I here argue for a particular formulation of truth-deflationism, namely, the propositionally quantified formula, (Q) “For all p, <p> is true iff p”. The main argument consists of an enumeration of the other (five) possible formulations and criticisms thereof. Notably, Horwich’s Minimal Theory is found objectionable in that it cannot be accepted by finite beings. Other formulations err in not providing non-questionbegging, sufficiently direct derivations of the T-schema instances. I end by defending (Q) against various objections. In particular, I argue (...) that certain circularity charges rest on mistaken assumptions about logic that lead to Carroll’s regress. I show how the propositional quantifier can be seen as on a par with first-order quantifiers and so equally acceptable to use. While the proposed parallelism between these quantifiers is controversial in general, deflationists have special reasons to affirm it. I further argue that the main three types of approach the truth-paradoxes are open to an adherent of (Q), and that the derivation of general facts about truth can be explained on its basis. (shrink)

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

Standard compositionality is the doctrine that the semantic content of a compound expression is a function of the semantic contents of the contentful component expressions. In 1954 Hilary Putnam proposed that standard compositionality be replaced by a stricter version according to which even sentences that are synonymously isomorphic (in the sense of Alonzo Church) are not strictly synonymous unless they have the same logical form. On Putnam’s proposal, the semantic content of a compound expression is a function of: (i) the (...) contentful component expressions; and (ii) the expression’s logical form. Kit Fine recently expanded and modified Putnam’s idea into a sweeping theory in philosophy of language and philosophy of mind. The present paper is a detailed critique of Fine’s “semantic relationism.” Fine’s notion of coordination is explained in terms of the familiar pragmatic phenomenon of recognition. A serious error in Fine’s formal disproof of standard Millianism is exposed. It is demonstrated furthermore that Church’s original criticism of Putnam’s proposal can be extended to Fine’s semantic relationism. Finally, it is also demonstrated that the positive position Fine proffers to supplant standard Millianism is in fact exactly equivalent to standard Millianism, so that Fine’s overall position not only does not displace standard Millianism but is in fact inconsistent. (shrink)

This chapter introduces Kripke's work to advanced undergraduates, mainly focussing on his "A Puzzle About Belief" and "Wittgenstein on Rules and Private Language".

Philosophers of science have often favoured reductive approaches to how-possibly explanation. This article identifies three alternative conceptions making how-possibly explanation an interesting phenomenon in its own right. The first variety approaches “how possibly X?” by showing that X is not epistemically impossible. This can sometimes be achieved by removing misunderstandings concerning the implications of one’s current belief system but involves characteristically a modification of this belief system so that acceptance of X does not result in contradiction. The second variety offers (...) a potential how-explanation of X. It is usually followed by a range of further potential how-explanations of the same phenomenon. In recent literature the factual claims implied by the second variety have been downplayed whereas the heuristic role of mapping the space of conceptual possibilities has been emphasized. I will focus especially on this truth-bracketing sense of potentiality when looking closer at the second variety in the paper. The third variety has attracted less interest. It presents a partial how-explanation of X. Typically it aims to establish the existence of a mechanism by which X could be and was generated. The third conception stands out as the natural alternative for the advocate of ontic how-possibly explanations. This article transfers Salmon’s view that explanation-concepts can be broadly divided into epistemic, modal, and ontic to the context of how-possibly explanations. Moreover, it is argued that each of the three above-mentioned varieties of how-possibly explanation occurs in science. To recognize this may be especially relevant for philosophers. We are often misled by the promises of various why-explanation accounts, and seem to have forgotten nearly everything about the diversity of how-possibly explanations. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that Russell (...) did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

…entities? 2. How to be a nominalist 2.1. “Speak with the vulgar …” 2.2. “…think with the learned” 3. Arguments for nominalism 3.1. Intelligibility, physicalism, and economy 3.2. Causal..

