The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], (...) a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics. (shrink)

In this essay, I appeal to the mental file approach in order to give an anti-realist semantic analysis of statements containing fictional names. I claim that fictive and parafictive uses of them express conceptual, though not general, propositions constituted by mental files, anchored in the conceptual world of the corresponding fictional story. Moreover, by positing a referential shift determined by the presence of a simulative referential intention characteristic of those uses, it is possible to take them to be true with (...) respect to those conceptual worlds. As for metafictive uses, since they are grounded on mixed intentions, they are considered to express hybrid propositions, partly conceptual and partly referential. (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.

Abstract. On the Direct Reference thesis, proper names are what I call ‘genuine terms’, terms whose sole semantic contributions to the propositions expressed by their use are the terms’ semantic referents. But unless qualified, this thesis implies the false consequence that sentences containing names that fail to refer can never express true or false propositions. (Consider ‘The ancient Greeks worshipped Zeus’, for instance.) I suggest that while names are typically and fundamentally used as genuine terms, there is a small class (...) of names, which I call ‘descriptive names’, whose reference is fixed by commonly associated definite descriptions, and I also suggest that there is an idiom of natural language on which such names can be used as abbreviated definite descriptions in a limited set of sentential contexts, including (1) positive and negative existentials, (2) cognitive ascriptions, and (3) uses of names to talk about myth. Uses of empty descriptive names in such contexts can then be either true or false. Relying on Gregory Currie’s theory of truth in fiction, I also propose an idiom on which fictional names can be used as short for a certain type of description in talk about fiction. Along the way, I provide arguments that names are used as short for descriptions in substantive existential statements as well as in both metamythic and metafictive contexts. I also discuss and criticize alternative views of these matters, including the views of David Braun, Saul Kripke, Peter van Inwagen, and others. (shrink)

Frege's philosophical writings, including the “logistic project,” acquire a new insight by being confronted with Kant's criticism and Wittgenstein's logical and grammatical investigations. Between these two points a non-formalist history of logic is just taking shape, a history emphasizing the Greek and Kantian inheritance and its aftermath. It allows us to understand the radical change in rationality introduced by Gottlob Frege's syntax. This syntax put an end to Greek categorization and opened the way to the multiplicity of expressions producing their (...) own intelligibility. This article is based on more technical analyses of Frege which Claude Imbert has previously offered in other writings. (shrink)

According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of (...) this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence. (shrink)

The five English words—sentence, proposition, judgment, statement, and fact—are central to coherent discussion in logic. However, each is ambiguous in that logicians use each with multiple normal meanings. Several of their meanings are vague in the sense of admitting borderline cases. In the course of displaying and describing the phenomena discussed using these words, this paper juxtaposes, distinguishes, and analyzes several senses of these and related words, focusing on a constellation of recommended senses. One of the purposes of this paper (...) is to demonstrate that ordinary English properly used has the resources for intricate and philosophically sound investigation of rather deep issues in logic and philosophy of language. No mathematical, logical, or linguistic symbols are used. Meanings need to be identified and clarified before being expressed in symbols. We hope to establish that clarity is served by deferring the extensive use of formalized or logically perfect languages until a solid “informal” foundation has been established. Questions of “ontological status”—e.g., whether propositions or sentences, or for that matter characters, numbers, truth-values, or instants, are “real entities”, are “idealizations”, or are “theoretical constructs”—plays no role in this paper. As is suggested by the title, this paper is written to be read aloud. -/- I hope that reading this aloud in groups will unite people in the enjoyment of the humanistic spirit of analytic philosophy. (shrink)

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

Following the development of the selectionist theory of the immune system, there was an attempt to characterize many biological mechanisms as being "selectionist" as juxtaposed to "instructionist." But this broad definition would group Darwinian evolution, the immune system, embryonic development, and Chomsky's language-acquisition mechanism as all being "selectionist." Yet Chomsky's mechanism (and embryonic development) are significantly different from the selectionist mechanisms of biological evolution or the immune system. Surprisingly, there is a very abstract way using two dual mathematical logics to (...) make the distinction between genuinely selectionist mechanisms and what are better called "generative" mechanisms. This note outlines that distinction. (shrink)

