for a number of debates in philosophy.

Hybrid contingentism combines first-order contingentism, the view that it is contingent what individuals there are, with higher-order necessitism, the view that it is non-contingent what properties and propositions there are (where these are conceived as entities in the range of appropriate higher-order quantifiers). This combination of views avoids the most delicate problems afflicting alternative contingentist positions while preserving the central contingentist claim that ordinary, concrete entities exist contingently. Despite these attractive features, hybrid contingentism is usually faced with rejection. The main (...) reason for this is an objection that crucially involves haecceitistic properties, properties such as being identical to Plato or being identical to Aristotle. The objection alleges that by accepting the necessary existence of such haecceities, hybrid contingentists incur an explanatory commitment that they are unable to discharge, namely that of explaining how it is that certain haecceities ‘lock onto’ their target individuals even when those individuals are absent. To defend hybrid contingentism against this charge, I first clarify the haecceities objection in several respects and consider, in particular, what notion of explanation the objection is operating with. After arguing that it can be fruitfully understood as a challenge to provide metaphysical grounds for certain haecceity facts, I develop a contingentist response to the objection that draws on recent work on the connection between ground and essence. (shrink)

Analytic philosophy in the mid-twentieth century underwent a major change of direction when a prior consensus in favour of extensionalism and descriptivism made way for approaches using direct reference, the necessity of identity, and modal logic. All three were first defended, in the analytic tradition, by one woman, Ruth Barcan Marcus. But analytic philosophers now tend to credit them to Kripke, or Kripke and Carnap. I argue that seeing Barcan Marcus in her historical context – one dominated by extensionalism and (...) descriptivism – allows us to see how revolutionary she was, in her work and influence on others. I focus on her debate with Quine, who found himself retreating to softened, and more viable, versions of his anti-modal arguments as a result. I make the case that Barcan's formal logic was philosophically well-motivated, connected to her views on reference, and well-matched to her overall views on ontology. Her nominalism led her to reject posits which could not be directly observed and named, such as possibilia. She conceived of modal calculi as facilitating counterfactual discourse about actual existents. I conclude that her contributions ought to be recognized as the first of their kind. Barcan Marcus must be awarded a central place in the canon of analytic philosophy. (shrink)

ABSTRACT Quine insisted that the satisfaction of an open modalised formula by an object depends on how that object is described. Kripke's ‘objectual’ interpretation of quantified modal logic, whereby variables are rigid, is commonly thought to avoid these Quinean worries. Yet there remain residual Quinean worries for epistemic modality. Theorists have recently been toying with assignment-shifting treatments of epistemic contexts. On such views an epistemic operator ends up binding all the variables in its scope. One might worry that this yields (...) the undesirable result that any attempt to ‘quantify in’ to an epistemic environment is blocked. If quantifying into the relevant constructions is vacuous, then such views would seem hopelessly misguided and empirically inadequate. But a famous alternative to Kripke's semantics, namely Lewis' counterpart semantics, also faces this worry since it also treats the boxes and diamonds as assignment-shifting devices. As I'll demonstrate, the mere fact that a variable is bound is no obstacle to binding it. This provides a helpful lesson for those modelling de re epistemic contexts with assignment sensitivity, and perhaps leads the way toward the proper treatment of binding in both metaphysical and epistemic contexts: Kripke for metaphysical modality, Lewis for epistemic modality. (shrink)

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

The Modal Predicate Calculus gives rise to issues surrounding the Barcan formulas, their converses, and necessary existence. I examine these issues by means of the Quantified Argument Calculus, a recently developed, powerful formal logic system. Quarc is closer in syntax and logical properties to Natural Language than is the Predicate Calculus, a fact that lends additional interest to this examination, as Quarc might offer a better representation of our modal concepts. The validity of the Barcan formulas and their converses is (...) shown by Quarc to be a result of the specific incorporation of quantification in the Predicate Calculus, and not as reflecting a feature of the interaction of quantification and modality more generally. Necessary existence is shown to follow from the identification, in the Predicate Calculus on its canonical interpretation, of particular quantification, ascription of existence and the ‘there is’ construction, three constructions which are distinguished in both Quarc and Natural Language. The issues surrounding the Barcan formulas, their converses and necessary existence are thus shown to be an artefact of a specific logic system, not an essential feature of our relevant modal concepts or of formal logic. (shrink)

