Descartes's view of modality is analyzed by contrast to two earlier models: the ancient realist one, defended by Boethius, where possibility and necessity are connected to natural potency, and the modern intensionalist one, which dissociates necessary and possible truths from any ontological foundation, treating them as conceptual, a priori given preconditions for any intellect. The emergence of this view is traced from Gilbert of Poitiers to duns Scotus, Ockham and Suarez. The Cartesian theory of the creation of eternal truths, (...) it is argued, involves a rejection of this idea of absolute conceivability and can be seen as a constructivist view of intelligibility and rationality. (shrink)

This book develops an argument for a foundational theory of modality using higher-order logic. The use of higher-order logic in metaphysics is motivated, and a particular higher-order logic is introduced. Fine-grained theories of propositional individuation are shown to be problematic, and a course-grained theory of propositional individuation is defended. On the basis of this theory, it is argued that the metaphysical necessities can be delineated using purely logical terms; by adding an actuality operator, it is shown that the logic (...) of metaphysical necessity is S5. The theoretical role of possible worlds is articulated, and shown to be satisfied by maximal propositions if it satisfied by anything; the existence of such maximal propositions is demonstrated using plural propositional quantification. The resulting theory reduces metaphysical modal terms to higher-order logical terms, and thereby vindicates a widespread modal conception of metaphysics. (shrink)

“Modal logic was conceived in sin: the sin of confusing use and mention.” So quips Quine. The stigma stuck with modal logic for a while. But by the mid-sixties, a whole cluster of mathematically elegant interpretations of modal logic became available. All are natural extensions of the classical Tarskian semantics of predicate logic. By the mid-seventies, Quine’s criticisms seemed obsolete. Today, we teach the model theory of modal logic as a matter of course. Quine’s “interpretive problem” is just forgotten. The (...) purpose of my dissertation is to revamp Quine’s thesis: the very foundations of modal logic—simple propositional modal logic, even before quantifiers are added—are shaky. Quine holds that the natural interpretation of the operator “necessarily” is in terms of a full-blooded metaphysical notion of real necessity. Such a notion, Quine conjectures, is committed to the “metaphysical jungle of Aristotelian essentialism”. Hence, the intended interpretation is to be rejected. Thus Quine: so much the worse for real necessity. An alternative comes to mind: a meta-level, formal rather than real, semantic interpretation of the operator of necessity. In the late forties, Carnap developed this line of interpretation which culminates in the aforementioned model theory (“Kripke’s possible world semantics”) in the sixties. On this basis a formal interpretation is introduced: it grounds necessity in the notion of model theoretic validity. As pointed by friends of the formal interpretation (Marcus, Parsons and Kaplan), no invidious essentialist claim is verified. But have Quine’s concerns been answered? Yes and no. Technical problems regarding quantification across modal operators, substitutivity, matters of scope, and the basis of cross-world identification have indeed been solved. However, no real interpretation of “necessarily” has been provided. I argue that the seeds of such an interpretation are present in Kripke’s philosophical discussion of de re necessities in Naming and Necessity. It is not model theoretic, and it does not reduce necessity to some non-modal notion, just as Quine predicted. Even more in his vein, the real interpretation is (i) metaphysically, grounded indeed in essentialist theses, and (ii) epistemologically, forced to give up Kant’s ideal that all necessities are known a priori. (shrink)

The book also sheds new light on the nature of metaphysical theorizing by exploring the interaction of semantic and metaphysical issues, the connections between different metaphysical issues, and the nature of ontological commitment.

(2013). Thomas Bradwardine on God and the Foundations of Modality. British Journal for the History of Philosophy: Vol. 21, No. 2, pp. 368-380. doi: 10.1080/09608788.2012.689754.

Prominent thinkers such as Kripke and Rescher hold that Russell has no modal logic, even that Russell was indisposed toward modal logic. In Part I, I show that Russell had a modal logic which he repeatedly described and that Russell repeatedly endorsed Leibniz's multiplicity of possible worlds. In Part II, I describe Russell's theory as having three ontological levels. In Part III, I describe six Parmenidean theories of being Russell held, including: literal in 1903; universal in 1912; timeless in 1914; (...) transcendental in 1918–1948. The transcendental theory underlies the primary level of Russell's modal logic. In Part IV, I examine Rescher's view that Russell and modal logic did not mix. (shrink)

