The Barcan formula (BF) is commonly paraphrased as the schematic conditional that if it is possible that there be a phi then something or other is possibly a phi. It is validated by the most straightforward systems of quantified modal logic. It is also widely considered to pose a threat to the commonsensical metaphysical view that there are no non-actual (or ‘merely possible’) things. I show how BF can be cleared of such a charge by construing it as a (...) bridge principle connecting modality de dicto and modality de re while retaining a Russellian robust sense of reality in modal matters. (shrink)

This paper presents structural similarities and historical connections between Prior’s rejection of the Barcan formula and his critique of Berkeley’s master argument for idealism in his 1955 paper “Berkeley in Logical Form”. Making use of Mackie’s paper “Self-Refutation—A Formal Analysis”, it concludes with some suggestions concerning what is at stake in the debate between Prior and Berkeley and in structurally similar debates such as whether to accept the Barcan formula.

The paper criticizes the truthmaker principle that every truth is made true by something. If we interpret ‘something’ as quantifying into sentence position, we can interpret the principle as a harmless logical truth, but that is not what advocates of the principle intend. They interpret ‘something’ as quantifying into name position, and the principle as requiring the existence of truthmaking individuals. The paper argues that we have no reason to believe the principle on this interpretation. Moreover, the converse Barcan (...) formula is inconsistent with the existence of truthmaking individuals for contingent truths. Considerations about our ability to count possible truthmaking individuals are used to argue that we should prefer the converse Barcan formula. (shrink)

We present in this note a plea for Barcan formula. This view connects Barcan formula with a modal principle that expresses the -Introduction rule of first-order logic.

A (normal) system of propositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal (...) logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systems with the Barcan Formula which are incomplete, while those without are complete. In the second group it is those without the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence. (shrink)

It has been argued by Bernard Linsky and Edward Zalta, and independently by Timothy Williamson, that the best quantified modal logic is one that validates both the Barcan Formula and its converse. This requires that domains be fixed across all possible worlds. All objects exist necessarily; some – those we would usually consider contingent – are concrete at some worlds and non-concrete (but still existent) at others. Linsky and Zalta refer to such objects as ‘contingently non-concrete’. I defend the (...) standard usage of the word ‘exists’, and the view that many objects exist only contingently. I argue that the Linsky/Zalta analysis, and to a lesser extent Williamson’s, suffers not only from a peculiar ontology but also from two related formal difficulties. Their analysis gives either counter-intuitive or ad hoc results about essences, and it fails to accommodate contingently existing abstracta. (shrink)

info:eu-repo/semantics/publishedVersion.

The Barcan formula is a theorem of quantified modal logic. Its most straightforward interpretation appears to commit one to “possibilism,” the view that merely possible things exist. Alternative systems of logic revise the formal semantics to preclude the theorem and its consequences. The crux, however, is the modal metaphysics presupposed by the formal semantics. This paper presents an alternative metaphysics of possibility that follows Augustine’s suggestion that God’s plan is only one of a range of alternative histories for a (...) creation. The metaphysics is a version of “trace actualism”—neither pure possibilism nor pure actualism. (shrink)

The paper is intended to provide a full-scale defence of the infamous Barcan Formulae. Not only do I put forth some arguments, both semantic and metaphysical, against recent criticism; I also take pains at supplying some rationale in favour of the formal semantics underlying the Formulae, namely Possibilist quantification. Such a task is carried out through an argument for Compositional Nihilism, according to which nothing but mereological simples ever exists, and consequently through an informal sketch of the (...) metaphysics of possible worlds grounded upon Compositional Nihilism. The paper closes by surveying the main reasons why the account championed here isn’t defective with respect to other possibilist accounts of existence and modality presently on stage. (shrink)

The revolution in semantics in the late 1960s and 1970s overturned an earlier competing paradigm, ‘translational’ semantics. I revive and defend Prior’s translational semantics for modals and tense-modals. I also show how to extend Prior’s propositional modal semantics to quantificational modal logic, and use the resulting semantics to formalize Prior’s own counterexample to the Barcan Formula.