Edward Nieznański developed two logical systems to deal with the problem of evil and to refute religious determinism. However, when formalized in first-order modal logic, two axioms of each system contradict one another, revealing that there is an underlying minimal set of axioms enough to settle the questions. In this article, we develop this minimal system, called N3, which is based on Nieznański’s contribution. The purpose of N3 is to solve the logical problem of evil through the defeat of a (...) version of religious determinism. On the one hand, these questions are also addressed by Nieznański’s systems, but, on the other hand, they are obtained in N3 with fewer assumptions. Our approach can be considered a case of logic of religion, that is, of logic applied to religious discourse, as proposed by Józef Maria Bocheński; in this particular case, it is a discourse in theodicy, which is situated in the context of the philosophy of religion. (shrink)

Modal Meinongianism provides the semantics of sentences involving intentional verbs Priest. To that end, Modal Meinongianism employs a pointed non-normal quantified modal logic model. Like earlier Meinongian views Modal Meinongianism has a characterisation principle, that claims that any condition whatsoever is satisfied by some object in some world. Recently, Everett has proposed an argument against QCP that, if successful, gives rise to problems identical to those Russell raised for Naïve Meinongianism, namely that it allows for true contradictions, and allows us (...) to define anything into existence. Everett claims that the ordinary meanings of “actual” license an inference pattern, such that if an object satisfies Actual A at some world, then that object satisfies A in the actual world. Given that actual world is the designated point of evaluation for truth simpliciter, QCP would fall prey to Russell’s criticisms. As opposed to Everett, I argue that, even if we grant Everett the assumption that “actual” is a modal indexical that rigidly refers to the actual world, it does not conform to the inference pattern above. This is because when an object satisfies Actual A at some world, this alters the assertoric force of “actual”, because “actual” is interpreted in the scope of some modal or intentional operator. I also explain that Everett’s proposed example carries existential commitment because the problematic noun-phrase occurs outside the scope of a modal or intentional operator. (shrink)

In `Essence and Modality', Kit Fine proposes that for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all objects whatsoever. Call this view Fine's Thesis. This paper is a study of Fine's Thesis in the context of Fine's logic of essence (LE). Fine himself has offered his most elaborate defense of the thesis in the context of LE. His defense rests on the widely shared assumption that metaphysical necessity obeys the (...) laws of the modal logic S5. In order to get S5 for metaphysical necessity, he assumes a controversial principle about the nature of all objects. I will show that the addition of this principle to his original system E5 leads to inconsistency with an independently plausible principle about essence. In response, I develop a theory that avoids this inconsistency while allowing us to maintain S5 for meta- physical necessity. However, I conclude that our investigation of Fine's Thesis in the context of LE motivates the revisionary conclusion that metaphysical necessity obeys the principles of the modal logic S4, but not those of S5. I argue that this constitutes a distinctively essentialist challenge to the received view that the logic of metaphysical necessity is S5. (shrink)

Kurt Gödel’s version of the Ontological Proof derives rather than assumes the crucial Possibility Claim: the claim that it is possible that something God-like exists. Gödel’s derivation starts off with a proof of the Possible Instantiation of the Positive: the principle that, if a property is positive, it is possible that there exists something that has that property. I argue that Gödel’s proof of this principle relies on some implausible axiological assumptions but it can be patched so that it only (...) relies on plausible principles. But Gödel’s derivation of the Possibility Claim also needs a substantial axiological assumption, which is still open to doubt. (shrink)

