The original interest of this article lies in existential import. It provides a broader view on the problem by reference to modern, symbolic logic (ch.1). Gradually, however, our interest will change into the amalgamated expressions often used in logic; that is, why are such expressions as “x is a round triangle” applied in logic? We critically discuss this question from an extensional viewpoint, namely model theoretic semantics (ch.2). We also touch on Church’s λ-calculus in the appendix (app.2).

The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined even if it is not sufficiently formal and precise to belong to mathematics proper (in a narrow sense)—and even if (as many have asserted) for that reason the Church–Turing thesis is unprovable. It is proved logically that the class of effective procedures is not decidable, i.e., that no effective procedure is possible for ascertaining whether a given procedure is effective. This (...) undecidability result is proved directly from the notion itself of an effective procedure, without reliance on any (partly) mathematical lemma, conjecture, or thesis invoking recursiveness or Turing-computability. In fact, there is no reliance on anything very mathematical. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of an effective procedure. Though the result that effectiveness is undecidable is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure invariably terminates with the correct verdict. (shrink)

This is a non-technical version of "The Decision Problem for Effective Procedures." The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined, even if it does not have a purely mathematical definition—and even if (as many have asserted) for that reason, the Church–Turing thesis (that the effectively calculable functions on natural numbers are exactly the general recursive functions), cannot be proved. However, it is logically provable from the notion of an (...) effective procedure--without reliance on any (partially) mathematical thesis or conjecture concerning effective procedures, such as the Church–Turing thesis--that the class of effective procedures is undecidable, i.e., that no effective procedure is possible for ascertaining whether a given procedure is effective. The proof does not appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of such a procedure. Though this undecidability result itself is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure is, in fact, a decision procedure, i.e., effective—for example, that it invariably terminates with the correct verdict. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Formulations of Anselm’s ontological argument have been the subject of a number of recent studies. We examine these studies in light of Anselm’s text and (a) respond to criticisms that have surfaced in reaction to our earlier representations of the argument, (b) identify and defend a more refined representation of Anselm’s argument on the basis of new research, and (c) compare our representation of the argument, which analyzes that than which none greater can be conceived as a definite description, to (...) a representation that analyzes it as an arbitrary name. (shrink)

What is the relation of logic to thinking? My dissertation offers a new argument for the claim that logic is constitutive of thinking in the following sense: representational activity counts as thinking only if it manifests sensitivity to logical rules. In short, thinking has to be minimally logical. An account of thinking has to allow for our freedom to question or revise our commitments – even seemingly obvious conceptual connections – without loss of understanding. This freedom, I argue, requires that (...) thinkers have general abilities to respond to support and tension among their thoughts. And these abilities are constituted by following logical rules. So thinkers have to follow logical rules. But there isn’t just one correct logic for thinking. I show that my view is consistent with logical pluralism: there are a range of correct logics, any one of which a thinker might follow. A logic for thinking does, however, have to contain certain minimal principles: Modus Ponens and Non-Contradiction, and perhaps others. We follow logical rules by exercising logical capacities, which display a distinctive first-person/third-person asymmetry: a subject can find the instances of a rule compelling without seeing them as instances of a rule. As a result, there are two limits on illogical thinking. First, thinkers have to tend to find instances of logical rules compelling. Second, thinkers can’t think in obviously illogical ways. So thinking has to be logical – but not perfectly so. When we try to think, but fail, we produce nonsense. But our failures to think are often subjectively indistinguishable from thinking. To explain how this occurs, I offer an account of nonsense. To be under the illusion that some nonsense makes sense is to enter a pretence that the nonsense is meaningful. Our use of nonsense within the pretence relies on the role of logical form in understanding. Finally, while the normativity of logic doesn’t fall directly out of logical constitutivism, it’s possible to build an attractive account of logical normativity which has logical constitutivism as an integral part. I argue that thinking is necessary for human flourishing, and that this is the source of logical normativity. (shrink)