Almost entirely ignored in the linguistic theorising on names and descriptions is a hybrid form of expression which, like definite descriptions, begin with 'the' but which, like proper names, are capitalised and seem to lack descriptive content. These are expressions such as the following, 'the Holy Roman Empire', 'the Mississippi River', or 'the Space Needle'. Such capitalised descriptions are ubiquitous in natural language, but to which linguistic categories do they belong? Are they simply proper names? Or are they definite descriptions (...) with unique orthography? Or are they something else entirely? This paper assesses two obvious assimilation strategies: (i) assimilation to proper names and (ii) assimilation to definite descriptions. It is argued that both of these strategies face major difficulties. The primary goal is to lay the groundwork for a linguistic analysis of capitalised descriptions. Yet, the hope is that clearing the ground on capitalised descriptions may reveal useful insights for the on-going research into the semantics and syntax of their lower-case or 'the'-less relatives. (shrink)

In Writing the Book of the World, Ted Sider argues that David Lewis’s distinction between those predicates which are ‘perfectly natural’ and those which are not can be extended so that it applies to words of all semantic types. Just as there are perfectly natural predicates, there may be perfectly natural connectives, operators, singular terms and so on. According to Sider, one of our goals as metaphysicians should be to identify the perfectly natural words. Sider claims that there is a (...) perfectly natural first-order quantifier. I argue that this claim is not justified. Quine has shown that we can dispense with first-order quantifiers, by using a family of ‘predicate functors’ instead. I argue that we have no reason to think that it is the first-order quantifiers, rather than Quine’s predicate functors, which are perfectly natural. The discussion of quantification is used to provide some motivation for a general scepticism about Sider’s project. Shamik Dasgupta’s ‘generalism’ and Jason Turner’s critique of ‘ontological nihilism’ are also discussed. (shrink)

Properties and facts play a central role within metaphysics, yet there is no widely accepted account of what constitutes a property or a fact. Traditional conceptions of these metaphysical notions raise serious philosophical puzzles, making the existence of each seem dubious. Drawing on the minimalist theory of truth, I argue in favor of a minimalist conception of properties and facts. A minimalist theory of properties and facts explains these matters in terms of the acceptance of trivial schemas. To make the (...) case that minimalism is a superior approach to properties and facts, I argue against the standard views in the philosophical literature. I argue that the minimalist approach to properties has advantages over realism, nominalism, and the trope theory. I argue that the minimalist approach to facts is superior to the standard treatment of facts on correspondence theories. Metaphysical minimalism, a minimalist metaphysics of properties and facts, is a distinct and superior alternative to the theories of properties and facts currently on offer. (shrink)

Properties and facts play a central role within metaphysics, yet there is no widely accepted account of what constitutes a property or a fact. Traditional conceptions of these metaphysical notions raise serious philosophical puzzles, making the existence of each seem dubious. Drawing on the minimalist theory of truth, I argue in favor of a minimalist conception of properties and facts. A minimalist theory of properties and facts explains these matters in terms of the acceptance of trivial schemas. To make the (...) case that minimalism is a superior approach to properties and facts, I argue against the standard views in the philosophical literature. I argue that the minimalist approach to properties has advantages over realism, nominalism, and the trope theory. I argue that the minimalist approach to facts is superior to the standard treatment of facts on correspondence theories. Metaphysical minimalism, a minimalist metaphysics of properties and facts, is a distinct and superior alternative to the theories of properties and facts currently on offer.Iako svojstva i činjenice u metafizici igraju središnju ulogu, ne postoji usuglašen opis toga što sačinjava svojstvo ili činjenicu. Tradicionalne koncepcije tih metafizičkih pojmova povlače za sobom ozbiljne filozofske zagonetke, zbog kojih se postojanje i svojstava i činjenica čini dvojbenim. Oslanjajući se na minimalističku teoriju istine, iznosim argumente u prilog minimalističkoj koncepciji svojstava i činjenica. Minimalistička teorija objašnjava svojstva i činjenice na osnovi prihvaćanja trivijalnih shema. Kako bih dokazao da je minimalizam nadmoćan pristup svojstvima i činjenicama, iznosim argumente protiv standardnih gledišta u filozofskoj literaturi. Tvrdim da minimalistički pristup svojstvima ima prednosti u odnosu na realizam, minimalizam i teoriju tropa. Tvrdim da je minimalistički pristup činjenicama nadmoćan standardnom shvaćanju činjenica u teorijama korespondencije. Metafizički minimalizam, minimalistička metafizika svojstava i činjenica, neovisna je i nadmoćna alternativa postojećim teorijama svojstava i činjenica. (shrink)