In this paper, I will argue that there is a version of possibilism—inspired by the modal analogue of Kit Fine’s fragmentalism—that can be combined with a weakening of actualism. The reasons for analysing this view, which I call Modal Fragmentalism, are twofold. Firstly, it can enrich our understanding of the actualism/possibilism divide, by showing that, at least in principle, the adoption of possibilia does not correspond to an outright rejection of the actualist intuitions. Secondly, and more specifically, it can enrich (...) our understanding of concretism, by proving that, at least in principle, the idea that objects have properties in an absolute manner is compatible with transworld identity. (shrink)

Edward Nieznanski developed in 2007 and 2008 two different systems in formal logic which deal with the problem of evil. Particularly, his aim is to refute a version of the logical problem of evil associated with a form of religious determinism. In this paper, we revisit his first system to give a more suitable form to it, reformulating it in first-order modal logic. The new resulting system, called N1, has much of the original basic structure, and many axioms, definitions, and (...) theorems still remain; however, some new results are obtained. If the conclusions attained are correct and true, then N1 solves the problem of evil through the refutation of a version of religious determinism, showing that the attributes of God in Classical Theism, namely, those of omniscience, omnipotence, infallibility, and omnibenevolence, when adequately formalized, are consistent with the existence of evil in the world. We consider that N1 is a good example of how formal systems can be applied in solving interesting philosophical issues, particularly in Philosophy of Religion and Analytic Theology, establishing bridges between such disciplines. (shrink)

In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. (...) This result raises the question: For which normal modal logics. (shrink)

In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. (...) This result raises the question: For which normal modal logics L can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for L? (shrink)

The problem of hyperintensional contexts, and the problem of logical omniscience, shows the severe limitation of possible-worlds semantics which is employed also in standard epistemic logic. As a solution, we deploy here hyperintensional semantics according to which the meaning of an expression is an abstract structured algorithm, namely Tichý's construction. Constructions determine the denotata of expressions. Propositional attitudes are modelled as attitudes towards constructions of truth values. Such a model of belief is, of course, inferentially restrictive. We therefore also propose (...) a model of implicit knowledge, which is the collection of a possible agent's explicit beliefs which are related through a derivation system mastered by the agent. A derivation system consists of beliefs and derivation rules by means of which the agent may derive beliefs different from the beliefs she is actually related to. Conditions imposed on the set of base beliefs and the set of rules capture the limitations of the agent's deriving capabilities. (shrink)

In this paper, we introduce an extension of the modal language with what we call the global quantificational modality [∀p]. In essence, this modality combines the propositional quantifier ∀p with the global modality A: [∀p] plays the same role as the compound modality ∀pA. Unlike the propositional quantifier by itself, the global quantificational modality can be straightforwardly interpreted in any Boolean Algebra Expansion (BAE). We present a logic GQM for this language and prove that it is complete with respect to (...) the intended algebraic semantics. This logic enables a conceptual shift, as what have traditionally been called different “modal logics” now become [∀p]-universal theories over the base logic GQM: instead of defining a new logic with an axiom schema such as □φ→□□φ, one reasons in GQM about what follows from the globally quantified formula [∀p](□p→□□p). (shrink)

When it comes to Kripke-style semantics for quantified modal logic, there’s a choice to be made concerning the interpretation of the quantifiers. The simple approach is to let quantifiers range over all possible objects, not just objects existing in the world of evaluation, and use a special predicate to make claims about existence. This is the constant domain approach. The more complicated approach is to assign a domain of objects to each world. This is the varying domain approach. Assuming that (...) all terms denote, the semantics of predication on the constant domain approach is obvious: either the denoted object has the denoted property in the world of evaluation, or it hasn’t. On the varying domain approach, there’s a third possibility: the object in question doesn’t exist. Terms may denote objects not included in the domain of the world of evaluation. The question is whether an atomic formula then should be evaluated as true or false, or if its truth value should be undefined. This question, however, cannot be answered in isolation. The consequences of one’s choice depends on the interpretation of molecular formulas. Should the negation of a formula whose truth value is undefined also be undefined? What about conjunction, universal quantification and necessitation? The main contribution of this paper is to identify two partial semantics for logical operators, a weak and a strong one, which uniquely satisfy a list of reasonable constraints. I also show that, provided that the point of using varying domains is to be able to make certain true claims about existence without using any existence predicate, this result yields two possible partial semantics for quantified modal logic with varying domains. (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)

