This paper solves a natural but still open question: can abstract categorial grammars (ACGs) respresent usual categorial grammars? Despite their name and their claim to be a unifying framework, up to now there was no faithful representation of usual categorial grammars in ACGs. This paper shows that Non-Associative Lambek grammars as well as their derivations can be defined using ACGs of order two. To conclude, the outcome of such a representation are discussed.

This paper solves a natural but still open question: can abstract categorial grammars respresent usual categorial grammars? Despite their name and their claim to be a unifying framework, up to now there was no faithful representation of usual categorial grammars in ACGs. This paper shows that Non-Associative Lambek grammars as well as their derivations can be defined using ACGs of order two. To conclude, the outcome of such a representation are discussed.

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice (...) of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents and \), thus solving a problem left open by Pentus. (shrink)

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

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.

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)

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)

How should you decide what to do when you’re uncertain about basic normative principles? A natural suggestion is to follow some "second-order:" norm: e.g., obey the most probable norm or maximize expected choiceworthiness. But what if you’re uncertain about second-order norms too—must you then invoke some third-order norm? If so, any norm-guided response to normative uncertainty appears doomed to a vicious regress. This paper aims to rescue second-order norms from the threat of regress. I first elaborate and defend the claim (...) some philosophers have made that the regress problem forces us to accept normative externalism, the view that at least one norm is incumbent on all agents regardless of their normative beliefs. But, I then argue, we need not accept externalism about first-order norms, thus closing off any question of how agents should respond to normative uncertainty. Rather, we can head off the threat of regress by ascribing external force to a single second-order norm: the enkratic principle. (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)

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)

Decision-making under normative uncertainty requires an agent to aggregate the assessments of options given by rival normative theories into a single assessment that tells her what to do in light of her uncertainty. But what if the assessments of rival theories differ not just in their content but in their structure -- e.g., some are merely ordinal while others are cardinal? This paper describes and evaluates three general approaches to this "problem of structural diversity": structural enrichment, structural depletion, and multi-stage (...) aggregation. All three approaches have notable drawbacks, but I tentatively defend multi-stage aggregation as least bad of the three. (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.

The principle that rational agents should maximize expected utility or choiceworthiness is intuitively plausible in many ordinary cases of decision-making under uncertainty. But it is less plausible in cases of extreme, low-probability risk (like Pascal's Mugging), and intolerably paradoxical in cases like the St. Petersburg and Pasadena games. In this paper I show that, under certain conditions, stochastic dominance reasoning can capture most of the plausible implications of expectational reasoning while avoiding most of its pitfalls. Specifically, given sufficient background uncertainty (...) about the choiceworthiness of one's options, many expectation-maximizing gambles that do not stochastically dominate their alternatives "in a vacuum" become stochastically dominant in virtue of that background uncertainty. But, even under these conditions, stochastic dominance will not require agents to accept options whose expectational superiority depends on sufficiently small probabilities of extreme payoffs. The sort of background uncertainty on which these results depend looks unavoidable for any agent who measures the choiceworthiness of her options in part by the total amount of value in the resulting world. At least for such agents, then, stochastic dominance offers a plausible general principle of choice under uncertainty that can explain more of the apparent rational constraints on such choices than has previously been recognized. (shrink)

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)

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.

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)

Nudge is a popular public policy tool that harnesses well-known biases in human judgement to subtly guide people’s decisions, often to improve their choices or to achieve some socially desirable outcome. Thanks to recent developments in artificial intelligence (AI) methods new possibilities emerge of how and when our decisions can be nudged. On the one hand, algorithmically personalized nudges have the potential to vastly improve human daily lives. On the other hand, blindly outsourcing the development and implementation of nudges to (...) “black box” AI systems means that the ultimate reasons for why such nudges work, that is, the underlying human cognitive processes that they harness, will often be unknown. In this paper, we unpack this concern by considering a series of examples and case studies that demonstrate how AI systems can learn to harness biases in human judgment to reach a specified goal. Drawing on an analogy in a philosophical debate concerning the methodology of economics, we call for the need of an interdisciplinary oversight of AI systems that are tasked and deployed to nudge human behaviours. (shrink)