Modern logicians have sought to unlock the modal secrets of Aristotle's Syllogistic by assuming a version of essentialism and treating it as a primitive within the semantics. These attempts ultimately distort Aristotle's ontology. None of these approaches make full use of tests found throughout Aristotle's corpus and ancient Greek philosophy. I base a system on Aristotle's tests for things that can never combine (polarity) and things that can never separate (inseparability). The resulting system not only reproduces Aristotle's recorded results for (...) the apodictic syllogistic in the Prior Analytics but it also generates rather than assumes Aristotle's distinctions among 'necessary', 'essential' and 'accidental'. By developing a system around tests that are in Aristotle and basic to ancient Greek philosophy, the system is linked to a history of practices, providing a platform for future work on the origins of logic. (shrink)

This paper proposes a logic, motivated by modal interpretations, in which every quantum mechanics propositions has a truth-value. This logic is completely classical, hence violates the conditions of the Kochen-Specker theorem. It is shown how the violation occurs, and it is argued that this violation is a natural and acceptable consequence of modal interpretations. It is shown that despite its classicality, the proposed logic is empirically indistinguishable from quantum logic.

This paper proposes a logic, motivated by modal interpretations, in which every quantum mechanics propositions has a truth-value. This logic is completely classical, hence violates the conditions of the Kochen-Specker theorem. It is shown how the violation occurs, and it is argued that this violation is a natural and acceptable consequence of modal interpretations. It is shown that despite its classicality, the proposed logic is empirically indistinguishable from quantum logic.

In the past three decades there has been a rapid development of the formal machinery for modal logic. Quantified modal logic has developed along with a semantics and model theory that is appropriate to it. With this technical development there has been relatively little discussion of what modality is all about. There are two fundamental questions that have gone unanswered. First, to what does necessity amount? Is this a new logical notion, or is it something that can be further (...) analyzed in terms of other notions that we already have at our disposal? The second question is what makes truths involving necessity true? What is their ontological grounding in the world? This essay is directed at the first of these questions. ;There are three possible reductions of metaphysical modality. First, there is the reduction suggested by the semantics for modal logic: a reduction in terms of possible worlds. Second, there is the reduction in terms of cognition. What is possible is what is conceivable. Third, there is the reduction in terms of language. Necessities arise from the structures of language and/or language use. ;I argue that none of these is successful. All of them suffer either from presupposing some form of metaphysical necessity in the reductive base, rendering them circular and not truly reductive, or they do not do justice to the modality being reduced. ;Since necessity is not reducible to any non-modal features of the world, we need to determine whether we ought to reject it as unacceptable on this basis, or whether we ought to retain it, but treat is as a new primitive that is introduced into a philosophical or logical theory. I urge the latter course is the one that we ought to take because without it too many things will likewise be deemed unacceptable. Causal necessity and explanation, normally accepted theories or knowledge and perception, logic and logical validity, and one common manner of distinguishing sets from properties will be deemed unacceptable if de re necessity is unacceptable. All of these presuppose de re metaphysical necessity in some way. I conclude that de re metaphysical necessity ought to be retained in our theory as primitive, unanalyzable notion. (shrink)

The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.

Why is two-dimensional semantics important? One can think of it as the most recent act in a drama involving three of the central concepts of philosophy: meaning, reason, and modality. First, Kant linked reason and modality, by suggesting that what is necessary is knowable a priori, and vice versa. Second, Frege linked reason and meaning, by proposing an aspect of meaning (sense) that is constitutively tied to cognitive signi?cance. Third, Carnap linked meaning and modality, by proposing an (...) aspect of meaning (intension) that is constitutively tied to possibility and necessity. (shrink)

正しいから証明できるのか、証明できるから正しいのか。数学にとって証明とは何か、正しさとは何なのかは数学基礎論の根本的な問題である。様相論理を軸とした、証明と真理に関わる数学基礎論の古典的な結果から最先 端の議論までを解説した。.

