Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

This compelling reevaluation of the relationship between logic and knowledge affirms the key role that the notion of judgement must play in such a review. The commentary repatriates the concept of judgement in the discussion, banished in recent times by the logical positivism of Wittgenstein, Hilbert and Schlick, and the Platonism of Bolzano. The volume commences with the insights of Swedish philosopher Per Martin-Löf, the father of constructive type theory, for whom logic is a demonstrative science in which judgement is (...) a settled feature of the landscape. His paper opens the first of four sections that examine, in turn, historical philosophical assessments of judgement and reason; their place in early modern philosophy; the notion of judgement and logical theory in Wolff, Kant and Neo-Kantians like Windelband; their development in the Husserlian phenomenological paradigm; and the work of Bolzano, Russell and Frege. The papers, whose authors include Per Martin-Löf, Göran Sundholm, Michael Della Rocca and Robin Rollinger, represent a finely judged editorial selection highlighting work on philosophers exercised by the question of whether or not an epistemic notion of judgement has a role to play in logic. The volume will be of profound interest to students and academicians for its application of historical developments in philosophy to the solution of vexatious contemporary issues in the foundation of logic. ​. (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

Sortal concepts are at the center of certain logical discussions and have played a significant role in solutions to particular problems in philosophy. Apart from logic and philosophy, the study of sortal concepts has found its place in specific fields of psychology, such as the theory of infant cognitive development and the theory of human perception. In this monograph, different formal logics for sortal concepts and sortal-related logical notions are characterized. Most of these logics are intensional in nature and possess, (...) in addition, a bidimensional character. That is, they simultaneously represent two different logical dimensions. In most cases, the dimensions are those of time and natural necessity, and, in other cases, those of time and epistemic necessity. Another feature of the logics in question concerns second-order quantification over sortal concepts, a logical notion that is also represented in the logics. Some of the logics adopt a constant domain interpretation, others a varying domain interpretation of such quantification. Two of the above bidimensional logics are philosophically grounded on predication sortalism, that is, on the philosophical view that predication necessarily requires sortal concepts. Another bidimensional logic constitutes a logic for complex sortal predicates. These three sorts of logics are among the important novelties of this work since logics with similar features have not been developed up to now, and they might be instrumental for the solution of philosophically significant problems regarding sortal predicates. The book assumes a modern variant of conceptualism as a philosophical background. For this reason, the approach to sortal predicates is in terms of sortal concepts. Concepts, in general, are here understood as intersubjective realizable cognitive capacities. The proper features of sortal concepts are determined by an analysis of the main features of sortal predicates. Posterior to this analysis, the sortal-related logical notions represented in the above logics are discussed. There is also a discussion on the extent to which the set-theoretic formal semantic systems of the book capture different aspects of the conceptualist approach to sortals. These different semantic frameworks are also related to realist and nominalist approaches to sortal predicates, and possible modifications to them are considered that might represent those alternative approaches. (shrink)