This work is compiled for the students, research scholars, academicians, who are interested in logic, philosophy, mathematics and critical thinking. The main objective of this book is to provide basics or fundamental knowledge for those who have chosen logic as their subject in order to develop analytical and critical ideas. It has been primarily developed to serve as an introductory piece of work which includes explanatory notes on different courses like Inductive logic, Deductive logic, propositional logic, Symbolic logic, Quantification logic, (...) Modal logic and Critical thinking. Besides this, it also includes illustrations in decision making and scientific research methods in logic. This book is mainly devised to clear fundamental problems of logic. It contains eight chapters which are simply described and elaborated. (shrink)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants Kripke-Feferman truth. The first, KF+CONS, features a consistent but partial truth predicate. The second, KF+COMP, an inconsistent but complete truth predicate. It is well-known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability under natural assumptions coincides with definitional (...) equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF+CONS and KF+COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (shrink)

A version of intuitionistic type theory is extended with opposite types, allowing a different formalization of negation and obtaining a paraconsistent type theory (⁠|$\textsf{PTT} $|⁠). The rules for opposite types in |$\textsf{PTT} $| are based on the rules of the so-called constructible falsity. A propositions-as-types correspondence between the many-sorted paraconsistent logic |$\textsf{PL}_\textsf{S} $| (a many-sorted extension of López-Escobar’s refutability calculus presented in natural deduction format) and |$\textsf{PTT} $| is proven. Moreover, a translation of |$\textsf{PTT} $| into intuitionistic type theory is (...) presented and some properties of |$\textsf{PTT} $| are discussed. (shrink)

In a recent article, P. Roger Turner and Justin Capes argue that no one is, or ever was, even partly morally responsible for certain world-indexed truths. Here we present our reasons for thinking that their argument is unsound: It depends on the premise that possible worlds are maximally consistent states of affairs, which is, under plausible assumptions concerning states of affairs, demonstrably false. Our argument to show this is based on Bertrand Russell’s original ‘paradox of propositions’. We should then opt (...) for a different approach to explain world-indexed truths whose upshot is that we may be morally responsible for some of them. The result to the effect that there are no maximally consistent states of affairs is independently interesting though, since this notion motivates an account of the nature of possible worlds in the metaphysics of modality. We also register in this article, independently of our response to Turner and Capes, and in the spirit of Russell’s aforementioned paradox and many other versions thereof, a proof of the claim that there is no set of all true propositions one can render false. (shrink)

The use-mention distinction is elaborated into a four-way distinction between use, formal mention, material mention and pragmatic mention. The notion of pragmatic mention is motivated through the problem of monsters in Kaplanian indexical semantics. It is then formalized and applied in an account of schemata in formalized languages.

Prior’s puzzle is standardly taken to be the puzzle of why, given the assumption that that-clauses denote propositions, substitution of “the proposition that P” for “that P” within the complements of many propositional attitude verbs is invalid. I show that Prior’s puzzle is much more general than is ordinarily supposed. There are two variants on the substitutional form of the puzzle—a quantificational variant and a pronominal variant—and all three forms of the puzzle arise in a wide range of grammatical positions, (...) rather than merely in the complements of propositional attitude verbs. The generalized puzzle shows that a range of proposed solutions to the original puzzle fail, or are radically incomplete, and also reveals the connections between Prior’s puzzle and debates over the nature of semantic types and higher-order quantification. I go on to develop a novel, higher-order solution to the generalized form of the puzzle, and I argue that this higher-approach is superior to its first-order alternatives. (shrink)

In linguistics, the dominant approach to the semantics of plurals appeals to mereology. However, this approach has received strong criticisms from philosophical logicians who subscribe to an alternative framework based on plural logic. In the first part of the article, we offer a precise characterization of the mereological approach and the semantic background in which the debate can be meaningfully reconstructed. In the second part, we deal with the criticisms and assess their logical, linguistic, and philosophical significance. We identify four (...) main objections and show how each can be addressed. Finally, we compare the strengths and shortcomings of the mereological approach and plural logic. Our conclusion is that the former remains a viable and well-motivated framework for the analysis of plurals. (shrink)

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)

Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between (...) this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes. (shrink)