In this two-parts paper we present paranormal modal logic: a modal logic which is both paraconsistent and paracomplete. Besides using a general framework in which a wide range of logics  including normal modal logics, paranormal modal logics and classical logic can be defined and proving some key theorems about paranormal modal logic (including that it is inferentially equivalent to classical normal modal logic), we also provide a philosophical justification for the view that paranormal modal logic is a formalization of (...) the notions of skeptical and credulous plausibility. (shrink)

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

It is a truth universally acknowledged that a claim of metaphysical modality, in possession of good alethic standing, must be in want of an essentialist foundation. Or at least so say the advocates of the reductive-essence-first view, according to which all modality is to be reductively defined in terms of essence. Here, I contest this bit of current wisdom. In particular, I offer two puzzles—one concerning the essences of non-compossible, complementary entities, and a second involving entities whose essences (...) are modally ‘loaded’—that together strongly call into question the possibility of reducing modality to essence. (shrink)

In general, a given object could have been different in certain respects. For example, the Great Pyramid could have been somewhat shorter or taller; the Mona Lisa could have had a somewhat different pattern of colours; an ordinary table could have been made of a somewhat different quantity of wood. But there seem to be limits. It would be odd to suppose that the Great Pyramid could have been thimble-sized; that the Mona Lisa could have had the pattern of colours (...) that actually characterizes The Scream; or that the table could have been made of the very quantity of wood that in fact made some other table. However, there are puzzling arguments that purport to show that so long as an object is capable of being somewhat different in some respect, it is capable of being radically different in that respect. These arguments rely on two tempting thoughts: first, that an object’s capacity for moderate variation is a non-contingent matter, and second, that what is possibly possible is simply possible. This book systematically investigates competing strategies for resolving these puzzles, and defends one of them. Along the way it engages with foundational questions about the metaphysics of modality. (shrink)

This volume covers the period from the beginning of Russell's work on Volume Two of the Principles of Mathematics to the critical discovery of the theory of descriptions in 1905. Foundations of Logic gives a vivid picture of Russell wrestling with the logical paradoxes, often unsuccessfully, as he tries out one foundational scheme after another. This volume provides the key to both Bertrand Russell's philosophy of logic and philosophy of mathematics. It includes unpublished work on the theory of denoting (...) which predates Russell's famous article of 1905 and unpublished manuscripts on the so-called "zig-zag" theory with which Russell attempted to provide a type-free foundation for mathematics. The volume also gathers together for the first time a number of reviews and survey articles, along with two talks on modality and truth. It will be an essential addition to any Bertrand Russell collection. (shrink)

It is a truth universally acknowledged that a claim of metaphysical modality, in possession of good alethic standing, must be in want of an essentialist foundation. Or at least so say the advocates of thereductive-essence-firstview, according to which all modality is to be reductively defined in terms of essence. Here, I contest this bit of current wisdom. In particular, I offer two puzzles—one concerning the essences of non-compossible, complementary entities, and a second involving entities whose essences are modally (...) ‘loaded’—that together strongly call into question the possibility of reducing modality to essence. (shrink)

The paper analyzes, from a perspective which is itself existential-ontological, the way in which in an early text of Martin Heidegger, Phänomenologische Interpretationen zu Aristoteles (Anzeige der hermeneutischen Situation) [1922] – which had already outlined some determinative elements of the ideas expounded in Being and Time –, the meditation on the always living and current conditions and hermeneutical situation of philosophizing expanded in fact into an inquiry about the origins, grounds, essence and sense of philosophy as such. Meditation in and (...) through which philosophy identifies itself and is founded on the one hand exactly as a mode of existence of the mortal “human Dasein” (menschliches Dasein), that is a factic mode of existence of this, philosophy, on the other hand, itself originates from and in man’s factic life exactly with the aim of being the modality through which this being – namely ourselves – returns towards the problematization of his existential possibilities even by taking upon himself the burden and “weight” of radical interrogation. Which therefore goes and must go itself and resolutely – because if this entirely “without God” and consequently a-theist – to the historical and ontological roots of a present con-temporarized (mitzeitigt) both with the past and the future existential horizons of the assumed factic possibilities. (shrink)