Os tópicos e problemas filosóficos discutidos no volume são de natureza bastante variada: a natureza da complexidade computacional no processamento de uma língua natural; a relação entre o significado linguístico e o sentido Fregeano; as conexões entre a a agência e o poder; o conteúdo semântico da ficção; a explicação dos impasses éticos; a natureza dos argumentos cépticos; as conexões entre as dissociações cognitivas e o carácter modular da mente; a relação entre a referência e o significado. Estes tópicos deixam-se (...) subsumir num tema mais geral, o tema das ligações múltiplas entre a cognição e o conteúdo, mental ou linguístico. O tópico do conteúdo, a questão de determinar como é que muitas das nossas elocuções e muitos dos nossos estados mentais são dotados de conteúdo, representam algo (correcta ou incorrectamente), e o tópico da cognição, a investigação da natureza e dos mecanismos envolvidos na cognição humana, no processamento de informação tipicamente proveniente do exterior, são inegavelmente tópicos centrais da investigação filosófica, presente ou passada.Dado o carácter inclusivo e pluridisciplinar dos tópicos cobertos, não é surpreendente que os ramos da Filosofia representados no presente volume sejam igualmente diversificados: a Filosofia Moral, a Filosofia da Linguagem, a Teoria do Conhecimento, a Filosofia da Mente, os Fundamentos da Ciência Cognitiva. E também não é surpreendente que nele também estejam representados outros ramos do conhecimento cuja relevância para o estudo do conteúdo e da cognição é conspícua: a Psicologia Cognitiva; a Linguística Computacional e a Inteligência Artificial. Os autores dos ensaios são de proveniências diversas. De um lado, há um conjunto de especialistas nacionais a trabalhar na tradição analítica em Filosofia, como João Branquinho, ou a trabalhar em algumas áreas filosoficamente importantes da Ciência Cognitiva, como António Branco (Processamento da Língua Natural), José Frederico Marques (Psicologia Cognitiva) e Helder Coelho (Inteligência Artificial). Do outro lado, há um conjunto de especialistas estrangeiros de elevada reputação internacional: Allan Gibbard, da Universidade de Michigan, autor de livros influentes na área da Ética e Filosofia Moral; Stephen Schiffer, da New York University, uma das principais figuras actuais da Filosofia da Linguagem e da Semântica Geral; Paolo Leonardi, da Universidade de Bolonha, um especialista nas áreas da Filosofia da Linguagem e Filosofia da Comunicação; e Manuel García-Carpintero, da Universidade de Barcelona, um dos grandes vultos da Filosofia Analítica praticada em Espanha e na Europa Continental. (shrink)

In this paper we introduce a theoretical framework and a logical application for analysing the semantics and pragmatics of contrastive conjunctions in natural language. It is shown how expressions like although, nevertheless, yet, and but are semantically definable as connectives using an operator for implication in natural language and how similar pragmatic principles affect the behaviour of both contrastive conjunctions and indicative conditionals. Following previous proposals, conditions on contrast in a conjunction are analysed as presuppositions of die conjunction. Further linguistic (...) evidence leads to a distinction between restrictive and non-restrictive connectives of contrast, and consequently between direct and indirect contrast, which are given a precise definition. A general interface for a theory of contrast using possible world semantics for implication is then presented. As a test case, we show how this interface is applicable to the semantics for conditionals that was introduced by Veltman in his article 'Data semantics and the pragmatics of indicative conditionals' (1986). This application yields an extension of Veltman's Data Logic, called Contrastive Data Logic Once appropriate modifications are added to Veltman's pragmatic considerations, we show that contrastive data logic provides an adequate tool for the analysis of substantial linguistic data concerning contrast and implication in natural language. (shrink)

Imagination has recently attracted considerable attention from epistemologists and is recognized as a source of belief and even knowledge. One remarkable feature of imagination is that it is often and typically agentive: agents decide to imagine. In cases in which imagination results in a belief, the agentiveness of imagination may be taken to give rise to indirect doxastic control and epistemic responsibility. This observation calls for a proper understanding of agentive imagination. In particular, it calls for the development of a (...) semantics of imagination ascriptions. In the present paper an earlier suggestion by Ilkka Niiniluoto for a logic of imagination is considered. This proposal does not capture the agentive nature of imagination, and an alternative semantics is suggested. The new semantics combines the modal logic of agency with the neighbourhood semantics from alethic modal logic. (shrink)