The triplet model treats a concept as complex structure that expresses three kinds of information. The first is about entities subsumed under a concept,their properties and relations. The second is about means and ways of representing the first information in intelligent systems. The third is about linkage between the first and second ones and methods of its constructing. The application of triplet models to generalization and development of concept models in philosophy, logic, cognitive psychology, cognitive science, linguistics, artificial intelligence has (...) been demonstrated. The componential structure of the triplet model and extended concept classification have been given. Theoretical considerations are combined with historical case-studies of concept history. For scientists and students with an interest in problems of knowledge structure and representation. Abstract and Extended Summary in English. (shrink)

In the first exposition of the doctrine of indeterminacy of translation, Quine asserted that the individuation and translation of truth-functional sentential connectives like 'and', 'or', 'not' are not indeterminate. He changed his mind later on, conjecturing that some sentential connectives might be interpreted in different non-equivalent ways. This issue has not been debated much by Quine, or in the subsequent literature, it is, as it were, an unsolved problem, not well understood. For the sake of the argument, I will adopt (...) Quine's background assumption that all the semantic features of a language can be reduced to the speakers' dispositions toward assent and dissent, as far as only the truth-conditional core of the meaning of sentences is concerned. I will put forward an argument to the effect that the speech dispositions of most, if not all, English (French, Italian, etc.) speakers constrain a unique translation of their connectives. This argument crucially relies on an empirical conjecture concerning the behaviour of these operators. (shrink)

Propositionalism is the view that intentional attitudes, such as belief, are relations to propositions. Propositionalists argue that propositionalism follows from the intuitive validity of certain kinds of inferences involving attitude reports. Jubien (2001) argues powerfully against propositions and sketches some interesting positive proposals, based on Russell’s multiple relation theory of judgment, about how to accommodate “propositional phenomena” without appeal to propositions. This paper argues that none of Jubien’s proposals succeeds in accommodating an important range of propositional phenomena, such as the (...) aforementioned validity of attitude-report inferences. It then shows that the notion of a predication act-type, which remains importantly Russellian in spirit, is sufficient to explain the range of propositional phenomena in question, in particular the validity of attitude-report inferences. The paper concludes with a discussion of whether predication act-types are really just propositions by another name. (shrink)

To talk about simple concepts presupposes that the notion of concept has been aptly explicated. I argue that a most adequate explication should abandon the set-theoretical paradigm and use a procedural approach. Such a procedural approach is offered by Tichý´s Transparent Intensional Logic (TIL). Some main notions and principles of TIL are briefly presented, and as a result, concepts are explicated as a kind of abstract procedure. Then it can be shown that simplicity, as applied to concepts, is well definable (...) as a property relative to conceptual systems, each of which is determined by a finite set of simple (‘primitive’) concepts. Refinement as a method of replacing simple concepts by compound concepts is defined. (shrink)

