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Wyświetlanie 1-11 z 11
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
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 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:
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:
Categorical Abstract Algebraic Logic: Referential π-Institutions
Autorzy:
Voutsadakis, George
Powiązania:
https://bibliotekanauki.pl/articles/749990.pdf
Data publikacji:
2015
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
Referential Logics
Selfextensional Logics
Leibniz operator
Tarski operator
Suszko operator
π-institutions
Opis:
Wójcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wójcicki asserts that a logic has a referential semantics if and only if it is selfextensional. Referential semantics was subsequently studied in detail by Malinowski and the concept of selfextensionality has played, more recently, an important role in the field of abstract algebraic logic in connection with the operator approach to algebraizability. We introduce and review some of the basic definitions and results pertaining to the referential semantics of π-institutions, abstracting corresponding results from the realm of propositional logics.
Źródło:
Bulletin of the Section of Logic; 2015, 44, 1-2
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
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:
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
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:
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:
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:
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
    Wyświetlanie 1-11 z 11

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