SUMMARY1 The rational soul becomes the constant and dimensionless Euclidean point in all experience - defining the situations in which it finds itself, but itself undefined and undefinable in any situation. It is in nature but not of nature. Just as the dimensionless Euclidean point can occupy infinite positions on a line and yet remain unaltered, so the immortal, active intellect remains unaffected by the world in which it finds itself. It is not influenced by age, sense data, sickness or (...) death. It is the knower whose nature is pure knowing or reason. This is in great part the basic assumption that runs through the history of Aristotelean and modern symbolic logics24.The rational soul grasps identity, reasons consistently, and knows objectively for though it is in the world it is not of the world. Thus it is unnecessary and impossible to relate its truth to time or place or condition. As the source of understanding it need not be understood. It understands in that it knows. It is this which places the objective knower and observer outside of all systems of knowledge that it creates in philosophy, science and theology, and frees it from subjective interpretation. It is the absolute observer beyond self-observation.252 Language, values, and meanings reflect the absolute viewpoint of the dimensionless observer beyond the necessity of observation. The task of this observer is to know trie world just as the task of the animal and plant soul is motion and reproduction. The rational soul performs its task beyond nature as surely as animals and plants perform their tasks in nature. This is why it is possible to discuss the necessary evolution of living organisms in nature, but unnecessary to discuss the necessary or unnecessary nature of the evolution of this discussion. This is one of the reasons why determinism or predestination which seem so clearly applicable to nature, need not be applied to theories of determinism and predestination. These are principles grasped in reality beyond nature and totally inapplicable to the non-natural spirit that comprehends their essences. Thus self-reflexive questions about the questioner are as irrelevant as would be questions historically of “in what sense might the definition of dimensionless point itself be dimensioned or dimension-less?”26 Questions, on the other hand, that relate to the observed world are important for they can be truly answered (where questions as to the nature of the questioner cannot). In this view reality is not nature, but rather it is from reality that the objective observer, philosopher, scientist, and theologian understands and controls nature.3 Logic, the science of reason, is the science of all sciences. It reflects the pure and innate rational nature of the knowing intellect or soul. It is pure form beyond nature. Its realm is that of first principles or axioms. These too are in the world but not of it, as with Plato's eidos and Aristotle's universals. They are clear, true and self-evident. This explains the on going use of axiomatized revelations in studies in the natural and social sciences, the arts and the humanities. In symbolic logics, natural languages are in and of the world, but meta-languages are the constructs of pure intellectual activity and thus not natural. The abstract tends to become the more real, the less it can be found in nature. Rational souls, however, share a common reality that is as universal for Frege as it was for Aristotle.274 The occidental object tradition in general breaks out of the nature continuum by withdrawing into reality where the intellect knows and manipulates the world in philosophy, science, and theology from its position in and understanding of reality. Nature is a sub-set of reality and not vice versa. Axioms are the laws and determinants of nature and logic, but they are not nature.28 This means that at the basis of the occidental continuum of nature there is a qualitative and irreducible dichotomy of reality vs. nature. This has been most useful in the history of occidental science, but it produces a world view in which nature may lose the race with reality. SUMMARY IIIn the first portion of my paper, I discussed the implications of the Euclidean egg for philosophy and science as nature is understood and manipulated from reality. I shall now turn to what I see as the implications of the three-legged chicken view that the human intellect is in and of nature and that reality is a perspective in nature and is not simply the other of an irreducible qualitative dichotomy. Although there are an endless number of such implications that I could explore, the following, it seems to me are the most important:1 In the natural continuum there is no “dimensionless real” which can serve either as the locus of an external observer or a fixed position in nature. There is no perspective which is not an aspect of that upon which it is a perspective. The Chinese sage, for example, is one who sees himself/herself as one with or within natural process. This is expressed in a number of ways. Fung Yu-lan speaks of the difference between having no knowledge and having no-knowledge.34 Within such a continuum there can be no abstracting out of the absolute and unchanging. The nameless is the source of the named. First principles are the process itself and these principles are the alternation of opposites:“The Nameless is the origin of Heaven and Earth;The Named is the mother of all things.Therefore let there always be non-being so we may see their subtlety, And let there always be being so we may see their outcome. The two are the same.“352 An empty mind is not a blank mind. Where Aristotle viewed the mind as a blank tablet at birth to be filled by sense experience, Hsun Tzu speaks of its emptiness. It is the emptiness of the Tao itself:“The mind is constantly storing up things, and yet it is said to be empty. The mind is constantly marked by diversity, and yet it is said to be unified. The mind is constantly moving, and yet it is said to be still. Inevitably it will see things for itself. And although the objects it perceives may be many and diverse, if its acuity is of the highest level, it cannot become divided within itself.”36The nature in which and of which the mind is both perspective and aspect remains a totality. Being a totality, it is empty of itself just as any part of a whole, while a part of the whole, is empty of the whole. My ideas of myself are both of myself and empty of myself. This point can be illustrated with the simple example of knowing and speaking about two hands – one of which is mine, and the second of which belongs to someone else. 1 can objectify the second hand and fill my blank mind with information and observations about it, but I continue to remain outside