Priority Theory is an increasingly popular view in metaphysics. By seeing metaphysical questions as primarily concerned with what explains what, instead of merely what exists, it promises not only an interesting approach to traditional metaphysical issues but also the resolution of some outstanding disputes. In a recent paper, Louis deRosset argues that Priority Theory isn’t up to the task: Priority Theory is committed to there being explanations that violate a formal constraint on any adequate explanation. This paper critically examines deRosset’s (...) challenge to Priority Theory. We argue that deRosset’s challenge ultimately fails: his proposed constraint on explanation is neither well-motivated nor a general constraint. Nonetheless, lurking behind his criticism is a deep problem for prominent ways of developing Priority Theory, a problem which we develop. (shrink)

Este livro se propõe a ser uma introdução fácil e acessível, porém rigorosa e tecnicamente precisa, à lógica. Prioridade é dada à clareza e lucidez na explicação das definições e teoremas, bem como à aplicação prática da lógica na análise de argumentos. O livro foi concebido de forma a permitir sua utilização por qualquer pessoa interessada em aprender lógica, independentemente de sua área de atuação ou bagagem teórica prévia. Em especial, ele deve ser útil a estudantes e professores de filosofia, (...) computação e matemática. (shrink)

Ingineria ontologică, împreună cu tehnologiile Web semantice, permit modelarea și dezvoltarea semantică a fluxului operațional necesar pentru proiectarea TB. Cel mai utilizat sistem de modelare blockchain prin reprezentarea abstractă, descrierea și definirea structurii, a proceselor, a informațiilor și a resurselor, este modelarea intreprinderilor. Modelarea intreprinderii utilizează ontologiile de domeniu folosind limbaje de reprezentare a modelului. Bitcoin este principalul sistem de plată peer-to-peer şi monedă digitală care folosește tehnologia blockchain. DOI: 10.13140/RG.2.2.35908.99201 .

L'ingénierie ontologique, associé aux technologies du Web sémantique, permet la modélisation sémantique et le développement du flux opérationnel requis pour la conception de la technologie blockchain. Le système de modélisation blockchain le plus utilisé, par la représentation abstraite, la description et la définition de la structure, des processus, des informations et des ressources, est la modélisation des entreprises. La modélisation d'entreprise utilise des ontologies de domaine utilisant des langages de représentation du modèle. Bitcoin est le principal système de paiement pair-à-pair (...) et cryptomonnaie qui utilise la technologie de la blockchain. DOI: 10.13140/RG.2.2.25379.50726. (shrink)

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...) :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (shrink)

Despre necesitatea şi utilitatea dezvoltării unei filosofii specifice tehnologiei blockchain, accentuând pe aspectele ontologice. După o Introducere în care evidenţiez principalele direcţii filosofice pentru această tehnologie emergentă, în Tehnologia blockchain explicitez modul de funcţionare al blockchain, punând în discuţie direcţiile ontologice de dezvoltare în Proiectarea şi Modelarea acestei tehnologii. Următoarea secţiune este dedicată principalei aplicaţii a tehnologiei blockchain, Bitcoin, cu implicaţiile sociale ale acestei criptovalute. Urmează o secţiune de Filosofie în care identific tehnologia blockchain cu conceptul de heterotopie dezvoltat de (...) Michel Foucault şi o interpretez în lumina tehnologiei notaţionale dezvoltată de Nelson Goodman ca sistem notaţional. În secţiunea Ontologii dezvolt două direcţii de dezvoltare considerate de mine ca importante: Ontologia narativă, bazată pe ideea de ordine şi structură a istoriei transmise prin naraţia istoriei a lui Paul Ricoeur, şi un sistem ontologic bazat pe Ontologiile de intreprindere, bazat pe conceptele şi modelele de intreprindere specifice webului semantic, şi pe care eu îl consider cel mai bine dezvoltat şi cel care va deveni, probabil, sistemul ontologic formal, cel puţin în ceea ce priveşte aspectele economice şi juridice ale tehnologiei blockchain. În Concluzii vorbesc despre direcţiile viitoare de dezvoltare a filosofiei tehnologiei blockchain în general ca o teorie explicativă şi robustă din punct de vedere fenomenologic, consistentă şi care să permită testabilitatea, şi a ontologiilor în special, argumentând pentru necesitatea unei adopţii globale a unui sistem ontologic pentru a dezvolta soluţii transversale şi a rentabiliza tehnologia. -/- Cuvinte cheie: filosofie, blockchain, ontologie, bitcoin, proiectare, modele -/- CUPRINS -/- Abstract Introducere Tehnologia blockchain - - - Proiectare - - - Modele Bitcoin Filosofia Ontologii - - - Ontologii narative - - - Ontologii de intreprindere Concluzii Note Bibliografie -/- DOI: 10.13140/RG.2.2.25492.35204. (shrink)

