In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language (...) of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificial intelligence and mathematical linguistics will also find this an accessible introduction to a subject of increasing application to these disciplines. (shrink)

The authors' novel approach to some interesting mathematical concepts - not normally taught in other courses - places them in a historical and philosophical setting. Although primarily intended for mathematics undergraduates, the book will also appeal to students in the sciences, humanities and education with a strong interest in this subject. The first part proceeds from about 1800 BC to 1800 AD, discussing, for example, the Renaissance method for solving cubic and quartic equations and providing rigorous elementary proof that certain (...) geometrical problems posed by the ancient Greeks cannot be solved by ruler and compass alone. The second part presents some fundamental topics of interest from the past two centuries, including proof of G del's incompleteness theorem, together with a discussion of its implications. (shrink)

It may be argued that the language of mathematics is about the category\nof sets, although the definite article requires some justification.\nAs possible worlds of mathematics we may admit all models of type\ntheory, by which we mean all local toposes. For an intuitionist,\nthere is a distinguished local topos, namely the so-called free topos,\nwhich may be constructed as the Tarski–Lindenbaum category of intuitionistic\ntype theory. However, for a classical mathematician, to pick a distinguished\nmodel may be as difficult as to define the notion of (...) truth in classical\ntype theory, which Tarski has shown to be impossible. Author Keywords:\nAuthor Keywords: Type theory; Categorical logic; Category of sets;\nTopos theory *1 I wish to thank Elaine Landry for organizing a symposium\nat Brock University in 1996, at which an earlier version of this\npaper was presented, and Phil Scott for helpful discussions. Support\nfrom the Social Sciences and Humanities Research Council of Canada\nis herewith acknowledged. (shrink)

A version of intuitionistic type theory is presented here in which all logical symbols are defined in terms of equality. This language is used to construct the so-called free topos with natural number object. It is argued that the free topos may be regarded as the universe of mathematics from an intuitionist's point of view.

RésuméUn paradigme en vogue en informatique théorique exploite l'analogie entre les formules et les types et traite une deduction Al… An→ B comme une opération plurisortale. On propose ici d'étendre cette analogie aux déductions de la forme Al… An→, où la place à droite de la flèche est vide. D'un point de vue logique, une telle déduction constitue une réfutation de la conjonction desformules qui se trouvent à gauche de la flèche. On défend l'idée qu'ilfaut, selon ce paradigme étendu, interpréter (...) la flèche comme assignant à n'importe quelle séquence al… an, où aiest de type Ai, le type des valeurs de vérite correspondantes. Pour que cela fonctionne, cependant, il faut abandonner la règle structurale de l'affaiblissement à droite de la flèche. La négation de A est alors interprétée comme l'ensemble puissance de A et la complétude fonctionnelle habituelle s'identifie au schème de compréhension avec extensionnalitè. (shrink)

Categories may be viewed as deductive systems or as algebraic theories. We are primarily interested in the interplay between these two views and trace it through a number of structured categories and their internal languages, bearing in mind their relevance to the foundations of mathematics. We see this as a common thread running through the six contributions to this issue of Studia Logica.

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formulaHaving graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules were (...) replaced by pocket calculators and personal computers, but I stuck to my original decision.My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem, which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functionsf: ℕn→ ℕ are computable?While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:f is recursive,f is computable on an abstract machine,f is definable in the untyped λ-calculus.These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance. (shrink)

Pregroups are partially ordered monoids in which each element has two “adjoints”. Pregroup grammars provide a computational approach to natural languages by assigning to each word in the mental dictionary a type, namely an element of the pregroup freely generated by a partially ordered set of basic types. In this expository article, the attempt is made to introduce linguists to a pregroup grammar of English by looking at Chomsky’s earliest examples.

We explore a computational algebraic approach to grammar via pregroups, that is, partially ordered monoids in which each element has both a left and a right adjoint. Grammatical judgements are formed with the help of calculations on types. These are elements of the free pregroup generated by a partially ordered set of basic types, which are assigned to words, here of English. We concentrate on the object pronoun who(m).

Adapting Wittgenstein's concept of the human species as 'a ceremonial animal', Wendy James discusses in a readable and lively style the conceptual ordering of space, time, and rhythm; the mutualities of language, consciousness, ritual and religious practice; the dialectics of gender and generation; power, war, and peace; and large-scale modern social formations such as the city and the nation. The Foreword is by Michael J. Lambek, Professor of Anthropology, University of Toronto.

A 'Kripke-style' semantics is given for combinatory logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for relevance logic. We prove by algebraic means a completeness theorem for combinatory logic, by proving a representation theorem for 'combinatory posets.' A philosophical interpretation is given of the models, showing that an element of a combinatory poset can be understood simultaneously as a set of states and as a set of actions on states. This double interpretation allows for (...) one such element to be applied to another . Application turns out to be modeled the same way as 'fusion' in relevance logic.We also introduce 'dual combinators' that apply from the right. We then explore relationships to some well-known substructural logics, showing that each can be embedded into the structurally free. non-associative Lambek calculus, with the embedding taking a theorem ω to a statement of the form Γ ⊨ ω, where Γ is some fusion of the combinators needed to justify the structural assumptions of the given substructural logic. This builds on earlier ideas from Belnap and Meyer about a Gentzen system wherein structural rules are replaced with rules for introducing combinators. We develop such a system and prove a cut theorem. (shrink)

Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.