of it –the arbiter in my statements of its nature and reality. However, in knowing my own hand, the hand known is also the knower and what I know represents a perspective upon an aspective of a totality of which “I–my–hand” are aspects. Being an aspect of the whole is being empty of the whole. If mind is in and of nature, any perspective on nature is an aspect of the whole. A blank mind is an unnatural mind outside of nature, filling up with impressions of nature. An empty mind is nature viewing nature as aspect and perspective. The occidental would be ashamed to reach old age and still have a blank mind. The Chinese sage would be ashamed to reach old age and still not recognize that the mind is empty.3 If mind is in nature and like nature, then language too reflects natural flow and change and process. It is one thing for me to talk about myself and another for me to talk about someone else. In the case of the latter, I may insist that objectively this other subject fits into this or that category and even administer tests to determine that it does do indeed so. But when I talk about myself, the speaker and the object of speech are one and the same. The terms I select do represent a dichotomy of sorts, but it is one of aspect and perspective and not subject and object. When I say “this is my hand” for example, the words create distinctions which are aspects of awareness but do not create an / that is or is not my hand. Language does not model or even represent reality as opposed to nature. It makes reality possible as something that nature says or thinks. Reality through language is an aspect of nature and thus a perspective upon itself as aspect. This is an important distinction that translators often miss.4 Logic in this natural continuum is concerned with distinctions in a very different sense. As a consequence of the unity of nature and mind, three kinds of distinction in experience are manifested: (a) Those distinctions which come into being as aspects of the changing world.The 10,000 things are in this sense aspects and not separate “identities” as some philosophers and scientists presuppose.37 (b) Those distinctions which come into being as aspects of the human mind. These are impressions of and thoughts about the infinite distinctions of which the restless mind is capable, (c) Those distinctions which come into being by way of the mind's use of symbols. These are the distinctions which permit us to stabilize and store our experiences, preserve the answers to our questions, and name the world into compartments and relationships.Implicit in these three assumptions which connect the distinguishing mind to the unity of a nature of'10,000‘ aspects, is an inferential process of aspect to perspective to knowledge to understanding to attitude to action/response. The reasoning mind synthesizes some aspect of experience in nature with a perspective upon this aspect. This synthesis, in turn, is an aspect of mind and the reasoning process proceeds from it to a perspective upon it which is knowledge and understanding. Knowledge and understanding in turn are aspects of thinking and the reasoning process proceeds from them to a perspective upon them which is an attitude. Attitude in turn is an aspect of thinking and the reasoning process proceeds from it to a perspective upon it which is reflected in the reasoning individual's actions and responses. This inferential process is a continuum which proceeds from aspects of experience to perspectives in thought to action and response. In this sequence an aspect is always connected by a perspective which becomes an aspect, until action/non-action, response occurs. This is as applicable to the details of life as it is to mastering life as a whole. When this inference process is incomplete many problems may result, ranging from action without understanding to the confusion of aspect and perspective with one another or to “obsession with distinctions.”38In practice this is a logic of paradox. The human being is aware that he or she is a living-dying aspect of nature, but an aspect which is also a perspective upon nature's aspects of living and dying. This paradox is the process around which a profound logic of aspect/perspective is developed by the sage. This view of logic within a continuum does not mean that there is no principle of identity, contradiction or excluded middle present – as some Chinese and European philosophers assume.These principles are also implicit in the knowing of my hand versus the knowing of the hand of another. However, the content, meaning, and application of these principles may vary just as when I say propriety, I mean propriety, and when the Chinese says li he means li but the context, meaning and application of the term may differ in each case.5 Aspect logic as it relates mind to nature in a cosmological totality is a series of syntheses in a cyclic continuum organizing thinking in phases. Each phase is an aspect/perspective synthesis which becomes an aspect of a further aspect/perspective synthesis. This progression might be described thusly: The synthesis of nature/human which is Tao (non-being) translates into the aspects of being (the 10,000 things), and as such is a synthesis of aspect and perspective. This synthesis of aspect and perspective translates into an aspect of ideas and as such is a synthesis of perception and conception. This synthesis of perception and conception translates into an aspect of knowledge, and as such is a synthesis of knowledge and understanding (understanding is a perspective upon knowledge) i.e., no-knowledge. This synthesis of knowledge and understanding translates into an aspect of purpose, and as such is a synthesis of purpose and action. The perspective here on action is aesthetic and/or moral. This synthesis of purpose and actions translates again into an aspect of nature and as such is a synthesis beyond individual aspects and perspective, i.e. Tao. Thus there is a cyclic dynamic in the system which remains open to further development.Tai Chen in his Inquiry Into Goodness connects this cycle of syntheses with wisdom and benevolence:Because there are mind and intelligence, the superior man knows that the perfection of the Way of Man depends completely upon nature. When he looks at the signs of the natural (Cheng), he knows the beginning of things; when he cherishes the virtue of supreme illumination, (Shen-ming), he knows the end of things. Proceeding from mind and intelligence to the attainment of the virtue of supreme intelligence, [the superior man] in his treatment of things in the world will cause the world to return to [the principles of] benevolence; and in his treatment of the capability inherent in things, he will cause the world to follow wisdom (Chih).39. (shrink)

