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    Wyświetlanie 1-8 z 8
    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:
    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:
    Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term π-Institutions
    Autorzy:
    Voutsadakis, George
    Powiązania:
    https://bibliotekanauki.pl/articles/749924.pdf
    Data publikacji:
    2016
    Wydawca:
    Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
    Tematy:
    Behavioral Equivalence
    Hidden Logic
    Multi-Sorted Logic
    Multi-term π-Institutions
    Interpretability
    Deductive Equivalence
    Opis:
    Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the π-institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-F´erez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-F´erez pertaining to arbitrary multi-term π-institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools.
    Źródło:
    Bulletin of the Section of Logic; 2016, 45, 2
    0138-0680
    2449-836X
    Pojawia się w:
    Bulletin of the Section of Logic
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Categorical Abstract Algebraic Logic: Coordinatization is Algebraization
    Autorzy:
    Voutsadakis, George
    Powiązania:
    https://bibliotekanauki.pl/articles/1368600.pdf
    Data publikacji:
    2012
    Wydawca:
    Uniwersytet Jagielloński. Wydawnictwo Uniwersytetu Jagiellońskiego
    Opis:
    The methods of categorical abstract algebraic logic are employed to show that the classical process of the coordinatization of abstract (affine plane) geometry can be viewed under the light of the algebraization of logical systems. This link offers, on the one hand, a new perspective to the coordinatization of geometry and, on the other, enriches abstract algebraic logic by bringing under its wings a very well-known geometric process, not known hitherto to be related or amenable to its methods and techniques. The algebraization takes the form of a deductive equivalence between two institutions, one corresponding to affine plane geometry and the other to Hall ternary rings.
    Źródło:
    Reports on Mathematical Logic; 2012, 47; 125-145
    0137-2904
    2084-2589
    Pojawia się w:
    Reports on Mathematical Logic
    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:
    Categorical Abstract Algebraic Logic: Pseudo-Referential Matrix System Semantics
    Autorzy:
    Voutsadakis, George
    Powiązania:
    https://bibliotekanauki.pl/articles/750034.pdf
    Data publikacji:
    2018
    Wydawca:
    Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
    Tematy:
    Referential Logics
    Selfextensional Logics
    Referential Semantics
    Referential π-institutions
    Selfextensional π-institutions
    Pseudo- Referential Semantics
    Discrete Referential Semantics
    Opis:
    This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics.
    Źródło:
    Bulletin of the Section of Logic; 2018, 47, 2
    0138-0680
    2449-836X
    Pojawia się w:
    Bulletin of the Section of Logic
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A categorical model of predicate linear logic
    Autorzy:
    Demeterová, E.
    Mihályi, D.
    Novitzká, V.
    Powiązania:
    https://bibliotekanauki.pl/articles/122570.pdf
    Data publikacji:
    2015
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    linear type theory
    predicate linear logic
    symmetric monoidal closed category
    Opis:
    Linear logic is one of the logical systems with special properties suitable for describing real processes used in computer science. It enables one to specify dynamics, non determinism, consecutive processes and important resources as memory and time on syntactic level. Moreover, its deduction system enables one to verify specified properties. Constructing an appropriate model based on categories can serve for modeling various program systems in the wide spectrum of computer science. Mainly, propositional linear logic is used for these purposes. The expression power of linear logic significantly grows by extending propositional logic with predicates and quantifiers. Our paper concerns itself with defining predicate linear logic together with its deduction system and our main aim is to construct a categorical model of predicate linear logic as a symmetric monoidal closed category.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2015, 14, 1; 27-42
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-8 z 8

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