One of Tarski’s stated aims was to give an explication of the classical conception of truth—truth as ‘saying it how it is’. Many subsequent commentators have felt that he achieved this aim. Tarski’s core idea of defining truth via satisfaction has now found its way into standard logic textbooks. This paper looks at such textbook definitions of truth in a model for standard first-order languages and argues that they fail from the point of view of explication of the classical notion (...) of truth. The paper furthermore argues that a subtly different definition—also to be found in classic textbooks but much less prevalent than the kind of definition that proceeds via satisfaction—succeeds from this point of view. (shrink)

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)

Following Henkin’s discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or “cardinality” quantifiers, e.g., “most”, “few”, “finitely many”, “exactly α ”, where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, Henkin definition first to a (...) general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise’s 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional 1st-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages. (shrink)

True beliefs and truth-preserving inferences are, in some sense, good beliefs and good inferences. When an inference is valid though, it is not merely truth-preserving, but truth-preserving in all cases. This motivates my question: I consider a Modus Ponens inference, and I ask what its validity in particular contributes to the explanation of why the inference is, in any sense, a good inference. I consider the question under three different definitions of ‘case’, and hence of ‘validity’: the orthodox definition given (...) in terms of interpretations or models, a metaphysical definition given in terms of possible worlds, and a substitutional definition defended by Quine. I argue that the orthodox notion is poorly suited to explain what's good about a Modus Ponens inference. I argue that there is something good that is explained by a certain kind of truth across possible worlds, but the explanation is not provided by metaphysical validity in particular; nothing of value is explained by truth across all possible worlds. Finally, I argue that the substitutional notion of validity allows us to correctly explain what is good about a valid inference. (shrink)

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant Bermúdez’s version (...) of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. (shrink)

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

Thoroughgoing relativists typically dismiss the realist conviction that competing theories describe just one definite and mind-independent world-structure on the grounds that such theories fail to be relatively translatable even though they are equally correct. This line of argument allegedly brings relativism into direct conflict with the metaphysics of realism. I argue that this relativist line of reasoning is shaky by deriving a theorem about relativistic inquiry in formal epistemology—more specifically, in the approach Kevin Kelly has dubbed “logic of reliable inquiry”. (...) According to the theorem, two scientists, who share some background knowledge but follow different appropriately reliable methods, will converge to relatively formally translatable competing theories, even if meaning, truth, logic and evidence are allowed to vary in time depending on each scientist’s conjectures, actions, or conceptual choices. Some final remarks on the relevance of the theorem to the incommensurability thesis that has vexed twentieth century philosophy of science are adduced. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

One logic or many? I say—many. Or rather, I say there is one logic for each way of specifying the class of all possible circumstances, or models, i.e., all ways of interpreting a given language. But because there is no unique way of doing this, I say there is no unique logic except in a relative sense. Indeed, given any two competing logical theories T1 and T2 (in the same language) one could always consider their common core, T, and settle (...) on that theory. So, given any language L, one could settle on the minimal logic T0 corresponding to the common core shared by all competitors. That would be a way of resisting relativism, as long as one is willing to redraw the bounds of logic accordingly. However, such a minimal theory T0 may be empty if the syntax of L contains no special ingredients the interpretation of which is independent of the specification of the relevant L-models. And generally—I argue—this is indeed the case. (shrink)

Bob Hale’s distinguished record of research places him among the most important and influential contemporary analytic metaphysicians. In his deep, wide ranging, yet highly readable book Necessary Beings, Hale draws upon, but substantially integrates and extends, a good deal his past research to produce a sustained and richly textured essay on — as promised in the subtitle — ontology, modality, and the relations between them. I’ve set myself two tasks in this review: first, to provide a reasonably thorough (if not (...) exactly comprehensive) overview of the structure and content of Hale’s book and, second, to a limited extent, to engage Hale’s book philosophically. I approach these tasks more or less sequentially: Parts I and 2 of the review are primarily expository; in Part 3 I adopt a somewhat more critical stance and raise several issues concerning one of the central elements of Hale’s account, his essentialist theory of modality. (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)

