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    Wyświetlanie 1-6 z 6
    Tytuł:
    Abstract logical structuralism
    Autorzy:
    Marquis, Jean-Pierre
    Powiązania:
    https://bibliotekanauki.pl/articles/1047597.pdf
    Data publikacji:
    2020-12-29
    Wydawca:
    Copernicus Center Press
    Tematy:
    philosophy
    logic
    structuralism
    categorical logic
    Opis:
    Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latter can be seen—and probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood.
    Źródło:
    Zagadnienia Filozoficzne w Nauce; 2020, 69; 67-110
    0867-8286
    2451-0602
    Pojawia się w:
    Zagadnienia Filozoficzne w Nauce
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The homunculus brain and categorical logic
    Autorzy:
    Awodey, Steve
    Heller, Michał
    Powiązania:
    https://bibliotekanauki.pl/articles/1047588.pdf
    Data publikacji:
    2020-12-29
    Wydawca:
    Copernicus Center Press
    Tematy:
    categorical logic
    syntax-semantics
    mind-brain
    Opis:
    The interaction between syntax (formal language) and its semantics (meanings of language) is one which has been well studied in categorical logic. The results of this particular study are employed to understand how the brain is able to create meanings. To emphasize the toy character of the proposed model, we prefer to speak of the homunculus brain rather than the brain per se. The homunculus brain consists of neurons, each of which is modeled by a category, and axons between neurons, which are modeled by functors between the corresponding neuron-categories. Each neuron (category) has its own program enabling its working, i.e. a theory of this neuron. In analogy to what is known from categorical logic, we postulate the existence of a pair of adjoint functors, called Lang and Syn, from a category, now called BRAIN, of categories, to a category, now called MIND, of theories. Our homunculus is a kind of “mathematical robot”, the neuronal architecture of which is not important. Its only aim is to provide us with the opportunity to study how such a simple brain-like structure could “create meanings” and perform abstraction operations out of its purely syntactic program. The pair of adjoint functors Lang and Syn model the mutual dependencies between the syntactical structure of a given theory of MIND and the internal logic of its semantics given by a category of BRAIN. In this way, a formal language (syntax) and its meanings (semantics) are interwoven with each other in a manner corresponding to the adjointness of the functors Lang and Syn. Higher cognitive functions of abstraction and realization of concepts are also modelled by a corresponding pair of adjoint functors. The categories BRAIN and MIND interact with each other with their entire structures and, at the same time, these very structures are shaped by this interaction.
    Źródło:
    Zagadnienia Filozoficzne w Nauce; 2020, 69; 253-280
    0867-8286
    2451-0602
    Pojawia się w:
    Zagadnienia Filozoficzne w Nauce
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Teoria kategorii i niektóre jej logiczne aspekty
    Category theory and some of its logical aspects
    Autorzy:
    Stopa, Mariusz
    Powiązania:
    https://bibliotekanauki.pl/articles/690940.pdf
    Data publikacji:
    2018
    Wydawca:
    Copernicus Center Press
    Tematy:
    category theory
    topos theory
    categorical logic
    propositional logic
    intuitionistic logic
    non-classical logic
    Opis:
    This article is intended for philosophers and logicians as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tautologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.
    Źródło:
    Zagadnienia Filozoficzne w Nauce; 2018, 64; 7-58
    0867-8286
    2451-0602
    Pojawia się w:
    Zagadnienia Filozoficzne w Nauce
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Syntax-Semantics Interaction in Mathematics
    Autorzy:
    Heller, Michael
    Powiązania:
    https://bibliotekanauki.pl/articles/561342.pdf
    Data publikacji:
    2018
    Wydawca:
    Polskie Towarzystwo Semiotyczne
    Tematy:
    philosophy of mathematics
    categorical logic
    syntax-semantic interaction
    Bell’s program
    Gödel-like limitations
    Opis:
    Mathematical tools of category theory are employed to study the syntaxsemantics problem in the philosophy of mathematics. Every category has its internal logic, and if this logic is sufficiently rich, a given category provides semantics for a certain formal theory and, vice versa, for each (suitably defined) formal theory one can construct a category, providing a semantics for it. There exists a pair of adjoint functors, Lang and Syn, between a category (belonging to a certain class of categories) and a category of theories. These functors describe, in a formal way, mutual dependencies between the syntactical structure of a formal theory and the internal logic of its semantics. Bell’s program to regard the world of topoi as the univers de discours of mathematics and as a tool of its local interpretation, is extended to a collection of categories and all functors between them, called “categorical field”. This informal idea serves to study the interaction between syntax and semantics of mathematical theories, in an analogy to functors Lang and Syn. With the help of these concepts, the role of Gödel-like limitations in the categorical field is briefly discussed. Some suggestions are made concerning the syntax-semantics interaction as far as physical theories are concerned.
    Źródło:
    Studia Semiotyczne; 2018, 32, 2; 87-105
    0137-6608
    Pojawia się w:
    Studia Semiotyczne
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On the validity of the definition of a complement-classifier
    Autorzy:
    Stopa, Mariusz
    Powiązania:
    https://bibliotekanauki.pl/articles/1047622.pdf
    Data publikacji:
    2020-12-29
    Wydawca:
    Copernicus Center Press
    Tematy:
    category theory
    topos theory
    categorical logic
    Heyting algebras
    co-Heyting algebras
    intuitionistic logic
    dual to intuitionistic logic
    complement-classifier
    Opis:
    It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier (and thus of a co-topos as well) is, at least in general and within the conceptual framework of category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier (and thus of a co-topos as well).
    Źródło:
    Zagadnienia Filozoficzne w Nauce; 2020, 69; 111-128
    0867-8286
    2451-0602
    Pojawia się w:
    Zagadnienia Filozoficzne w Nauce
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    La linguistique peut-elle sortir de son état pré-galiléen ?
    How can linguistics get out of its pre-Galilean state?
    Autorzy:
    Desclés, Jean-Pierre
    Powiązania:
    https://bibliotekanauki.pl/articles/31341185.pdf
    Data publikacji:
    2021-11-10
    Wydawca:
    Wydawnictwo Uniwersytetu Śląskiego
    Tematy:
    Operators
    Categorical grammars
    Tenses, aspects and modalities
    Mathematization of linguistic concepts
    Types of Church
    Combinatorial logic
    Opis:
    The future of linguistics implies a better definition of concepts, especially in the semantic analysis. The notion of operator plays an important role in several areas of linguistics, for instance categorical grammars and representations of the meanings of grammatical categories. The general topology makes it possible to mathematize the grammatical concepts (time, aspects, modalities, enunciative operations) by means of operators. Curry’s Combinatorial Logic is an adequate formalism for composing and transforming operators at different levels of analysis that connect the semiotic expressions of languages (the observables) with their semantico-cognitive interpretations. The article refers to many studies that develop the points discussed.
    Źródło:
    Neophilologica; 2021, 33; 1-23
    0208-5550
    2353-088X
    Pojawia się w:
    Neophilologica
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-6 z 6

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