On its usual interpretation, the Barcan Formula—◊∃_xB_ → ∃_x_◊_B_—says that, if there could have been something that is such and such a way, then there is something that could have been that way. It is traditionally held that contingentist actualists should—indeed, must—reject the Barcan Formula. I argue that contingentist actualists should—indeed, must—endorse the Barcan Formula, at least assuming a standard, Tarskian conception of truth and truth preservation. I end by proposing a logic for contingentist actualists that validates (...) the Barcan Formula. This logic has the surprising feature of also validating the Converse Barcan Formula, □∀_xB_ → ∀_x_□_B_, while still invalidating related formulas—such as □∀_x_□∃_y x_ = _y_ (NNE)—that contingentist actualists should reject. It does this by employing models with fixed domains but assignments to the identity predicate that vary across worlds. (shrink)

Based on her earlier ground-breaking axiomatization of quantified modal logic, the papers collected here by the distinguished philosopher Ruth Barcan Marcus cover much ground in the development of her thought, spanning from 1961 to 1990. The first essay here introduces themes initially viewed as iconoclastic, such as the necessity of identity, the directly referential role of proper names as "tags", the Barcan Formula about the interplay of possibility and existence, and alternative interpretations of quantification. Marcus also addresses the (...) putative puzzles about substitutivity and about essentialism. The collection also includes influential essays on moral conflict, on belief and rationality, and on some historical figures. Many of her views have been incorporated into current theories, while others remain part of a continuing debate. (shrink)

En esta nota discutimos una forma generaldel Teorema de Deducción (TD) para sistemas modales de primer orden, la cual permite derivar varias otras formulaciones del mismoque aparecen en la literatura, así como su relación con la Fórmula de Barcan.

RESUMEN Se ha destacado que la Fórmula de Barcan -FB- y la Conversa de la Fórmula de Barcan -CFB- para lógica modal cuantificacional de orden superior parecen válidas. Si se interpreta que los cuantificadores tienen como rango propiedades, la validez de FB y CFB parece implicar la existencia de universales trascendentes, que no requieren estar instanciados para existir en un mundo posible. Se discute esta argumentación, porque la semántica, en la que los resultados de validez se siguen, no (...) requiere que las ‘intensiones' asignadas a las variables de orden superior estén instanciadas en un mundo posible para que la intensión exista ahí. En una semántica modificada, más neutral, FB y CFB de orden superior ya no son válidas. Además se sostiene que, incluso si FB y CFB de orden superior fuesen válidas, no se obtendrían resultados metafísicos sustantivos, pues diferentes formas de nominalismo y teorías de tropos las aceptarían. ABSTRACT It has been pointed out that the Barcan Formula -BF- and the Converse Barcan Formula -CBF- seem to be valid for higher-order modal quantificational logic. If quantifiers are construed as ranging over properties, the validity of BF and CBF seems to entail the existence of transcendent universals that do not require instantiation in order to exist in a possible world. The paper discusses this argument given that semantics, in which validity results follow from one another, do not require that the "intensions" assigned to higher- order variables be instantiated in a possible world in order for the intension to exist in such a world. In a modified, more neutral semantics, higher-order BF and CBF are no longer valid. The paper also argues that even if higher-order BF and CBF were valid, no substantive metaphysical results would be obtained, since different forms of nominalism and theories of tropes would accept them. (shrink)

In 2003 Zia Movahed pointed to a passage of Avicenna, written probably in 1022, which Movahed claimed anticipated the modal formula of Barcan (that ‘For every |$x$| necessarily |$\phi $|’ entails ‘Necessarily for every |$x$||$\phi $|’), and its converse. Since 2003, examination of early logical writings of Avicenna has clarified how he understood entailments between modal sentences, using his own new temporal language to provide a kind of semantics. In the light of that, Movahed’s claim for the Barcan (...) formula needs some tidying up but is basically correct. But by his semantics Avicenna should not have claimed the converse Barcan formula and probably didn’t claim it. (shrink)

The great contribution Marcus has made to several of intensely discussed topics in philosophy might not have been noticed fully without this collection of some of her most important articles that makes it evident that her achievement is not limited to inventing the famous Barcan formula.

Se ha destacado recientemente que la Fórmula de Barcan y la Conversa de la Fórmula de Barcan para lógica modal cuantificacional de orden superior parecen válidas. Si se interpreta que los cuantificadores tienen como rango propiedades, la validez de y de orden superior parece implicar la existencia de universales trascendentes, esto es, universales que no requieren estar instanciados para existir en un mundo posible. Este trabajo discute esta línea de argumentación. En primer lugar, se sostiene que la semántica (...) en la que los resultados de validez se siguen no requiere que las ‘intensiones’ asignadas a las variables de orden superior estén instanciadas en un mundo posible para que la intensión exista ahí. El defensor de universales inmanentes objetaría esta suposición. En una semántica modificada, más neutral, y de orden superior ya no son válidas. En segundo lugar, se sostiene que, incluso si se concediese la validez de y de orden superior, no se obtendrían resultados metafísicos sustantivos, pues diferentes formas de nominalismo y diferentes teorías de tropos no tendrían dificultades en aceptar y de orden superior. (shrink)

Obituary. Ruth Barcan Marcus' contributions to modal logic and to semantics are discussed.