This book is dedicated to the life and work of the mathematician Joachim Lambek. The editors gather together noted experts to discuss the state of the art of various of Lambek’s works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist. In the 1960s, he began to work in category theory, categorical (...) algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard’s linear logic. In a surprising development, he introduced a novel and deeper algebraic framework for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists. (shrink)

Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system (...) and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)

A biresiduation algebra is a 〈/,\,1〉-subreduct of an integral residuated lattice. These algebras arise as algebraic models of the implicational fragment of the Full Lambek Calculus with weakening. We axiomatize the quasi-variety B of biresiduation algebras using a construction for integral residuated lattices. We define a filter of a biresiduation algebra and show that the lattice of filters is isomorphic to the lattice of B-congruences and that these lattices are distributive. We give a finite basis of terms for generating (...) filters and use this to characterize the subvarieties of B with EDPC and also the discriminator varieties. A variety generated by a finite biresiduation algebra is shown to be a subvariety of B. The lattice of subvarieties of B is investigated; we show that there are precisely three finitely generated covers of the atom. (shrink)

We study the computational complexity of the universal theory of residuated ordered groupoids, which are algebraic structures corresponding to Nonassociative Lambek Calculus. We prove that the universal theory is co $$\textsf {NP}$$ -complete which, as we observe, is the lowest possible complexity for a universal theory of a non-trivial class of structures. The universal theories of the classes of unital and integral residuated ordered groupoids are also shown to be co $$\textsf {NP}$$ -complete. We also prove the co $$\textsf (...) {NP}$$ -completeness of the universal theory of classes of residuated algebras, algebraic structures corresponding to Generalized Lambek Calculus. (shrink)

In 1958 J. Lambek introduced a calculusL of syntactic types and defined an equivalence relation on types: x y means that there exists a sequence x=x1,...,xn=y (n 1), such thatx i x i+1 or xi+ x i (1 i n). He pointed out thatx y if and only if there is joinz such thatx z andy z. This paper gives an effective characterization of this equivalence for the Lambeck calculiL andLP, and for the multiplicative fragments of Girard's and Yetter's (...) linear logics. Moreover, for the non-directed Lambek calculusLP and the multiplicative fragment of Girard's linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are. (shrink)

Information can be public among a group. Whether or not information is public matters, for example, for accounts of interdependent rational choice, of communication, and of joint intention. A standard analysis of public information identifies it with (some variant of) common belief. The latter notion is stipulatively defined as an infinite conjunction: for p to be commonly believed is for it to believed by all members of a group, for all members to believe that all members believe it, and so (...) forth. This analysis is often presupposed without much argument in philosophy. Theoretical entrenchment or intuitions about cases might give some traction on the question, but give little insight about why the identification holds, if it does. The strategy of this paper is to characterize a practical-normative role for information being public, and show that the only things that play that role are (variants of) common belief as stipulatively characterized. In more detail: a functional role for “taking a proposition for granted” in non-isolated decision making is characterized. I then present some minimal conditions under which such an attitude is correctly held. The key assumption links this attitude to beliefs about what is public. From minimal a priori principles, we can argue that a proposition being public among a group entails common commitment to believe among that group. Later sections explore partial converses to this result, the factivity of publicity and publicity from the perspective of outsiders to the group, and objections to the aprioricity of the result deriving from a posteriori existential presuppositions. (shrink)

Bringing together ethnographic exposition with philosophical concepts and arguments and effectively transcending subdisciplinary boundaries between cultural and ...

The achievements of the textile industry in Roman Britain are often underestimated as a result of the meagreness of our available evidence. The Edict on maximum prices issued by Diocletian in A.D. 301 shows that British capes commanded high prices on the markets of the Empire, and that in the late third century A.D. British rugs were the best in the world. In view of the competition from the traditional centres of rug manufacture in the East, this is an astonishing (...) achievement. (shrink)

The received translation and interpretation of many of the technical terms current in the textile industry of the Roman Empire are inaccurate, because lexicographers have either fought shy of being precise, or have thought that they recognized in the ancient world technical processes which originated at a much later date. The evidence is often equivocal or insufficient, but may still yield details that have been overlooked. The textile expression scutulatus, to take an example, deserves more attention than Blümner has devoted (...) to it. When it refers to the appearance of a garment, Lewis and Short, s.v. scutulatus, translate ‘diamond or lozenge-shaped, checkered’. It seems worth while to attempt a more precise translation, and, if possible, to estimate the significance in context of the goods so designated. (shrink)