This paper offers a critical assessment of the current state of the debate about the identity and individuality of material objects. Its main aim, in particular, is to show that, in a sense to be carefully specified, the opposition between the Leibnizian ‘reductionist’ tradition, based on discernibility, and the sort of ‘primitivism’ that denies that facts of identity and individuality must be analysable has become outdated. In particular, it is argued that—contrary to a widespread consensus—‘naturalised’ metaphysics supports both the acceptability (...) of non-qualitatively grounded (both ‘contextual’ and intrinsic) identity and a pluralistic approach to individuality and individuation. A case study is offered that focuses on non-relativistic quantum mechanics, in the context of which primitivism about identity and individuality, rather than being regarded as unscientific, is on the contrary suggested to be preferable to the complicated forms of reductionism that have recently been proposed. More generally, by assuming a plausible form of anti-reductionism about scientific theories and domains, it is claimed that science can be regarded as compatible with, or even as suggesting, the existence of a series of equally plausible grades of individuality. The kind of individuality that prevails in a certain context and at a given level can be ascertained only on the basis of the specific scientific theory at hand. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...) universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

Relevantism, as a matter of definition, rejects classical logic as incorrect and adopts instead a relevant logic as encapsulating correct inference. It rejects classical logic on the grounds that the rule of Material Detachment, from A and not A or B to infer B,, sometimes leads from truth to falsity. Relevantism — although promoted by some relevant logicians, and an integral part of ultralogic — has recently encountered heavy, but interesting, criticism from relevance logicians themselves.

This book presents novel formalizations of three of the most important medieval logical theories: supposition, consequence and obligations. In an additional fourth part, an in-depth analysis of the concept of formalization is presented - a crucial concept in the current logical panorama, which as such receives surprisingly little attention.Although formalizations of medieval logical theories have been proposed earlier in the literature, the formalizations presented here are all based on innovative vantage points: supposition theories as algorithmic hermeneutics, theories of consequence analyzed (...) with tools borrowed from model-theory and two-dimensional semantics, and obligations as logical games. For this reason, this is perhaps the first time that these medieval logical theories are made fully accessible to the modern philosopher and logician who wishes to obtain a better grasp of them, but who has always been held back by the lack of appropriate ‘translations' into modern terms.Moreover, the book offers a reflection on the very nature of logic, a reflection that is prompted by the comparisons between medieval and modern logic, their similarities and dissimilarities. It is thus a contribution not only to the history of logic, but also to the philosophy of logic, the philosophy of language and semantics.The analysis of medieval logic is also relevant for the modern philosopher and logician in that, being the unifying methodology used across all disciplines at that time, logic really provided unity to science. It thus presents a unified model of scientific investigation, where logic plays the aggregating role. (shrink)

Whether meaning is compositional has been a major issue in linguistics and formal philosophy of language for the last 2 decades. Semantic holism is widely and plausibly considered as an objection to the principle of semantic compositionality therein. It comes as a surprise that the holistic peculiarities of scientific language have been rarely addressed in formal accounts so far, given that semantic holism has its roots in the philosophy of science. For this reason, a model-theoretic approach to semantic holism in (...) the language of science is presented here. This approach preserves compositionality to a large extent. *Received September 2009; revised February 2010. †To contact the author, please write to: Seminar for Philosophy, Logic, and Theory of Science, Hauspostfach 49, Ludwig-Maximilians-Universität, Munich 80539, Germany; e-mail: [email protected]. (shrink)

A first-order formulation of Leśniewski's ontology is formulated and shown to be interpretable within a free first-order logic of identity extended to include nominal quantification over proper and common-name concepts. The latter theory is then shown to be interpretable in monadic second-order predicate logic, which shows that the first-order part of Leśniewski's ontology is decidable.