This thesis is a study on the nature of modality in Hegel’s Logic and contemporary modal metaphysics. The thesis has two aims: Firstly, it examines Lewisian modal realism, as well as the post-Lewisian modal metaphysical accounts of modal actualism and modal essentialism in order to reveal that each position appeals to a non-modal foundation to make modal concepts explicit. Each position thus falls under what Hegel regards as pre-critical metaphysics by suggesting a modally unaccountable ground for modal concepts. The (...) very idea of a foundation to modality is problematic because as a foundation, it is necessarily assumed without further justification, and it directs our attention away from the nature of modality itself to its supposed foundations. These non-modal foundational take the form of possible worlds for modal realism, the form of properties for modal actualism and the form of essence for modal essentialism. To eschew foundationalism in modal metaphysics and provide a less questionable basis for modality, the thesis suggests Hegel’s treatment of modality as a viable alternative in virtue of its being free from the concerns stemming from the pre-critical way of doing metaphysics. The thesis situates Hegel in contemporary modal metaphysics by undertaking two types of exposition. By examining Hegel's treatment of essence and property, the thesis will show negatively that essence and property cannot be a foundation for modality, while by examining Hegel’s treatment of modality, the thesis positively proves that modal concepts are analysable and explainable within the domain of actuality, a domain that can only be made explicit by modal determinations. These two expositions provide an idea of a self-sustaining and self-explanatory modal metaphysics without appealing to non-modal foundationalist metaphysical commitments. More precisely, I argue that Hegel’s logical derivation of modal concepts constitutes a Hegelian modal metaphysics, which contains a comprehensive account of modal concepts that does not rely on any non-modal primitive. (shrink)

I provide an analysis of sentences of the form ‘To be F is to be G’ in terms of exact truth-maker semantics—an approach that identifies the meanings of sentences with the states of the world directly responsible for their truth-values. Roughly, I argue that these sentences hold just in case that which makes something F is that which makes it G. This approach is hyperintensional, and possesses desirable logical and modal features. These sentences are reflexive, transitive and symmetric, and, if (...) they are true, then they are necessarily true, and it is necessary that all and only Fs are Gs. I close by defining an asymmetric and irreflexive notion of analysis in terms of the reflexive and symmetric one. (shrink)

I provide an analysis of sentences of the form ‘To be F is to be G’ in terms of exact truth-maker semantics—an approach that identifies the meanings of sentences with the states of the world directly responsible for their truth-values. Roughly, I argue that these sentences hold just in case that which makes something F also makes it G. This approach is hyperintensional and possesses desirable logical and modal features. In particular, these sentences are reflexive, transitive, and symmetric, and if (...) they are true, then they are necessarily true, and it is necessary that all and only Fs are Gs. I motivate my account over Correia and Skiles’ [11] prominent alternative and close by defining an irreflexive and asymmetric notion of analysis in terms of the symmetric and reflexive notion. (shrink)

In this pioneering book Kasia Jaszczolt lays down the foundations of an original theory of meaning in discourse, reveals the cognitive foundations of discourse interpretation, and puts forward a new basis for the analysis of discourse processing. She provides a step-by-step introduction to the theory and its application, and explains new terms and formalisms as required. Dr Jaszczolt unites the precision of truth-conditional, dynamic approaches with insights from neo-Gricean pragmatics into the role of speaker's intentions in communication. She (...) shows that the compositionality of meaning may be understood as merger representations combining information from various sources including word meaning and sentence structure, various kinds of default interpretations, and conscious pragmatic inference. Among the applications the author discusses are constructions that pose problems in semantic analysis such as referring expressions, propositional attitude constructions, presupposition, modality, numerals, and sentential connectives. She proposes solutions to cutting edge problems in the semantics/pragmatics interface - for example, how many levels of meaning should be distinguished; the status of underspecification; how much contextual information should be placed in the representation of the speaker's meaning; whether there are default interpretations; the stage of utterance interpretation at which pragmatic inference begins; and whether compositionality is a necessary feature of the theory of meaning and if so how it is to be defined.The book is for students and researchers in semantics, pragmatics, computational linguistics, and philosophy of language at advanced undergraduate level and above. (shrink)

In this paper we show that a variety of modal algebras of finite type is semisimple iff it is discriminator iff it is both weakly transitive and cyclic. This fact has been claimed already in [4] (based on joint work by the two authors) but the proof was fatally flawed.

In this article I defend a form of classical possibilism with an actualist foundation. As a matter of fact, I believe that this position is more in keeping with the classical metaphysical tradition. According to this form of possibilism, I construe possible objects as possible non-existing objects of an existing producing power. Consequently, they are nothing vis-à -vis the modality of their own actual being, although they do exist with regard to the modality of the producing power’s being. (...) The actualist requirement prescribed by the Frege-Quinean criterion of the quantification domain is thus fulfilled; indeed, really possible objects are not actual objects, but their possibility is actual. (shrink)

In the modal literature various notions of "completeness" have been studied for normal modal logics. Four of these are defined here, viz. completeness, first-order completeness, canonicity and possession of the finite model property -- and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and (...) preservation under ultrafilter extensions are introduced and studied as well. (shrink)