The paper presents a semantics for quantified modal logic which has a weaker axiomatization than the usual Kripke semantics. In particular, the Barcan Formula and its converse are not valid with the proposed semantics. Subclasses of models which validate BF and other interesting formulas are presented. A completeness theorem is proved, and the relation between this result and completeness with respect to Kripke models is investigated.

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)

In "An Alternative Semantics for Quantified Relevant Logic" (JSL 71 (2006)) we developed a semantics for quantified relevant logic that uses general frames. In this paper, we adapt that model theory to treat quantified modal logics, giving a complete semantics to the quantified extensions, both with and without the Barcan formula, of every proposi- tional modal logic S. If S is canonical our models are based on propositional frames that validate S. We employ frames in which not every set (...) of worlds is an admissible proposition, and an alternative interpretation of the uni- versal quantifier using greatest lower bounds in the lattice of admissible propositions. Our models have a fixed domain of individuals, even in the absence of the Barcan formula. For systems with the Barcan formula it is possible to preserve the usual Tarskian reading of the quantifier, at the expensive of sometimes losing va- lidity of S in the underlying propositional frames. We apply our results to a number of logics, including S4.2, S4M and KW, whose quantified extensions are incomplete for the standard semantics. (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)

In his book Modal Logic as Metaphysics, Timothy Williamson defends first-order necessitism using simplicity as a powerful argument. However, simplicity is decomposed into two different, even antagonistic, sides: elegance and parsimony. On the one hand, elegance is the property of theories possessing few and simple principles that allow them to deploy all their theoretical power; on the other hand, parsimony is the property of theories having the fair and necessary number of ontological entities that allow such theories give an account (...) of themselves. Since necessitism endorses Barcan Formulae for the sake of elegance, it is committed to a vast number of contingently non-concrete objects, so one may think that it is not qualitatively parsimonious. I argue that necessitism could be viewed as an additive case in the sense that Alan Baker characterizes the adjective so quantitative parsimony should be considered when it comes to necessitism instead of qualitative parsimony. (shrink)

En este artículo, presentamos un sistema de lógica modal que permite representar relaciones entre conjuntos o clases de individuos definidos por una propiedad específica. Introducimos dos operadores modales, [a] y, que se utilizan respectivamente para expresar "para todo A" y "existe un A". Tanto la sintaxis como la semántica del sistema tienen dos niveles que evitan el anidamiento del operador modal. La semántica se basa en una variante de la semántica de Kripke, en donde los operadores modales se indexan sobre (...) fórmulas de lógica proposicional (“pre-fórmulas” en el trabajo). Además, presentamos un conjunto de axiomas y reglas que rigen el sistema, y demostramos que el sistema es correcto y completo con relación a los modelos de Kripke. En la sección final del artículo, discutimos posibles trabajos futuros. Consideramos la posibilidad de combinar nuestro operador modal con otras modalidades, como necesidad o conocimiento. Además, como ejemplo de la utilidad de nuestro operador modal, analizamos brevemente la fórmula de Barcan adaptada de manera conveniente dentro del marco de nuestro sistema. En resumen, proponemos la combinación de nuestro operador modal con otros como una forma más simple y compacta, aunque con un menor poder expresivo, para abordar la lógica modal cuantificada. (shrink)

The purpose of this paper is to describe a set of quantified temporal alethic-deontic systems, i.e., systems that combine temporal alethicdeontic logic with predicate logic. We consider three basic kinds of systems: constant, variable and constant and variable domain systems. These systems can be augmented by either necessary or contingent identity, and every system that includes identity can be combined with descriptors. All logics are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by (...) the so-called T × W semantics, to characterize them semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics. (shrink)

The theorems of the simplest and strongest sensible quantified modal logic include the Barcan Formula and its converse. Both formulas face strong intuitive objections. This paper develops a theory of possibilia to meet those objections.

The paper is a critique of the widespread conception of logic as a neutral arbiter between metaphysical theories, one that makes no `substantive’ claims of its own (David Kaplan and John Etchemendy are two recent examples). A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds (however mistaken), with a consequent proliferation of alternative logics. However, this apparent contentiousness of logic is often treated as though it were (...) neutralized by the possibility of studying all these alternative logics within an agreed metalogical framework, typically that of first-order logic with set theory. In effect, metalogic is given the role of neutral arbiter. The paper will consider a variety of examples in which deep logical disputes re-emerge at the meta-level. One case is quantified modal logic, where some varieties of actualism require a modal meta-language (as opposed to the usual non-modal language of possible worlds model theory) in order not to make their denial of the Barcan formula self-defeating. Similarly, on some views the intended model theory for second-order logic can only be given in a second-order metalanguage—this may be needed to avoid versions of Russell’s paradox when the first-order quantifiers are read as absolutely unrestricted. It can be shown that the phenomenon of higher-order vagueness eventually forces fuzzy logical treatments of vagueness to use a fuzzy metalanguage, with consequent repercussions for what first-order principles are validated. The difficulty of proving the completeness of first-order intuitionistic logic on its intended interpretation by intuitionistically rather than just classically valid means is a more familiar example. These case studies will be discussed in some detail to reveal a variety of ways in which even metalogic is metaphysically contested, substantial and non-neutral. (shrink)