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (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)

Frege's theory of indirect contexts and the shift of sense and reference in these contexts has puzzled many. What can the hierarchy of indirect senses, doubly indirect senses, and so on, be? Donald Davidson gave a well-known 'unlearnability' argument against Frege's theory. The present paper argues that the key to Frege's theory lies in the fact that whenever a reference is specified (even though many senses determine a single reference), it is specified in a particular way, so that giving a (...) reference implies giving a sense; and that one must be 'acquainted' with the sense. It is argued that an indirect sense must be 'immediately revelatory' of its reference. General principles for Frege's doctrine of sense and reference are sated, for both direct and indirect quotation, to be understood iteratively. I also discuss Frege's doctrine of tensed and first person statements in the light of my analysis. The views of various other authors are examined. The conclusion is to ascribe to Frege an implicit doctrine of acquaintance similar to that of Russell. (shrink)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does not discuss (...) many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

-/- A schema (plural: schemata, or schemas), also known as a scheme (plural: schemes), is a linguistic template or pattern together with a rule for using it to specify a potentially infinite multitude of phrases, sentences, or arguments, which are called instances of the schema. Schemas are used in logic to specify rules of inference, in mathematics to describe theories with infinitely many axioms, and in semantics to give adequacy conditions for definitions of truth. -/- 1. What is a Schema? (...) 2. Uses of Schemas 3. The Ontological Status of Schemas 4. Schemas in the History of Logic Bibliography. (shrink)

We critically investigate and refine Dunn's relevant predication, his formalisation of the notion of a real property. We argue that Dunn's original dialectical moves presuppose some interpretation of relevant identity, though none is given. We then re-motivate the proposal in a broader context, considering the prospects for a classical formalisation of real properties, particularly of Geach's implicit distinction between real and ''Cambridge'' properties. After arguing against these prospects, we turn to relevance logic, re-motivating relevant predication with Geach's distinction in mind. (...) Finally we draw out some consequences of Dunn's proposal for the theory of identity in relevance logic. (shrink)

The literature on conditionals is rife with alternate formulations of the abstract semantics of conditional logic. Each formulation has its own advantages in terms of applications and generalizations; nevertheless, they are for the most part equivalent, in the sense that they underwrite the same range of logical systems. The purpose of the present note is to bring under this umbrella the partition semantics introduced by Brian Skyrms in (Skyrms, 1984).

The paper deals with the semantics of mathematical notation. In arithmetic, for example, the syntactic shape of a formula represents a particular way of specifying, arriving at, or constructing an arithmetical object (that is, a number, a function, or a truth value). A general definition of this sense of "construction" is proposed and compared with related notions, in particular with Frege's concept of "function" and Carnap's concept of "intensional isomorphism." It is argued that constructions constitute the proper subject matter of (...) both logic and mathematics, and that a coherent semantic account of mathematical formulas cannot be given without assuming that they serve as names of constructions. (shrink)

The representation of quantification over relations in monadic third-order logic is discussed; it is shown to be possible in numerous special cases of foundational interest, but not in general unless something akin to the Axiom of Choice is assumed.

Mathematicians and physical scientists depend heavily on the formal symbolism of mathematics in order to express and develop their theories. For this and other reasons the last hundred years has seen a growing interest in the nature of formal language and the way it expresses meaning; particularly the objective, shared aspect of meaning as opposed to subjective, personal aspects. This dichotomy suggests the question: do the objective philosophical theories of meaning offer concepts which can be applied in psychological theories of (...) meaning? In recent years cognitive scientists such as Chomsky [1980], Fodor [1981] and MacNamara [1982] have used philosophical approaches to the meaning of formal language expressions as the basis for their psychological theories. Following this lead it seems appropriate to review some of the main treatments of meaning with a view to their transferability. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Intentionality is the power of minds to be about, to represent, or to stand for, things, properties and states of affairs. The puzzles of intentionality lie at the interface between the philosophy of mind and the philosophy of language. The word itself, which is of medieval Scholastic origin, was rehabilitated by the philosopher Franz Brentano towards the end of the nineteenth century. ‘Intentionality’ is a philosopher's word. It derives from the Latin word intentio, which in turn derives from the verb (...) intendere, which means being directed towards some goal or thing. The entry falls into eleven sections. (shrink)