This paper is about the metaphysical debate whether objects persist over time by the selfsame object existing at different times (nowadays called “endurance” by metaphysicians), or by different temporal parts, or stages, existing at different times (called “perdurance”). I aim to illuminate the debate by using some elementary kinematics and real analysis: resources which metaphysicians have, surprisingly, not availed themselves of. There are two main results, which are of interest to both endurantists and perdurantists. (1) I describe a precise formal (...) equivalence between the way that the two metaphysical positions represent the motion of the objects of classical mechanics (both point-particles and continua). (2) I make precise, and prove a result about, the idea that the persistence of objects moving in a void is to be analysed in terms of tracking the continuous curves in spacetime that connect points occupied by matter. The result is entirely elementary: it is a corollary of the Heine–Borel theorem. (shrink)

This article was written jointly by a philosopher and a mathematician. It has two aims: to acquaint mathematicians with some of the philosophical questions at the foundations of their subject and to familiarize philosophers with some of the answers to these questions which have recently been obtained by mathematicians. In particular, we argue that, if these recent findings are borne in mind, four different basic philosophical positions, logicism, formalism, platonism and intuitionism, if stated with some moderation, are in fact reconcilable, (...) although with some reservations in the case of logicism, provided one adopts a nominalistic interpretation of Plato's ideal objects. This eclectic view has been asserted by Lambek and Scott (LS 1986) on fairly technical grounds, but the present argument is meant to be accessible to a wider audience and to provide some new insights. (shrink)

I propose a way of thinking aboout content, and a related way of thinking about ontological commitment. (This is part of a series of four closely related papers. The other three are ‘On Specifying Truth-Conditions’, ‘An Actualist’s Guide to Quantifying In’ and ‘An Account of Possibility’.).

The positions of Frege, Russell and Wittgenstein on the priority of complexes over (propositional) functions are sketched, challenging those who take the "judgment centered" aspects of the Tractatus to be inherited from Frege not Russell. Frege's views on the priority of judgments are problematic, and unlike Wittgenstein's. Russell's views on these matters, and their development, are discussed in detail, and shown to be more sophisticated than usually supposed. Certain misreadings of Russell, including those regarding the relationship between propositional functions and (...) universals, are exposed. Wittgenstein's and Russell's views on logical grammar are shown to be very similar. Russell's type theory does not countenance types of genuine entities nor metaphysical truths that cannot be put into words, contrary to conventional wisdom. I relate this to the debate over "inexpressible truths" in the Tractatus. I lastly comment on the changes to Russell's views brought about by Wittgenstein's influence. (shrink)

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

A new approach to truth is offered which dispenses with the truth predicate, and replaces it with a special kind of quantifier which simultaneously binds variables in sentential and nominal positions. The resulting theory of truth for a (first-order) language is shown to be able to handle blind truth ascriptions, and is shown to be compatible with a characterization of the semantic and syntactic principles governing that language. Comparisons with other approaches to truth are drawn. An axiomatization of AU-quantifiers and (...) a model theory for them is given, and an appendix contains a completeness proof. (shrink)

This paper presents two interpretations of the fiber bundle formalism that is applicable to all gauge field theories. The constructionist interpretation yields a substantival spacetime. The analytic interpretation yields a structural spacetime, a third option besides the familiar substantivalism and relationalism. That the same mathematical formalism can be derived in two different ways leading to two different ontological interpretations reveals the inadequacy of pure formal arguments.

Russell is often taken as a forerunner of the Quinean position that “to be is to be the value of a bound variable”, whereupon the ontological commitment of a theory is given by what it quantifies over. Among other reasons, Russell was among the first to suggest that all existence statements should be analyzed by means of existential quantification. That there was more to Russell’s metaphysics than what existential quantifications come out as true is obvious in the earlier period where (...) Russell still made a distinction between existence and being/subsistence. But even the later Russell, including that of the Logical Atomism lectures period, would not have understood ontological questions to be first and foremost questions of quantification. He would take fundamentality to be important too, which explains in part his assertions to the effect the the values of individual variables have a reality not attributable to values of higher-order variables, even ineliminable higher-order variables. (shrink)

There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...) We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. Indeed, we argue that developing a fully compositional semantics in the tradition of Frege leads to a typographic variant of the most radical of Tarskian views: variabilism, the view that names should be modeled as Tarskian variables. We conclude with the consequences of this result for Frege's distinction between sense and reference. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