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation (...) of quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

Relativized Metaphysical Modality (RMM: Murray and Wilson, 'Relativized metaphysical modality', Oxford Studies in Metaphysics, 2012; Murray, Perspectives on Modal Metaphysics, 2017) exploits 'two-dimensionalist' resources to metaphysical, rather than epistemological, ends: the second dimension offers perspective-dependence without contingency, diverting attacks on 'Classical' analyses of modals (in effect, analyses validating S5 and the Barcan Formulae). Here, we extend the RMM program in two directions. First, we harvest resources for RMM from Lewis's 1980 'Context--Index' (CI) framework: (a) the ban in CI (...) on binding into context-arguments (akin to Kaplan's 'monstrosity' ban) projects a bright line between perspective-dependence and contingency; and (b) CI-postulated connections among meaning, content, truth, argument-structure, context, and modality collectively generate a 'Generalized Humphrey Problem' for any non-Classical analysis (examples covered include appeals to accessibility, contingent domains, and counterpart relations). Second, we sharpen the tools of RMM-based metaphysical analysis, and extend their domain of coverage across familiar anomalies for Classical modals: we revisit earlier RMM-based bulwarks for S5 (against 'Chisholm's Paradox' for moderate flexibility of essence, and nomological necessitarianism); and we now similarly shore up the Barcan Formulae (against the apparent contingency of existence and nonexistence). (shrink)

This paper investigates the metaphysics in higher-order counterfactual logic. I establish the necessity of identity and distinctness and show that the logic is committed to vacuism, which entails that all counteridenticals are true. I prove the Barcan, Converse Barcan, Being Constraint and Necessitism. I then show how to derive the Identity of Indiscernibles in counterfactual logic. I study a form of maximalist ontology which has been claimed to be so expansive as to be inconsistent. I show that it (...) is equivalent to the collapse of the counterfactual into the material conditional---which is itself equivalent to the modal logic TRIV. TRIV is consistent, from which it follows that maximalism is, surprisingly, consistent. I close by arguing that stating the limit assumption requires a higher-order logic. (shrink)

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

In this extended critical discussion of 'Kant's Modal Metaphysics' by Nicholas Stang (OUP 2016), I focus on one central issue from the first chapter of the book: Stang’s account of Kant’s doctrine that existence is not a real predicate. In §2 I outline some background. In §§3-4 I present and then elaborate on Stang’s interpretation of Kant’s view that existence is not a real predicate. For Stang, the question of whether existence is a real predicate amounts to the question: ‘could (...) there be non-actual possibilia?’ (p.35). Kant’s view, according to Stang, is that there could not, and that the very notion of non-actual or ‘mere’ possibilia is incoherent. In §5 I take a close look at Stang’s master argument that Kant’s Leibnizian predecessors are committed to the claim that existence is a real predicate, and thus to mere possibilia. I argue that it involves substantial logical commitments that the Leibnizian could reject. I also suggest that it is danger of proving too much. In §6 I explore two closely related logical commitments that Stang’s reading implicitly imposes on Kant, namely a negative universal free logic and a quantified modal logic that invalidates the Converse Barcan Formula. I suggest that each can seem to involve Kant himself in commitment to mere possibilia. (shrink)

In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between $\textbf {QwGrz}$ and $\textbf {QGL.3}$ or between $\textbf {QwGrz}$ and $\textbf {QGrz.3}$ is $\Pi ^1_1$-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single (...) proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ are not Kripke complete. Both $\Pi ^1_1$-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula. (shrink)

This paper examines two logical principles that combine modality with quantification and with identity: a weaker version of the converse Barcan formula, and the principle of the necessity, not of identity, but of distinctness. It is argued that there are conceptual assumptions that lie behind and help the independence of these principles, and a semantics with some conceptual interest that invalidates them. A qualified converse Barcan formula is discussed. It is shown that the necessity of distinctness is an (...) independent principle, even in a theory that includes the qualified converse Barcan formula. (shrink)