Living out of silence, out of a fully functioning, lovingly attentive mind, and not just out of thought, out of a partially functioning mind, is requisite for depth or profundity in living or relating. A fully attentive, truly silent or meditative mind sees that there is real dualism of time and the timeless and that time and the timeless each has its own unique value. The timeless, or real silence, that which alone can make for depth in one's living and (...) relating, can be there only when, in one's awareness, time and all that is related to time, have come to an end. An examination of a wide variety of statements made by writers, both past and present, shows that these understandings are seldom there in contemporary writings, just as they were not there in writings that have come down to us from the past. 'Living beyond the One and the Many' invites readers to consider the possibility that what is presented in it can bring deep meaning to one's way of living in the world"--Publisher's description, back cover. (shrink)

This article presents a reading of Nietzsche’s use of rhetoric as inseparable from his philosophical project. I provide an exegesis of Nietzsche’s own reflections on rhetoric and consider its actual deployment, arguing that Nietzsche’s rhetoric is often deliberately dissonant and oriented toward facilitating receptive effects. The aim, I suggest, is to shift politics of possibility—to alter what can and cannot be done and said politically. Dissonant rhetoric, rhetoric that marries aesthetic attunement with affective turbulence, helps to accomplish this end by (...) shaping the way that rhetoric is received by audiences. I conclude by suggesting that Nietzsche’s rhetoric has implications for contemporary theory, shifting how we might view critical political engagement in the public sphere. Understood in this way, Nietzsche’s rhetoric provides a perhaps surprising model for a critically robust form of rhetoric. (shrink)

This article presents a reading of Nietzsche’s use of rhetoric as inseparable from his philosophical project. I provide an exegesis of Nietzsche’s own reflections on rhetoric and consider its actual deployment, arguing that Nietzsche’s rhetoric is often deliberately dissonant and oriented toward facilitating receptive effects. The aim, I suggest, is to shift politics of possibility—to alter what can and cannot be done and said politically. Dissonant rhetoric, rhetoric that marries aesthetic attunement with affective turbulence, helps to accomplish this end by (...) shaping the way that rhetoric is received by audiences. I conclude by suggesting that Nietzsche’s rhetoric has implications for contemporary theory, shifting how we might view critical political engagement in the public sphere. Understood in this way, Nietzsche’s rhetoric provides a perhaps surprising model for a critically robust form of rhetoric. (shrink)

If the world itself is metaphysically indeterminate in a specified respect, what follows? In this paper, we develop a theory of metaphysical indeterminacy answering this question.

Anthropology has recently seen a lively interest in the subject of ethics and comparative notions of morality and freedom. This masterclass brings together four of the most eminent anthropologists working in this field--Michael Lambek, Veena Das, Didier Fassin, and Webb Keane--to discuss, via lectures and responses, important topics facing anthropological ethics and the theoretical debates that surround it. The authors explore the ways we understand morality across many different cultural settings, asking questions such as: How do we recognize the (...) ethical in different ethnographic worlds? What constitutes agency and awareness in everyday life? What might an anthropology of ordinary ethics look like? And what happens when ethics approaches the political in both Western and non-Western societies. Contrasting perspectives and methods- and yet in complimentary ways- this masterclass will serve as an essential guide for how an anthropology of ethics can be formulated in the twenty-first century. (shrink)

Husserlian phenomenology, as the study of conscious experience, has often been accused of solipsism. Husserl’s method, it is argued, does not have the resources to provide an account of consciousness of other minds. This chapter will address this issue by providing a brief overview of the multiple angles from which Husserl approached the theme of intersubjectivity, with specific focus on the details of his account of the concrete interpersonal encounter – “empathy.” Husserl understood empathy as a direct, quasi-perceptual form of (...) intentionality through which the sense of the Other is constituted. Furthermore, his account of empathy is holistically integrated with his overall theory of intersubjectivity, including his discussions of the objectivity of nature, and the social, historical, and communal aspects of subjectivity. Husserl’s theory of empathy continues to cast a long shadow, influencing both the analytic and continental approaches to the problem of other minds, as well as contemporary account of social cognition in the cognitive sciences. (shrink)

The Wrong Question? is the response by an anthropologist to a question posed by a philosopher concerning the intelligibility of alien forms of thought. I argue that it is wrong to describe the problem of intelligibility as one of logic or rationality. Indeed, foreign practices (no less than our own) may become intelligible only once they are not evaluated according to abstract criteria of rationality. To ask of a given practice or form of life whether it is rational is an (...) error of grammar (nonsense) in Wittgenstein’s sense. I describe how intelligibility emerges over the course of ethnographic fieldwork but also argue that we must work on our own concepts in order to make foreign ones intelligible. The response draws from both Gadamer and Wittgenstein as well as anthropologists Geertz, Evans-Pritchard, and Lévi-Strauss. Following Cora Diamond, I suggest further that the ethical and rational dimensions of understanding another are indissociable. (shrink)

Grammar can be formulated as a kind of substructural propositional logic. In support of this claim, we survey bare Gentzen style deductive systems and two kinds of non-commutative linear logic: intuitionistic and compact bilinear logic. We also glance at their categorical refinements.

Accession Number: ATLA0001712194; Hosting Book Page Citation: p 271-289.; Language(s): English; General Note: Bibliography: p 287-289.; Issued by ATLA: 20130825; Publication Type: Essay.