This book deals with an old conundrum: if God knows what we will choose tomorrow, how can we be free to choose otherwise? If all our choices are already written, is our freedom simply an illusion? This book provides a precise analysis of this dilemma using the tools of modern ontology and the logic of time. With a focus on three intertwined concepts - God's nature, the formal structure of time, and the metaphysics of time, including the relationship between temporal (...) entities and a timeless God - the chapters analyse various solutions to the problem of foreknowledge and freedom, revealing the advantages and drawbacks of each. Building on this analysis, the authors advance constructive solutions, showing under what conditions an entity can be omniscient in the presence of free agents, and whether an eternal entity can know the tensed futures of the world. The metaphysics of time, its topology and the semantics of future tensed sentences are shown to be invaluable topics in dealing with this issue. Combining investigations into the metaphysics of time with the discipline of temporal Logic this monograph brings about important advancements in the philosophical understanding of an ancient and fascinating problem. The answer, if any, is hidden in the folds of time, in the elusive nature of this feature of reality and in the infinite branching of our lives. (shrink)

En diciembre de 1996 fui invitado a impartir una sesión a profesores de lógica en los estudios institucionales del Studium Generale de la Prelatura del Opus Dei. En aquella ocasión preparé concienzudamente un texto escrito que pasé a mi querido y admirado colega Ángel Luis González para su revisión. Pocos días después Ángel Luis me lo devolvió con unas pocas correcciones y sugerencias y un alentador “¡Mucho ánimo!” en su encabezamiento. Durante muchos años conservé ese texto con sus anotaciones manuscritas. (...) Por este motivo, me ha parecido que podría ser adecuado reproducir en este volumen en homenaje de Ángel Luis aquella exposición en forma abreviada con unas pocas correcciones y actualizaciones de detalle. Mi exposición se divide en tres partes: 1) Situación de la lógica en la filosofía contemporánea; 2) El papel de la lógica en los estudios institucionales; y termina con 3) A modo de conclusión, una reflexión más personal. (shrink)

O objetivo geral da pesquisa da qual esse artigo faz parte é investigar o sistema metafísico que emerge dos trabalhos de David Lewis. Esse sistema pode ser decomposto em pelo menos duas teorias. A primeira nomeada como realismo modal genuíno (RMG) e a segunda como mosaico neo-humeano. O RMG é, sem dúvida, mais popular e defende a hipótese metafísica da existência de uma pluralidade de mundos possíveis. A principal razão em favor dessa hipótese é a sua aplicabilidade na discussão de (...) problemas losó cos, esse motivo não será diretamente abordado neste artigo. Pois, o foco está em compreender o mosaico neo-humea- no e como ele relaciona-se com a metafísica de mundos possíveis. Em outras palavras, o meu objetivo é entender as bases losó cas que sus- tentam o realismo modal genuíno. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions about (...) mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

The textbook-like history of analytic philosophy is a history of myths, re-ceived views and dogmas. Though mainly the last few years have witnessed a huge amount of historical work that aimed to reconsider our narratives of the history of ana-lytic philosophy there is still a lot to do. The present study is meant to present such a micro story which is still quite untouched by historians. According to the received view Kripke has defeated all the arguments of Quine against quantified (...) modal logic and thus it became a respectful tool for philosophers. If we accept the historical interpreta-tion of the network between Quine, Kripke and modal logic, which is to be presented here, we have to conclude that Quine’s real philosophical animadversions against the modalities are still on the table: though Kripke has provided some important (formal-logical) answers, Quine’s animadversions are still viable and worthy of further consideration. (shrink)

The aim of this dissertation is to offer and defend a correspondence theory of truth. I begin by critically examining the coherence, pragmatic, simple, redundancy, disquotational, minimal, and prosentential theories of truth. Special attention is paid to several versions of disquotationalism, whose plausibility has led to its fairly constant support since the pioneering work of Alfred Tarski, through that by W. V. Quine, and recently in the work of Paul Horwich. I argue that none of these theories meets the correspondence (...) intuition---that a true sentence or proposition in some way corresponds to reality---despite the explicit claims by each to capture this intuition. I distinguish six versions of the correspondence theory, and defend two against traditional objections, standardly taken as decisive against them, and show, plainly, that these two theories capture the correspondence intuition. Due to the importance of meeting this intuition, only these two theories stands a chance of being a satisfactory theory of truth. I argue that the version of the correspondence theory incorporating a simple semantic representation relation is preferable to its rival, for which the representation relation is complex. I present and argue for a novel version of this correspondence theory according to which truth is a correspondence property sensitive to semantic context. One consequence of this context-sensitivity is that an ungrounded sentence does not express a proposition. In addition to accounting for the similarity between the Liar and Truth-Teller sentences, this theory of truth is immune to the Liar Paradox, including empirical versions. It is argued that the Liar Paradox is devastating to all of the other theories above, and even formal theories of truth designed to solve it, such as the revision and vagueness theories. Customized versions of the Liar Paradox besetting this theory are handled by its context-sensitivity, and by enforcing the distinction between truth and truth value. This same pair of considerations also yields solutions to Lob's Paradox and Grelling's Paradox. Arguments similar to those given to defend this correspondence theory show that with one minor alteration, Kripke's fixed point theory may be used to model this correspondence notion of truth. (shrink)