According to sententialism, ‘Otto said that I am a fool’ expresses the holding of a relation between Otto and the sentence ‘I am a fool’. Sententialism is generally considered doomed, but I will show that a suitably developed sententialist account can surmount the many objections that have been raised. I will also show how important it is to have a fairer attitude towards sententialism. For if sententialist accounts are recognised as real options, it should also be recognised that the conclusion, (...) drawn by many, that sentences such as the report above force us to introduce Kaplanian monsters is unwarranted. (shrink)

Analytic philosophy was introduced in Latin America in the mid-twentieth century. Its development has been heterogeneous in different countries of the region but has today reached a considerable degree of maturity and originality, with a strong community working within the analytic tradition in Latin America. This entry describes the historical development of analytic philosophy in Latin America and offers some examples of original contributions by Latin American analytic philosophers.

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts (...) on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained). (shrink)

RésuméDans cet article, je soutiens qu'Aristote utilise une approche de la logique que l'on retrouve chez les praticiens de la logique informelle d'aujourd’hui. Même si sa position est complexe, Aristote privilégie des interprétations intensionnelles plutôt qu'extensionnelles de sa logique syllogistique. Il n'utilise pas les fonctions de vérité, mais adopte une version de la sémantique qui accentue le rôle du langage naturel qui ouvre, pour ainsi dire, une fenêtre sur la réalité métaphysique. Le mouvement de la logique informelle fut, en grande (...) partie, une initiative canadienne représentée par des auteurs bien connus tels Johnson, Blair, Govier et plusieurs autres. (shrink)

The paper is Part III of the large research, dedicated to both the revision of the system of basic logical categories and the generalization of modern predicate logic to functional logic. We determinate and contrapose modern Fregean logistics and proposed by the author ultra-Fregean logistics, next we describe values and arguments of functions, arguments of relations, relations themselves, sets (classes), and subsets (subclasses) as derivative categories (concepts) of ultrafregean logistics. Logistics is a part of metalogic, independent of semantics. Fregean logictics (...) is a metalogical theory, based on the quadruple <particular (individual), predicate, equality, sequence>; it generates predicate logic. Ultrafregean logictics is based on the quadruple <particular, function, representation, sequence>, where the notion of a function is a generalization of the notion of a predicate and the notion of representation is a generalization of the notion of equality; this logictics generates functional logic. For the completely correct denotation of the functional values, we need the Churchian symbolics with parenthesis. Predicates are usually identified with relations. A relation is the derived and even definable category of ultra-Fregean logictics. Namely, relations are representations by functions (of one of their arguments). We show that Frege could really establish this definition and the notion (category) of representation but, unfortunately, rejected this course of thought. Next, we show that every n-ary relation can be resolved for some of its arguments via some (n-1)-ary function. A set, or class, is a derived and not definable category of ultra-Fregean logistics. The universal way to introduce the sets is Frege’s abstraction principle. We formulate this principle for functional logic and show that the notion of a set is quantified, so there is the dual existential notion of a nonempty subset, involved by the same abstraction principle. (shrink)

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift (...) to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems. (shrink)

The Russell–Myhill theorem threatens a familiar structured conception of propositions according to which two sentences express the same proposition only if they share the same syntactic structure and their corresponding syntactic constituents share the same semantic value. Given the role of the principle of universal instantiation in the derivation of the theorem in simple type theory, one may hope to rehabilitate the core of the structured view of propositions in ramified type theory, where the principle is systematically restricted. We suggest (...) otherwise. The ramified core of the structured theory of propositions remains inconsistent in ramified type theory augmented with axioms of reducibility. This is significant because reducibility has been thought to be perfectly consistent with the ramified approach to the intensional antinomies. Nor is the addition of reducibility to ramified type theory sufficient to restore other intensional puzzles such as Prior’s paradox or Kripke’s puzzle about time and thought. (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 collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)