In Chapter One I present an interpretation of Descartes' theory of distinction. I argue that the best understanding of the notion of separate existence at stake in the real distinction between mind and body is not that each can exist without the other existing, nor that each can exist without a real union with the other, but that each can exist without the attributes of the other. However, the only notion of separate existence which can provide an adequate acccount of (...) all three kinds of distinction--real, modal, and of reason--is that of one thing existing without a real union with another thing. ;In Chapter Two I examine Descartes' claims about the dependence of created substances on God and their independence from other creatures. The pure substances, those which are not composed of parts, can be caused to come into existence and go out of existence only by God, and they can perish only by annihilation. These substances include individual created minds and the extended universe considered as a whole. In contrast, bodies and human beings, which are composed of parts, can be caused to go out of existence by other creatures, and they can perish without being annihilated. Nevertheless, I argue that Descartes thinks bodies and human beings are full-fledged substances. ;In Chapter Three I discuss the union of mind and body. I argue that Descartes thinks the human soul is the only substantial form and that it inheres in the body as form inheres in matter. I defend his account of how two substances can form a unity by comparing it with medieval accounts of the unity of composites. ;Chapter Four examines Descartes' arguments for rejecting substantial forms and the so-called real qualities. I argue that his earliest series of arguments, those from perception and scientific explanation, is the most successful. ;Finally, in Chapter Five, I consider Descartes' positive account of the status of those so-called real qualities which he asserts are merely sensations, for example, color, sound, and heat. I argue that he is best understood as holding that they are modes of both thought and extension which are property ascribed to the composite of mind and body. (shrink)

In the modal literature various notions of completeness have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability (...) and preservation under ultrafilter extensions are introduced and studied as well. (shrink)

Reviews of the papers mentioned in the title.

The propensity interpretation of fitness (PIF) holds that evolutionary fitness is an objectively probabilistic causal disposition (i.e., a propensity) toward reproductive success. I characterize this as the conceptual foundation of the PIF. Reproductive propensities are meant to explain trends in actual reproductive outcomes. In this paper, I analyze the minimal theoretical and ontological commitments that must accompany the explanatory power afforded by the PIF’s foundation. I discuss three senses in which these commitments are less burdensome than has typically been recognized: (...) the PIF’s foundation is (i) compatible with a principled pluralism regarding the mathematical relationship between measures of individual and trait reproductive success; (ii) independent of the propensity interpretation of probability; and (iii) independent of microphysical indeterminism. The most substantive ontological commitment of the PIF’s foundation is to objective modal structures wherein macrophysical probabilities and causation can be found, but I hedge against metaphysically inflationary readings of this modality. (shrink)

Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante rem (...) structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures. These research projects have guided a more systematic focus on philosophical topics related to mathematical structuralism, including the identity criteria for objects in structures, dependence relations between objects and structures, and also, more recently, structural abstraction principles. Parallel to these developments are approaches that describe mathematical structure in category-theoretic terms. Category-theoretic approaches have been further developed using tools from homotopy type theory. Here we find a strong relationship between mathematical structuralism and the univalent foundations project, an approach to the foundations of mathematics based on higher category theory. (shrink)

The word "modem" in the title of this book refers primarily to post-medieval discussions, but it also hints at those medieval mo dal theories which were considered modem in contradistinction to ancient conceptions and which in different ways influenced philosophical discussions during the early modem period. The me dieval developments are investigated in the opening paper, 'The Foundations of Modality and Conceivability in Descartes and His Predecessors', by Lilli Alanen and Simo Knuuttila. Boethius's works from the early sixth (...) century belonged to the sources from which early medieval thinkers obtained their knowledge of ancient thought. They offered extensive discus sions of traditional modal conceptions the basic forms of which were: (1) the paradigm of possibility as a potency striving to realize itself; (2) the "statistical" interpretation of modal no tions where necessity means actuality in all relevant cases or omnitemporal actuality, possibility means actuality in some rel evant cases or sometimes, and impossibility means omnitemporal non-actuality; and (3) the "logical" definition of possibility as something which, being assumed, results in nothing contradic tory. Boethius accepted the Aristotelian view according to which total possibilities in the first sense must prove their met tle through actualization and possibilities in the third sense are assumed to be realized in our actual history. On these presump tions, all of the above-mentioned ancient paradigms imply the Principle of Plenitude according to which no genuine possibility remains unrealized. (shrink)