In this paper we present tableau methods for two-dimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain first-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux modulate (...) over different notions of validity such as local, general, and diagonal, besides being general enough for several two-dimensional logics proposed in the literature. We also motivate the introduction of a new operator into two-dimensional languages and explore some of the philosophical questions raised by it concerning the relations there are between actuality, necessity, and the a priori, that seem to undermine traditional intuitive interpretations of two-dimensional operators. (shrink)

I examine the familiar quadruple of categorical statements “Every F is/is not G.”, “Some F is/is not G.” as well as the quadruple of their modal versions “Necessarily, every F is/is not G.”, “Possibly, some F is/is not G.”. I focus on their existential import and its impact on the resulting Squares of Opposition. Though my construal of existential import follows modern approach, I add some extra details which are enabled by framing my definition of existential import within expressively rich (...) higher-order partial type logic. As regards the modal categorical statements, I find that so-called void properties bring existential import to them, so they are the only properties which invalidate subalternation, and thus also contrariety and subcontrariety, in the corresponding Square of Opposition. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

The most common first- and second-order modal logics either have as theorems every instance of the Barcan and Converse Barcan formulae and of their second-order analogues, or else fail to capture the actual truth of every theorem of classical first- and second-order logic. In this paper we characterise and motivate sound and complete first- and second-order modal logics that successfully capture the actual truth of every theorem of classical first- and second-order logic and yet do not possess controversial instances of (...) the Barcan and Converse Barcan formulae as theorems, nor of their second-order analogues. What makes possible these results is an understanding of the individual constants and predicates of the target languages as strongly Millian expressions, where a strongly Millian expression is one that has an actually existing entity as its semantic value. For this reason these logics are called ‘strongly Millian’. It is shown that the strength of the strongly Millian second-order modal logics here characterised afford the means to resist an argument by Timothy Williamson for the truth of the claim that necessarily, every property necessarily exists. (shrink)

The paper examines Loar’s and Bach’s defence of Nominal Description Theory against Kripkean Modal Argument (MA). Using formal tools of hyperintensional logic, I discriminate three kinds of nominal description which are possible substitutes for a proper name, thus considering various readings of the MA. On its natural understanding, the MA is valid – contrary to what Loar and Bach say. On the other hand, the soundness of the MA remains doubtful, as pointed out already by Loar and Bach.

In this study, we examine modern reading of the Square of Opposition by means of Tichý's Transparent intensional logic. Explicit use of possible world semantics helps us to sharply discriminate between standard and modal readings of categorial statements. We thus get two basic versions of the Square, whereas the Modal Square has not been fully introduced in the contemporary debate yet. Some properties ascribed by mediaeval logicians to the Square require a shift from its Standard to Modal version. Not inevitably, (...) because for each of the two squares there exists its mate which can be easily confused with it. We thus investigate four distinct squares. The discrimination between the initial and modified versions of the Standard and Modal Square leads to better knowledge of the Square as well as a solution of various puzzles often mentioned in recent literature. (shrink)