Pace Necessitism – roughly, the view that existence is not contingent – essential properties provide necessary conditions for the existence of objects. Sufficiency properties, by contrast, provide sufficient conditions, and individual essences provide necessary and sufficient conditions. This paper explains how these kinds of properties can be used to illuminate the ontological status of merely possible objects and to construct a respectable possibilist ontology. The paper also reviews two points of interaction between essentialism and modal logic. First, we will briefly (...) see the challenge that arises against S4 from flexible essential properties; as well as the moves available to block it. After this, the emphasis is put on the Barcan Formula (BF), and on why it is problematic for essentialists. As we will see, Necessitism can accommodate both (BF) and essential properties. What necessitists cannot do at the same time is to continue to understanding essential properties as providing necessary conditions for the existence of individuals; against what might be for some a truism. (shrink)

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem''s modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap''s display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse (...) to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. (shrink)

Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to assess Williamson’s (...) arguments either for necessitism (although discussion of these arguments will play a central role in the dialectic) or for necessitism’s two famous corollaries. Instead, the focus shall be a general principle governing abstract objects—the abstract of principle (or AOP) —instances of which seems to be at work in some of Williamson’s central arguments for necessitism. The AOP can be straightforwardly formulated and applied within the neo-logicist framework—in fact, arguably the principle is most naturally formulated in neo-logicist terms. -/- After closely examining, and carefully formalizing, the AOP, the remainder of the paper focuses on arguments for necessitism-like claims (the exact meaning of “necessitism-like” will become clearer as the essay progresses) based on the AOP. In particular, we shall focus on the instance of the AOP that applies to the abstract objects governed by the most well-known and most fully studied abstraction principle: Hume’s Principle (HP). It turns out that, although we cannot reconstruct a valid argument for necessitism based on this numerical instance of the AOP, we can obtain valid arguments for weaker, but equally interesting conclusions. In particular, we shall show that, although HP combined with the AOP (and some additional, related assumptions) allows the contents of the domains of possible worlds to vary, the size of those domains must remain constant. The paper concludes by developing and critiquing some related arguments for necessitism based on applying relevant instances of the AOP to abstraction principles governing sets (or extensions), and to a simple objectual abstraction principle. (shrink)

This paper aims to provide two abductive considerations adducing in favor of the thesis of Necessitism in modal ontology. I demonstrate how instances of the Barcan formula can be witnessed, when the modal operators are interpreted 'naturally' -- i.e., as including geometric possibilities -- and the quantifiers in the formula range over a domain of natural, or concrete, entities and their contingently non-concrete analogues. I argue that, because there are considerations within physics and metaphysical inquiry which corroborate modal relationalist (...) claims concerning the possible geometric structures of spacetime, and dispositional properties are actual possible entities, the condition of being grounded in the concrete is consistent with the Barcan formula; and thus -- in the geometric setting -- merits adoption by the Necessitist. (shrink)

This paper aims to provide two abductive considerations adducing in favor of the thesis of Necessitism in modal ontology. I demonstrate how instances of the Barcan formula can be witnessed, when the modal operators are interpreted 'naturally' -- i.e., as including geometric possibilities -- and the quantifiers in the formula range over a domain of natural, or concrete, entities and their contingently non-concrete analogues. I argue that, because there are considerations within physics and metaphysical inquiry which corroborate modal relationalist (...) claims concerning the possible geometric structures of spacetime, and dispositional properties are actual possible entities, the condition of being grounded in the concrete is consistent with the Barcan formula; and thus -- in the geometric setting -- merits adoption by the Necessitist. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (...) (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

There are logics where necessity is defined by means of a given identity connective: \ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity \ can be defined by strict equivalence \}\). All these approaches to modality involve a principle that we call the Collapse Axiom : “There is only one necessary proposition.” In this paper, we consider a notion of PI which relies on the identity axioms of Suszko’s non-Fregean logic SCI. (...) Then S3 proves to be the smallest Lewis modal system where PI can be defined as SE. We extend S3 to a non-Fregean logic with propositional quantifiers such that necessity and PI are integrated as non-interdefinable concepts. CA is not valid and PI refines SE. Models are expansions of SCI-models. We show that SCI-models are Boolean prealgebras, and vice-versa. This associates non-Fregean logic with research on Hyperintensional Semantics. PI equals SE iff models are Boolean algebras and CA holds. A representation result establishes a connection to Fine’s approach to propositional quantifiers and shows that our theories are conservative extensions of S3–S5, respectively. If we exclude the Barcan formula and a related axiom, then the resulting systems are still complete w.r.t. a simpler denotational semantics. (shrink)