Biosemiotics argues that “sign” and “meaning” are two essential concepts for the explanation of life. Peircean biosemiotics, founded by Tomas Sebeok from Peirce’s semiotics and Jacob von Uexkül’s studies on animal communication, today makes up the mainstream of this discipline. Marcello Barbieri has developed an alternative account of meaning in biology based on the concept of code. Barbieri rejects Peircean biosemiotics on the grounds that this discipline opens the door to nonscientific approaches to biology through its use of the concept (...) of “interpretation.” In this article, it is noted that Barbieri does not adequately distinguish among Peirce’s semiotics, Peircean biosemiotics, and “interpretation-based” biosemiotics. Two key arguments of Barbieri are criticized: his limited view of science and his rejection of “interpretation-based” biosemiotics. My argument is based on tools taken from a different approach: Robert Rosen’s relational biology. Instead of “signs” and “meanings,” the study begins in this case from the “components” and “functions” of the organism. Rosen pursues a new definition of a law of nature, introduces the anticipatory nature of organisms, and defines the living being as a system closed to efficient cause. It is shown that Code Biosemiotics and Peircean biosemiotics can share a common field of study. Additionally, some proposals are suggested to carry out a reading of Rosen’s biology as a biosemiotic theory. (shrink)

There is a little known paradox the solution to which is a guide to a much more thoroughgoing solution to a whole range of classic paradoxes. This is shown in this paper with respect to Berry's Paradox, Heterologicality, Russell's Paradox, and the Paradox of Predication, also the Liar and the Strengthened Liar, using primarily the epsilon calculus. The solutions, however, show not only that the first-order predicate calculus derived from Frege is inadequate as a basis for a clear science, and (...) should be replaced with Hilbert and Bernays' conservative extension. Standard second-order logic, and quantified propositional logic also must be substantially modified, to incorporate, in the first place, nominalizations of predicates, and whole sentences. And further modifications must be made, so as to insist that predicates are parts of sentences rather than forms of them, and that truth is a property of propositions rather than their sentential expressions. In all, a thorough reworking of what has been called 'logic' in recent years must be undertaken, to make it more fit for use. (shrink)

The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition which has followed Frege. It is shown first in this paper that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the (...) paper then goes on to indicate the lines on which formal logic—and also the lambda calculus and set theory—needs to be restructured, to remove the difficulties. Throughout the study of what have come to be known as first-, second-, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well known paradoxes, we shall see, arose because of the elementary level of simplification which has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off’ any natural language reading. (shrink)

In this paper, I raise some interconnected concerns for Paul Horwich’s minimal theory of truth, framed by these three questions: How should the minimal theory be formulated? How does the minimal theory address the liar paradox? What is the explanatory role of the concept of truth? I conclude that we cannot be linguistic or conceptual deflationists about truth.

