140,767 research outputs found

    Reasoning about Knowledge in Linear Logic: Modalities and Complexity

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    In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has no

    Modal Linear Logic in Higher Order Logic, an experiment in Coq

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    The sequent calculus of classical modal linear logic KDT 4lin is coded in the higher order logic using the proof assistant COQ. The encoding has been done using two-level meta reasoning in Coq. KDT 4lin has been encoded as an object logic by inductively defining the set of modal linear logic formulas, the sequent relation on lists of these formulas, and some lemmas to work with lists.This modal linear logic has been argued to be a good candidate for epistemic applications. As examples some epistemic problems have been coded and proven in our encoding in Coq::the problem of logical omniscience and an epistemic puzzle: ’King, three wise men and five hats’

    Ethical judgment and radical business changes: the role of entrepreneurial perspicacity

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    This study examines the implications of practical reason for entrepreneurial activities. Our study is based on Thomas Aquinas’ interpretation of such virtue, with a particular focus on the partition of practical reason in potential parts such as synesis, or common sense, and gnome, or perspicacity. Since entrepreneurial acts and actions deal with extremely uncertain situations, we argue that only this perspicacity, as the ability of correctly judging in exceptional cases, has the power to find wisdom under such blurred conditions. Perspicacity frees entrepreneurs from their cognitive schemata rendering them able to be truly entrepreneurial. Based on this vision and thanks to a semantic analysis of the meaning of the Greek word gnome, we construct an interpretative model for entrepreneurial judgment composed of three dimensions, specifically, knowledge-cognitive, external-affective and personal-reflective. The model highlights how a ‘successful’ entrepreneurial judgment is also such from a holistic point of view

    Putting Unity in Its Place: Organic Unity in Plato’s Phaedrus

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    Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus

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    The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambek's pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambek's previous type-logic, the Lambek calculus, which has been extensively used to formalise and reason about various linguistic phenomena. The apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi

    Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support

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    A framework and methodology---termed LogiKEy---for the design and engineering of ethical reasoners, normative theories and deontic logics is presented. The overall motivation is the development of suitable means for the control and governance of intelligent autonomous systems. LogiKEy's unifying formal framework is based on semantical embeddings of deontic logics, logic combinations and ethico-legal domain theories in expressive classic higher-order logic (HOL). This meta-logical approach enables the provision of powerful tool support in LogiKEy: off-the-shelf theorem provers and model finders for HOL are assisting the LogiKEy designer of ethical intelligent agents to flexibly experiment with underlying logics and their combinations, with ethico-legal domain theories, and with concrete examples---all at the same time. Continuous improvements of these off-the-shelf provers, without further ado, leverage the reasoning performance in LogiKEy. Case studies, in which the LogiKEy framework and methodology has been applied and tested, give evidence that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure

    A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras

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    Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, then instantiate the abstract setting to sets and relations and to finite dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics
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