If I am aware that p, say, that it is raining, is it the case that I must be aware that I am aware that p? Does introspective or object-awareness entail the apprehension of mental states as being of some kind or another: self-monitoring or intentional? That is, are cognitive events implicitly self-aware or is “self-awareness” just another term for metacognition? Not surprisingly, intuitions on the matter vary widely. This paper proposes a novel solution to this classical debate by reframing (...) the problem of self-knowledge in terms of the relation between phenomenal concepts and phenomenal knowledge. Concepts of consciousness such as “introspective awareness” and “reflexive self-awareness” are grounded in phenomenal experiences rather than physical events and processes. As such they yield a different kind of self-knowledge than what can be gained by applying externalist conceptual schemas to understanding the mind. I argue that Dharmakīrti’s theory of content can be seen as endorsing the efficacy of phenomenal experience as a vehicle for self-knowledge. (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.

Today's digital revolution is a worldwide phenomenon, with profound and often differential implications for communities around the world and their relationships to one another. This book presents a new, explicitly international theory of media ethics, incorporating non-Western perspectives and drawing deeply on both moral philosophy and the philosophy of technology. Clifford Christians develops an ethics grounded in three principles - truth, human dignity, and non-violence - and shows how these principles can be applied across a wide range of cases and (...) domains. The book is a guide for media professionals, scholars, and educators who are concerned with the global ramifications of new technologies and with creating a more just world. (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.

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)

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)

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.

Many different methodological approaches have been proposed to infer the presence of consciousness in non-human systems. In this paper, a version of the theory-heavy approach is defended. Theory-heavy approaches rely heavily on considerations from theories of consciousness to make inferences about non-human consciousness. Recently, the theory-heavy approach has been critiqued in the form of Birch's (Noûs, 56(1): 133-153, 2022) dilemma of demandingness and Shevlin's (Mind & Language, 36(2): 297-314, 2021) specificity problem. However, both challenges implicitly assume an inapt characterization of (...) the theory-heavy approach. I argue that an alternative characterization of the approach, what I call a credence-based theory-heavy approach, avoids these challenges. Theorists can generate interpretations of their theory, at different levels of generality, and operationalize these into theory-informed markers. These theory-informed are assigned a likelihood and are used to assess the probability that a target system is conscious. In providing this characterization, and mapping out the possible ways in which a credence-based theory-heavy approach can be fleshed out, the aim is to situate the theory-heavy approach as a more compelling approach than it is currently being perceived as. Our attention, then, needs to towards remaining challenges such as the consensus problem and the problem of calibrating the likelihoods associated with theory-informed markers. I also explore methodological pluralism and assess how the credence-based theory-heavy approach can benefit from other methodological approaches. (shrink)

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.

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)

We study the monoid of primitive recursive functions and investigate a onestep construction of a kind of exact completion, which resembles that of the familiar category of modest sets, except that the partial equivalence relations which serve as objects are recursively enumerable. As usual, these constructions involve the splitting of symmetric idempotents.

This article considers the normative and critical value of popular comedy. I begin by assembling and evaluating a range of political theory literature on comedy. I argue that popular comedy can be conducive to both critical and transformative democratic effects, but that these effects are contingent on the way comedic performances are received by audiences. I illustrate this by means of a case study of a comedic climate change ‘debate’ from the television show, Last Week Tonight. Drawing from recent scholarship (...) on deliberation, judgment and rhetoric, I highlight both critical and transformative dimensions of the performance. I attribute these to the vignette’s likely reception, which I describe as ‘dissonant’ – unresolved, affectively turbulent and aesthetically attuned. I argue that comedy is uniquely positioned to spur such ‘dissonant’ modes of engagement and, in so doing, to promote acknowledgement and reflective judgment. (shrink)

This article considers the normative and critical value of popular comedy. I begin by assembling and evaluating a range of political theory literature on comedy. I argue that popular comedy can be conducive to both critical and transformative democratic effects, but that these effects are contingent on the way comedic performances are received by audiences. I illustrate this by means of a case study of a comedic climate change ‘debate’ from the television show, Last Week Tonight. Drawing from recent scholarship (...) on deliberation, judgment and rhetoric, I highlight both critical and transformative dimensions of the performance. I attribute these to the vignette’s likely reception, which I describe as ‘dissonant’ – unresolved, affectively turbulent and aesthetically attuned. I argue that comedy is uniquely positioned to spur such ‘dissonant’ modes of engagement and, in so doing, to promote acknowledgement and reflective judgment. (shrink)