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in this paper (...) have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent (...) formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other. (shrink)

This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question–Answer Semantics. We develop a generalized account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression ‘No’ is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators.

This paper aims to develop the implications of logical expressivism for a theory of dialogue coherence. I proceed in three steps. Firstly, certain structural properties of cooperative dialogue are identified. Secondly, I describe a variant of the multi-agent natural deduction calculus that I introduced in Piwek (J Logic Lang Inf 16(4):403–421, 2007 ) and demonstrate how it accounts for the aforementioned structures. Thirdly, I examine how the aforementioned system can be used to formalise an expressivist account of logical vocabulary that (...) is inspired by Brandom (Making it explicit: reasoning, representing, and discursive commitment, 1994 ; Articulating reasons: an introduction to inferentialism, 2000 ). This account conceives of the logical vocabulary as a tool which allows speakers to describe the inferential practices which underlie their language use, i.e., it allows them to make those practices explicit. The rewards of this exercise are twofold: (1) We obtain a more precise account of logical expressivism which can be defended more effectively against the critique that such accounts lead to cultural relativism. (2) The formalised distinction between engaging in a practice and expressing it, opens the way for a revision of the theory of dialogue coherence. This revision eliminates the need for logically complex formulae to account for certain structural properties of cooperative dialogue. (shrink)

In §8 of his Begriffsschrift (1879), Gottlob Frege discusses issues related to identity. Frege begins his most famous essay, “On Sense and Denotation” (1892), published 13 years later, by criticizing the view advocated in §8. He returns to these issues in the concluding paragraph. Controversies continue over these important passages. We offer an interpretation and discuss some alternatives. We defend that in the Begriffsschrift, Frege does not hold that identity is a relation between signs. §8 of the Begriffsschrift is motivated (...) by the conflict between two different criteria for sameness of conceptual content of sentences. To resolve that conflict, Frege introduces ‘≡’ in §8 and, thus, circumstances with names as constituents. To the same end, in “On Sense and Denotation,” Frege introduces senses and Thoughts and abandons both ‘≡’ and circumstances. He solves what we call the Co-instantiation problem, and disregards, but does not solve, the Name problem. (shrink)

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the prospects for justifying this (...) move are qualitatively worse in mathematics than they are in science. (shrink)

One of the central debates in the philosophy of language is that between defenders of the causal-historical and descriptivist theories of reference. Most philosophers involved in the debate support one or the other of the theories. Building on recent experimental work in semantics, we argue that there is a sense in which both theories are correct. In particular, we defend the view that natural kind terms can sometimes take on a causal-historical reading and at other times take on a descriptivist (...) reading. The meaning will shift depending on the conversational setting. The theoretical view has roots in work by Kitcher. We present some original experiments that support the thesis. (shrink)

This is the introductory essay to a collection commemorating the 100th anniversary of the publication in Mind of Bertrand Russell’s paper ‘On Denoting’.

This book sets out the foundations, methodology, and practice of a formal framework for the description of language. The approach embraces the trends of lexicalism and compositional semantics in computational linguistics, and theoretical linguistics more broadly, by developing categorial grammar into a powerful and extendable logic of signs. Taking Montague Grammar as its point of departure, the book explains how integration of methods from philosophy (logical semantics), computer science (type theory), linguistics (categorial grammar) and meta-mathematics (mathematical logic ) provides a (...) categorial foundation with coverage including intensionality, quantification, featural polymorphism, domains and constraints. For the first time, the book systematises categorial thinking into a unified program which is at once both logically secured, and a practical tool for pure lexical grammar development with type-theoretic semantics. It should be of interest to all those active in computational linguistics and formal grammar and is suitable for use at advanced undergraduate, postgraduate, and research levels. (shrink)

In this essay, I argue for an account of property possession as strict, numerical identity. According to this account, for an entity to possess a property is for that entity to be numerically identical to that property.