“Guns don’t kill people, people kill people.” This National Rifle Association slogan has a popular analogue in the philosophy of language: words don’t refer to things, people refer to things. This paper looks at how the question of what refers intersects with the phenomenon of opacity. The discussion is used to motivate an important revision to standard intention-based accounts of speaker reference, one that takes due account of the fact that distinct words can have the same referent.

The standard truth-conditional semantics for substitutional quantification, due to Saul Kripke, does not specify what proposition is expressed by sentences containing the particular substitutional quantifier. In this paper, I propose an alternative semantics for substitutional quantification that does. The key to this semantics is identifying an appropriate propositional function to serve as the content of a bound occurrence of a formula containing a free substitutional variable. I apply this semantics to traditional philosophical reasons for interest in substitutional quantification, namely, theories (...) of truth and ontological commitment. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

This study looks to the work of Tarski's mentors Stanislaw Lesniewski and Tadeusz Kotarbinski, and reconsiders all of the major issues in Tarski scholarship in light of the conception of Intuitionistic Formalism developed: semantics, truth, paradox, logical consequence.

Russell says that logical constructions are fictions. Does this show that he took them not to be real things?

This book is an inquiry into the philosophical concern with truth as one joint subject in philosophy of language and metaphysics and presents a theory of truth, substantive perspectivism (SP). Emphasizing our basic pre-theoretic understanding of truth (i.e., what is captured by the axiomatic thesis of truth that the nature of truth consists in capturing the way things are), and in the deflationism vs. substantivism debate background, SP argues for the substantive nature of non-linguistic truth and its notion’s indispensable substantive (...) explanatory role, both of which are not only intrinsically beyond what the linguistic function of the truth predicate can tell but are fundamentally related to the raison d’être of the truth predicate. Taking a holistic approach, SP endeavors to do justice to various reasonable perspectives, which are somehow contained in many competing accounts of truth, through a coordinate system: SP interprets such perspectives as distinct but related perspective-elaboration principles that distinctively (regarding distinct dimensions of the truth concern and/or for the sake of distinct purposes) elaborate, but are also unified by, the truth axiom thesis. To look at the issue from a broader vision, the book also takes a cross-tradition approach exploring the relationship between Daoist thinking of truth and thinking about truth in analytic philosophy.This book will enhance our systematic understanding of the issue through its holistic approach, broaden our vision on the issue via its cross-tradition approach, and enrich the conceptual and explanatory resources in treating the issue. (shrink)

In this article, the prospects of deflationism about the concept of truth are investigated. A new version of deflationism, called inferential deflationism, is articulated and defended. It is argued that it avoids the pitfalls of earlier deflationist views such as Horwich’s minimalist theory of truth and Field’s version of deflationism.

The theory that Kit Fine calls 'semantic relationism' replaces standard semantic compositionality with an alternative according to which statements of the form '... A … A ...’ and ‘... A … B ...’ (e.g., ‘Cicero admires Cicero’ and ‘Cicero admires Tully’) differ in semantic content—even where the two terms involved are exactly synonymous—simply in virtue of the recurrence that is present in the former statement and absent from the latter. A semantic-relationist alternative to standard compositionality was first explicitly proffered by (...) Hilary Putnam in 1954 and later by others in the latter part of the 20th Century. However, semantic relationism had already been implicitly rejected in Frege’s 1892 landmark masterpiece "Über Sinn und Bedeutung" (“On Sense and Reference”). The present compilation of three previously published essays cuts through the hype and thoroughly rebuts semantic relationism, particularly Fine’s more developed version (2007). A fatal fallacy is exposed in Fine's high-flown purported formal disproof of Millianism with standard compositionality. Standard Millianism does not implode; Fine's overzealous attempt to demolish it does. The failure of his purported disproof renders Fine's central notion of coordination either redundant or empty. So-called coordination is non-semantic if it is anything real. (shrink)

