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 Detailed historical references are provided throughout, and each section concludes with a set of exercises.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)

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.

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.

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)

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.

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)

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)

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)

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.

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)

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

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)

This book suggests a bold comparative approach to broad cultural differences between Africa and Melanesia. Its theme is personhood, understood in terms of what anthropologists call embodiment. These concepts are applied to questions ranging from the meanings of spirit possession, to the logics of witchcraft and kinship relations, the use of rituals in healing, and even the impact of capitalism. Questioning common assumptions about the huge differences among these discrete areas, the contributions document surprising continuities.

The Tanner Lectures are a collection of educational and scientific discussions relating to human values. Conducted by leaders in their fields, the lectures are presented at renowned institutions around the world, including the Universities of Oxford, Harvard, and Yale. In January 2019, University of Toronto's Michael Lambek, professor, former Canada Research Chair, and member of the Royal Society of Canada, delivered the Tanner Lecture at the University of Michigan's Department of Philosophy on the topic of "Concepts and Persons." As (...) well as tracing his career in social and cultural anthropology, Lambek's Tanner Lecture spoke on the intersection of anthropology and philosophy as a means of articulating the moral basis of human action. By elucidating where anthropology and philosophy might intersect, Lamberk's lecture is a profound examination of the human condition, and is beautifully captured in this publication. Concepts and Persons recounts the lecture as delivered at the prestigious event, the commentary of three distinguished respondents, and Lambek's own response to that commentary. The book's presentation of the lecture also includes a rich and layered set of notes that augment the lecture significantly, as well as additional clarification and thought that has developed since the event. (shrink)

Written over a thirty-year span, Michael Lambek’s essays in this collection point with definitive force toward a single central truth: ethics is intrinsic to social life. As he shows through rich ethnographic accounts and multiple theoretical traditions, our human condition is at heart an ethical one—we may not always be good or just, but we are always subject to their criteria. Detailing Lambek’s trajectory as one anthropologist thinking deeply throughout a career on the nature of ethical life, the (...) essays accumulate into a vibrant demonstration of the relevance of ethics as a practice and its crucial importance to ethnography, social theory, and philosophy. Organized chronologically, the essays begin among Malagasy speakers on the island of Mayotte and in northwest Madagascar. Building from ethnographic accounts there, they synthesize Aristotelian notions of practical judgment and virtuous action with Wittgensteinian notions of the ordinariness of ethical life and the importance of language, everyday speech, and ritual in order to understand how ethics are lived. They illustrate the multiple ways in which ethics informs personhood, character, and practice; explore the centrality of judgment, action, and irony to ethical life; and consider the relation of virtue to value. The result is a fully fleshed-out picture of ethics as a deeply rooted aspect of the human experience. (shrink)

Consequence is at the heart of logic, and an account of consequence offers a vital tool in the evaluation of arguments. This text presents what the authors term as 'logical pluralism' arguing that the notion of logical consequence doesn't pin down one deductive consequence relation; it allows for many of them.

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)

In this paper we consider the relations existing between four deductive systems that have been called categorial grammars and have relevant connections with linguistic investigations: the syntactic calculus, bilinear logic, compact bilinear logic and Curry''s semantic calculus.

Providing an original and systematic treatment of foundational issues in philosophy of religion, J. L. Schellenberg's new book addresses the structure of..

What happens when someone acts? A familiar answer goes like this. There is something that the agent wants, and there is an action that he believes conducive to its attainment. His desire for the end, and his belief in the action as a means, justify taking the action, and they jointly cause an intention to take it, which in turn causes the corresponding movements of the agent's body. I think that the standard story is flawed in several respects. The flaw (...) that will concern me in this paper is that the story fails to include an agent-or, more precisely, fails to cast the agent in his proper role. (shrink)

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

One way of coping with agreement of features in French is to perform two parallel computations, one in the free pregroup of syntactic types, the other in that of feature types. Technically speaking, this amounts to working in the direct product of two free pregroups.

Abstract I argue that meaning in life is importantly influenced by bioloical ties. More specifically, I maintain that knowing one's relatives and especially one's parents provides a kind of self-knowledge that is of irreplaceable value in the life-task of identity formation. These claims lead me to the conclusion that it is immoral to create children with the intention that they be alienated from their bioloical relatives?for example, by donor conception.

For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory. This Festschrift was put together on the occasion of his 90th birthday. The papers in it give a good picture of the multiple research areas where the impact of Jim Lambek's work can be felt. The volume includes contributions by prominent researchers and by their students, showing how Jim Lambek's ideas (...) keep inspiring upcoming generations of scholars. (shrink)