This book develops a novel generalization of possible world semantics, called ‘world line semantics’, which recognizes worlds and links between world-bound objects (world lines) as mutually independent aspects of modal semantics. Addressing a wide range of questions vital for contemporary debates in logic and philosophy of language and offering new tools for theoretical linguistics and knowledge representation, the book proposes a radically new paradigm in modal semantics. This framework is motivated philosophically, viewing a structure of world lines as a precondition (...) of modal talk. The author provides a uniform analysis of quantification over individuals (physical objects) and objects of thought (intentional objects). The semantic account of what it means to speak of intentional objects throws new light on accounts of intentionality and singular thought in the philosophy of mind and offers novel insights into the semantics of intensional transitive verbs. (shrink)

Necessitists hold that, necessarily, everything is such that, necessarily, something is identical to it. Timothy Williamson has posed a number of challenges to contingentism, the negation of necessitism. One such challenge is an argument that necessitists can more wholeheartedly embrace possible worlds semantics than can contingentists. If this charge is correct, then necessitists, but not contingentists, can unproblematically exploit the technical successes of possible worlds semantics. I will argue, however, that the charge is incorrect: contingentists can embrace possible worlds semantics (...) as wholeheartedly as necessitists. Williamson offers a criterion for a class of models of quantified modal logic to be intended, and argues on its basis that contingentists must deny that there is an intended class of models. I argue that Williamson’s criterion is objectionable, supply an alternative that does not support Williamson’s argument, and adapt Williamson’s construction of an intended model structure to the needs of contingentist metaphysics. (shrink)

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the (...) cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument. (shrink)

This paper is a study of higher-order contingentism – the view, roughly, that it is contingent what properties and propositions there are. We explore the motivations for this view and various ways in which it might be developed, synthesizing and expanding on work by Kit Fine, Robert Stalnaker, and Timothy Williamson. Special attention is paid to the question of whether the view makes sense by its own lights, or whether articulating the view requires drawing distinctions among possibilities that, according to (...) the view itself, do not exist to be drawn. The paper begins with a non-technical exposition of the main ideas and technical results, which can be read on its own. This exposition is followed by a formal investigation of higher-order contingentism, in which the tools of variable-domain intensional model theory are used to articulate various versions of the view, understood as theories formulated in a higher-order modal language. Our overall assessment is mixed: higher-order contingentism can be fleshed out into an elegant systematic theory, but perhaps only at the cost of abandoning some of its original motivations. (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)

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)

This special issue of the Canadian Journal of Philosophy is dedicated to Timothy Williamson's work on modality. It consists of a new paper by Williamson followed by papers on Williamson's work on modality, with each followed by a reply by Williamson. -/- Contributors: Andrew Bacon, Kit Fine, Peter Fritz, Jeremy Goodman, John Hawthorne, Øystein Linnebo, Ted Sider, Robert Stalnaker, Meghan Sullivan, Gabriel Uzquiano, Barbara Vetter, Timothy Williamson, Juhani Yli-Vakkuri.

In this paper ontological implications of the Barcan formula and its converse will be discussed at the conceptual and technical level. The thesis that will be defended is that sentential moods are not ontologically neutral since the rejection of ontological implications of Barcan formula and its converse is a condition of a possibility of the imperative mood. The paper is divided into four sections. In the first section a systematization of semantical systems of quantified modal logic is introduced for the (...) purpose of making explicit their ontological presuppositions. In this context Jadacki's ontological difference between being and existence is discussed and analyzed within the framework of hereby proposed system of quantified modal logic. The second section discusses ontological implications of the Barcan formula and its converse within the system accommodating the difference between being and existence. The third section presents a proof of incompatibility of the Barcan formula and its converse with the use of imperatives. In the concluding section, a thesis on logical pragmatics foreclosing the dilemma between necessitism and contingentism is put forward and defended against some objections. (shrink)

This paper outlines a formal account of tensed sentences that is consistent with Ockhamism, a view according to which future contingents are either true or false. The account outlined substantively differs from the attempts that have been made so far to provide a formal apparatus for such a view in terms of some expressly modified version of branching time semantics. The system on which it is based is the simplest quantified modal logic.