I explain why model theory is unsatisfactory as a semantic theory and has drawbacks as a tool for proofs on logic systems. I then motivate and develop an alternative, the truth-valuational substitutional approach (TVS), and prove with it the soundness and completeness of the first order Predicate Calculus with identity and of Modal Propositional Calculus. Modal logic is developed without recourse to possible worlds. Along the way I answer a variety of difficulties that have been raised against TVS and show (...) that, as applied to several central questions, model-theoretic semantics can be considered TVS in disguise. The conclusion is that the truth-valuational substitutional approach is an adequate tool for many of our logic inquiries, conceptually preferable over model-theoretic semantics. Another conclusion is that formal logic is independent of semantics, apart from its use of the notion of truth, but that even with respect to it its assumptions are minimal. (shrink)

The so-called New Theory of Reference (Marcus, Kripke etc.) is inspired by the insight that in modal and intensional contexts quantifiers presuppose nondescriptive unanalyzable identity criteria which do not reduce to any descriptive conditions. From this valid insight the New Theorists fallaciously move to the idea that free singular terms can exhibit a built-in direct reference and that there is even a special class of singular terms (proper names) necessarily exhibiting direct reference. This fallacious move has been encouraged by a (...) mistaken belief in the substitutional interpretation of quantifiers, by the myth of thede re reference, and a mistaken assimilation of direct reference to ostensive (perspectival) identification. Thede dicto vs.de re contrast does not involve direct reference, being merely a matter of rule-ordering (scope).The New Theorists' thesis of the necessity of identities of directly referred-to individuals is a consequence of an unmotivated and arbitrary restriction they tacitly impose on the identification of individuals. (shrink)

A collection of ten previously published essays on existence, nonexistence, empty names, fiction and myth, and free logic.

Quine’s criterion of theoretical ontological commitment is subject to a variety of interpretations, all of which save one yield incorrect verdicts. Moreover, the interpretation that yields correct verdicts is not what Quine meant. Instead the intended criterion unfairly imputes ontological commitments to theories that lack those commitments and fails to impute commitments to theories that have them. Insofar as Quine’s criterion is interpreted so that it yields only correct verdicts, it is trivial and of questionable utility. Moreover, the correct criterion (...) invokes analyticity, a notion that Quine spent most of his life tirelessly combating. This yields a dilemma for Quinean philosophy: Either his criterion of ontological commitment is incorrect, or else Quine is committed to a traditional philosophical notion that he emphatically rejected as disreputable. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

Predicate approaches to modality have been a topic of increased interest in recent intensional logic. Halbach and Welch :71–100, 2009) have proposed a new formal technique to reduce the necessity predicate to an operator, demonstrating that predicate and operator methods are ultimately compatible. This article concerns the question of whether Halbach and Welch’s approach can provide a uniform formal treatment for intensionality. I show that the monotonicity constraint in Halbach and Welch’s proof for necessity fails for almost all possible-worlds theories (...) of knowledge. The nonmonotonicity results demonstrate that the most obvious way of emulating Halbach and Welch’s rapprochement of the predicate and operator fails in the epistemic setting. (shrink)

This book is an exploration and defense of the coherence of classical theism’s doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide range of options (...) available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments. (shrink)

Despite the wide acceptance of standard modal logic, there has always been a temptation to think that ordinary modal discourse may be correctly analyzed and adequately represented in terms of predicates rather than in terms of operators. The aim of the formal model outlined in this paper is to capture what I take to be the only plausible sense in which ‘possible’ and ‘necessary’ can be treated as predicates. The model is built by enriching the language of standard modal logic (...) with a quantificational apparatus that is “substitutional” rather than “objectual”, and by obtaining from the language so enriched another language in which constants for such predicates apply to singular terms that stand for propositions. (shrink)