Whether a predicate is a referential expression depends upon what reference is conceived to be. Even if it is granted that reference is a relation between words and worldly items, the referents of expressions being the items to which they are so related, this still leaves considerable scope for disagreement about whether predicates refer. One of Frege's great contributions to the philosophy of language was to introduce an especially liberal conception of reference relative to which it is unproblematic to suppose (...) that predicates are referring expressions. According to this liberal conception, each significant expression in a language has its own distinctive semantic role or power, a power to effect the truth-value of the sentences in which it occurs. (shrink)

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples (...) from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization. (shrink)

In bilateral logic formulas are signed by + and –, indicating the speech acts assertion and denial. I argue that making an assumption is also speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Attempts to solve this problem are considered and rejected.

Part 1 sets out the logical/semantical background to ‘On Denoting’, including an exposition of Russell's views in Principles of Mathematics, the role and justification of Frege's notorious Axiom V, and speculation about how the search for a solution to the Contradiction might have motivated a new treatment of denoting. Part 2 consists primarily of an extended analysis of Russell's views on knowledge by acquaintance and knowledge by description, in which I try to show that the discomfiture between Russell's semantical and (...) epistemological commitments begins as far back as 1903. I close with a non-Russellian critique of Russell's views on how we are able to make use of linguistic representations in thought and with the suggestion that a theory of comprehension is needed to supplement semantic theory. (shrink)

Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different (...) formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences. (shrink)

Quantifiers in Frege's Grundgesetze like are not well-defined because the part Fx & Gx stands for a concept but the yoking conjunction is horizontalised and must stand for a truth-value. This standard interpretation is rejected in favor of a substitutional reading that, it is argued, both conforms better to the text and is well-defined. The theory of the horizontal is investigated in detail and the composite reading of Frege's connectives as made up of horizontals is rejected. The sense in which (...) the Grundgesetze has a many-valued but classical logic is explained by proving that its propositional fragment (under the standard interpretation) and Bochvar's 3-valued logic are instances of the same metatheoretic methods. Historically, the paper argues for a more naive but well-defined reading of the text. Theoretically, it provides a formally adequate statement of that semantics, as well as developing the abstract metatheory which embraces Bochvar's language and the Fregean fragment. (shrink)

There are two principles which bear the name Frege''sprinciple: the principle of compositionality, and the contextprinciple. The aim of this contribution is to investigate whether thisis justified: did Frege accept both principles at the same time, did hehold the one principle but not the other, or did he, at some moment,change his opinion? The conclusion is as follows. There is a developmentin Frege''s position. In the period of Grundlagen he followed to a strict form of contextuality. He repeatedcontextuality in later (...) writings, but became less strict. From 1914 on,pushed by the needs of research, he comes close to compositionality. Buthe could never make the final step toward compositionality forprincipled reasons, therefore he always would reject compositionality. (shrink)

§1 Analytic philosophers interested in depiction have focused for the most part on two problems: first, explaining how pictures represent; second, describing the distinctive kinds of artistic value pictures can possess, or the distinctive ways in which they can embody artistic values that extend more broadly across the arts. I shall discuss the first problem here. The main concepts I shall be concerned with are depiction, resemblance, sense and reference.