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) in his theory of axiomatics is presented. Specifically, it is shown that Carnap’s early model theory is based on a convention to simulate domain variation that is not identical but logically comparable to the modern account. (shrink)

Ce bel ouvrage, clair, aéré et spacieux, se caractérise à la fois par sa volonté de simplicité d’accès, et son ambition, puisqu’on y trouve notamment une démonstration de la complétude d’un certain système déductif S1 pour la logique classique des prédicats, ainsi qu’une version synoptique du théorème de Gödel, selon lequel toute théorie du premier ordre complète axiomatisable est décidable, d’où il s’ensuit que l’arithmétique, c’est-à-dire l’ensemble des énoncés du premier ordre vrais dans, n’est pas axiomatisable; ce qu’on exprime souvent (...) en disant que tout système formel pour l’arithmétique est incomplet, au sens où il y a des énoncés vrais qui ne sont pas des théorèmes du système. (shrink)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

In this article we discuss what constitutes a good choice of semantic representation, compare different approaches of constructing semantic representations for fragments of natural language, and give an overview of recent methods for employing inference engines for natural language understanding tasks.

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)

In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which (...) follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentenceσis false. Moreover, … we are concerned here with the concept of logical, i.e.,formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since. (shrink)

The implicational fragment of the relevance logic "ticket entailment" is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of $T_\rightarrow$ are in normal form and built in all but one casefrom B, B' and W only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define $FIT_\rightarrow$ - a Fitch-style calculus (related to (...) $FT_\rightarrow$ ) for the implicational fragment of ticket entailment. (shrink)

How might we philosophize about the metaverse? It is traditionally held that the four main branches of philosophy are metaphysics, epistemology, axiology, and logic. In this article, I shall demonstrate how virtual walt-fictionalism, a particular version of virtual irrealism, is able to offer a straightforward, internally consistent, and powerful response about the metaphysics, epistemology, and axiology (ethics) of the metaverse. I will first characterize the metaverse in terms of a reality-virtuality (RV) continuum and distinguish between virtual realism and virtual irrealism, (...) before elaborating on the explanatory power and potential of virtual walt-fictionalism relative to the metaphysics, epistemology, and axiology (ethics) of the metaverse. (shrink)

Quine maintained that philosophical and scientific theorizing should be conducted in an untyped language, which has just one style of variables and quantifiers. By contrast, typed languages, such as those advocated by Frege and Russell, include multiple styles of variables and matching kinds of quantification. Which form should our theories take? In this article, I argue that expressivity does not favour typed languages over untyped ones.

The Thousands of Problems for Theorem Provers (TPTP) World is a well-established infrastructure that supports research, development and deployment of automated theorem proving systems. This paper provides an overview of the logic languages of the TPTP World, from classical first-order form (FOF), through typed FOF, up to typed higher-order form, and beyond to non-classical forms. The logic languages are described in a non-technical way and are illustrated with examples using the TPTP language.

I critically discuss two kinds of argument in favour of ontological pluralism and argue that they fail to show that ways of being are explanatorily fruitful.

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

In this paper I try to answer four basic questions: (1) How the concept of God is to be represented? (2) Are there any logical principles governing it? (3) If so, what kind of logic lies behind them? (4) Can there be a logic of the concept of God? I address them by presenting a formal-logical account to the concept of God. I take it as a methodological desideratum that this should be done within the simplest existing logical formalism. I (...) start with first-order logic (FOL) with identity, and then show that its simplest modal extension (SQML, or the simplest quantified modal logic) is enough for us to formalize a minimally satisfactory theory of the concept of God. I focus exclusively on the monotheistic concept of God. (shrink)

We study the decidability of k-provability in \—the relation ‘being provable in \ with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton. The decidability of k-provability for the usual Hilbert-style formalisation of \ is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which (...) k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms. (shrink)