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Wyświetlanie 1-10 z 12
Tytuł:
Algebraic Characterization of the Local Craig Interpolation Property
Autorzy:
Gyenis, Zalán
Powiązania:
https://bibliotekanauki.pl/articles/749892.pdf
Data publikacji:
2018
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
Craig interpolation
Algebraic logic
Superamalgamation
Opis:
The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka– Németi and Sain. 
Źródło:
Bulletin of the Section of Logic; 2018, 47, 1
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The Modelwise Interpolation Property of Semantic Logics
Autorzy:
Gyenis, Zalán
Molnár, Zalán
Öztürk, Övge
Powiązania:
https://bibliotekanauki.pl/articles/43179744.pdf
Data publikacji:
2023
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
interpolation
algebraic logic
amalgamation
superamalgamation
Opis:
In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi\) formulated in the intersection of the vocabularies of \(\phi\) and \(\psi\), such that \(\mathfrak{M}\models\phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic.
Źródło:
Bulletin of the Section of Logic; 2023, 52, 1; 59-83
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Weak products of universal algebras
Autorzy:
Sain, Ildikó
Powiązania:
https://bibliotekanauki.pl/articles/1361096.pdf
Data publikacji:
1993
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
universal algebra
algebraic logic
cylindric algebras
Opis:
Weak direct products of arbitrary universal algebras are introduced. The usual notion for groups and rings is a special case. Some universal algebraic properties are proved and applications to cylindric and polyadic algebras are considered.
Źródło:
Banach Center Publications; 1993, 28, 1; 311-318
0137-6934
Pojawia się w:
Banach Center Publications
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Lifting Results for Finite Dimensions to the Transfinite in Systems of Varieties Using Ultraproducts
Autorzy:
Sayed Ahmed, Tarek
Powiązania:
https://bibliotekanauki.pl/articles/43189294.pdf
Data publikacji:
2024
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
algebraic logic
systems of varieties
ultraproducts
non-finite axiomaitizability
Opis:
We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let \(\bf Ord\) denote the class of all ordinals. Let \(\langle \mathbf{K}_{\alpha}: \alpha\in \bf Ord\rangle\) be a system of varieties definable by a schema. Given any ordinal \(\alpha\), we define an operator \(\mathsf{Nr}_{\alpha}\) that acts on \(\mathbf{K}_{\beta}\) for any \(\beta>\alpha\) giving an algebra in \(\mathbf{K}_{\alpha}\), as an abstraction of taking \(\alpha\)-neat reducts for cylindric algebras. We show that for any positive \(k\), and any infinite ordinal \(\alpha\) that \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k+1}\) cannot be axiomatized by a finite schema over \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k}\) given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of \(L_{\omega,\omega}\).
Źródło:
Bulletin of the Section of Logic; 2024, 53, 2; 145-154
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results
Autorzy:
Sayed Ahmed, Tarek
Powiązania:
https://bibliotekanauki.pl/articles/2033851.pdf
Data publikacji:
2021-07-21
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
Algebraic logic
relation algebras
cylindric algebras
polyadic algebras
complete representations
Opis:
Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n<\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.
Źródło:
Bulletin of the Section of Logic; 2021, 50, 4; 465-511
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the Leibniz congruences
Autorzy:
Font, Josep
Powiązania:
https://bibliotekanauki.pl/articles/1361077.pdf
Data publikacji:
1993
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
deductive system
protoalgebraic logic
Gentzen calculus
closure operator
abstract logic
algebraizable logic
Leibniz congruence
selfextensional logic
logical matrices
algebraic logic
Opis:
The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability relation modulo the theory). For protoalgebraic deductive systems the class of algebras associated in general turns out to be the class of algebra reducts of reduced matrices, which is the algebraic counterpart usually considered for this large class of deductive systems; but in the general case the new class of algebras shows a better behaviour.