Frege claims that the laws of logic are characterized by their “generality,” but it is hard to see how this could identify a special feature of those laws. I argue that we must understand this talk of generality in normative terms, but that what Frege says provides a normative demarcation of the logical laws only once we connect it with his thinking about truth and science. He means to be identifying the laws of logic as those that appear in every (...) one of the scientific systems whose construction is the ultimate aim of science, and in which all truths have a place. Though an account of logic in terms of scientific systems might seem hopelessly antiquated, I argue that it is not: a basically Fregean account of the nature of logic still looks quite promising. (shrink)

Having briefly sketched the aims of our paper, namely, to logically analyse the ascription of propositional attitudes to somebody else in terms, not of Fregean senses or of intensions-with-s, but of the intentional object of the person spoken about, say, the believer or intender (Section 1), we try to introduce the concept of an intentional object as simply as possible, to wit, as coming into view whenever two (or more) subjective belief-worlds strikingly diverge (Section 2). Then, we assess the pros (...) and cons of Frege’s view that the indirect reference of an expression is nothing but its customary sense (Sections 3–4), and call the reader’s attention to the fact that in belief ascriptions de re we take it for granted that the believer’s intentional object is at the same time a ‘citizen’ of the belief ascriber’s subjective world (or, for that matter, the real word), and that the idea of such a ‘dual citizenship’ is even more obviously presupposed in the cases of true belief and propositional knowledge (Section 5). Then, we try to argue that it is more fertile to take the belief ascriber’s intentional object to be, not the whole state of affairs the believer has in mind and thinks to exist, but the latter’s intentional object as such, that is, as being his intentional object (Section 6). Finally, we discuss the intricate and mostly neglected question of whether an intentional object’s inhabiting two or more subjective belief-worlds should be considered to be sort of a ‘transmundane identity’ in the numerical sense or rather in some deviant sense of the term, which can be specified as a momentous subcase of a ‘categorial difference’ (Section 7). (shrink)

Brahmānanda Sarasvatī has written an elaborate comment on the following inference cited in Advaitasiddhi: attribute etc. are identical to and different from attributee etc. since they are co-referential. There he wants to prove that every significant case of attribution is a case of identity that coexists with a difference between two demarcators. The identity that coexists with difference is called ‘equality’. This paper will argue that in each case of equality, the realist ontology chooses either identity over difference or the (...) reverse. In the case of ‘the black pot’, the realist ontology prefers difference over identity. In the case of ‘the qualified object is the mere object’, the same ontology prefers identity over difference. Accordingly the ontological ‘entity’ gets projected. This is perhaps because the realist ontology is guided by Ockham’s razor and the grammatical classification of objects. (shrink)

The translation of both ‘bedeuten’ and ‘Bedeutung’ in Frege's works remains sufficiently problematic that some contemporary authors prefer to leave these words untranslated. Here a case is made for returning to Russell's initial choice of ‘to indicate’ and ‘indication’ as better alternatives than the more usual ‘meaning’, ‘reference’, or ‘denotation’. It is argued that this choice has the philosophical payoff that Frege's controversial doctrines concerning the semantic values of sentences and predicative expressions are rendered far more comprehensible by it, and (...) that this translational strategy fulfills the desiderata of offering a translation which is acceptable both before and after Frege introduced the distinction between sense and reference or, as this paper would have it, between the sense of an expression and what it indicates. (shrink)

Frege seems to hold two incompatible theses:(i) that sentences differing in structure can yet express the same sense; and (ii) that the senses of the meaningful parts of a complex term are determinate parts of the sense of the term. Dummett offered a solution, distinguishing analysis from decomposition. The present paper offers an embellishment of Dummett?s distinction by providing a way of depicting the internal structures of complex senses?determinate structures that yield distinct decompositions. Decomposition is then shown to be adequate (...) as a foundation for the informativity and analyticity of logic. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

Koopman’s book revolves around the notion of transition, which he proposes is one of the central ideas of the pragmatist tradition but one which had not previously been fully articulated yet nevertheless shapes the pragmatist attitude in philosophy. Transition, according to Koopman, denotes “those temporal structures and historical shapes in virtue of which we get from here to there”. One of the consequences of transitionalism is the understanding of critique and inquiry as historical pro...

Book reviewed in this article:F.H. Bradley, Collected Works Volumes 1–5.