Źródło:
Banach Center Publications; 1993, 28, 1; 17-36
0137-6934
Pojawia się w:
Banach Center Publications
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On an episode in academic contacts of Jacek Hawranek and Jan Zygmunt with Professor Bogusław Wolniewicz
O pewnym epizodzie w kontaktach naukowych Jacka Hawranka i Jana Zygmunta z Profesorem Bogusławem Wolniewiczem
Autorzy:
Zygmunt, Jan
Powiązania:
https://bibliotekanauki.pl/articles/2097360.pdf
Data publikacji:
2018
Wydawca:
Polska Akademia Nauk. Czytelnia Czasopism PAN
Tematy:
semilattice
formal ontology of situations
algebraic logic
history of Polish logic
Bogusław Wolniewicz
półkrata
formalna ontologia sytuacji
logika algebraiczna
historia logiki polskiej
Opis:
W eseju przedstawione zostały wybrane kontakty naukowe Jacka Hawranka i Jana Zygmunta z Profesorem Bogusławem Wolniewiczem w okresie od końca lat osiemdziesiątych XX w. do początku XXI w. Kontakty dotyczyły algebraicznych aspektów ontologii sytuacji, a od pewnego momentu – jednego tylko pytania sformułowanego w nocie A question about join-semilattices (Wolniewicz 1990). Esej streszcza dyskusję naukową między B. Wolniewiczem a J. Hawrankiem i J. Zygmuntem, w rezultacie której powstał artykuł Wokół pewnego zagadnienia z dziedziny półkrat górnych z jednością (Hawranek, Zygmunt 1993), zawierający próbę odpowiedzi na pytanie Wolniewicza. Artykuł Hawranka i Zygmunta jest niżej przedrukowany, a niniejszy esej jest też pomyślany jako wstęp historyczno-analityczny do jego lektury. Historia kontaktów: Wolniewicz – Hawranek & Zygmunt została ukazana za pomocą zachowanej korespondencji, która jest dość obficie cytowana. W listach Profesor Wolniewicz jawi się jako badacz-pasjonat, otwarty na dyskusję, gotowy do dzielenia się z innymi swoimi trudnościami i sukcesami badawczymi.
Źródło:
Przegląd Filozoficzny. Nowa Seria; 2018, 3; 149-162
1230-1493
Pojawia się w:
Przegląd Filozoficzny. Nowa Seria
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Nilpotent Minimum Logic NM and Pretabularity
Autorzy:
Yang, Eunsuk
Powiązania:
https://bibliotekanauki.pl/articles/750006.pdf
Data publikacji:
2020
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
pretabularity
nilpotent minimum logic
algebraic semantics
fuzzy logic
finite model property
Opis:
This paper deals with pretabularity of fuzzy logics. For this, we first introduce two systems NMnfp and NM½, which are expansions of the fuzzy system NM (Nilpotent minimum logic), and examine the relationships between NMnfp and the another known extended system NM-. Next, we show that NMnfp and NM½ are pretabular, whereas NM is not. We also discuss their algebraic completeness.  
Źródło:
Bulletin of the Section of Logic; 2020, 49, 1
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Variable Sharing in Substructural Logics: an Algebraic Characterization
Autorzy:
Badia, Guillermo
Powiązania:
https://bibliotekanauki.pl/articles/750042.pdf
Data publikacji:
2018
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
relevant logic
algebraic characterizations of logical properties
variable sharing property
substructural logics
Opis:
We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.
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 Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions
Autorzy:
Cornejo, Juan Manuel
Sankappanavar, Hanamantagouda P.
Powiązania:
https://bibliotekanauki.pl/articles/43189647.pdf
Data publikacji:
2022
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
semi-intuitionistic logic
dually hemimorphic semi-Heyting logic
dually quasi-De Morgan semi-Heyting logic
De Morgan semi-Heyting logic
dually pseudocomplemented semi-Heyting logic
regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1
implicative logic
equivalent algebraic semantics
algebraizable logic
De Morgan Gödel logic
dually pseudocomplemented Gödel logic
Moisil's logic
3-valued Łukasiewicz logic
Opis:
The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
Źródło:
Bulletin of the Section of Logic; 2022, 51, 4; 555-645
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-10 z 12

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