Mathematical Logic Quarterly最新文献

The variety of weak Heyting algebras was introduced in 2005 by Celani and Jansana. This corresponds to the strict implication fragment of the normal modal logic $�K$ which is also known as the subintuitionistic local consequence of the class of all Kripke models. Subresiduated lattices are a generalization of Heyting algebras and particular cases of weak Heyting algebras. They were introduced during the 1970s by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. In this paper we study the class of implicative-infimum subreducts of weak Heyting algebras. In particular, we prove that this class is a variety by giving an equational base for it. We also present a topological duality for the algebraic category whose objects are the implicative-infimum subreducts of subresiduated lattices.

The Ramsey Choice principle for families of $�n$-element sets, denoted $�{RC}_n$, states that every infinite set $�X$ has an infinite subset $��Y�\subseteq �X�$ with a choice function on $��{�\left[�Y�\right]�}^n�:�=��\{�z�\subseteq �Y�:�|�z�|�=�n�\}��$. We investigate for which positive integers $�m$ and $�n$ the implication $��{RC}_m�\Rightarrow �{RC}_n�$ is provable in $�\mathrm{ZF}$

Let $�A$ be a countable structure such that each finite partial isomorphism of it can be extended to an automorphism. Evans asked if the age (set of finite substructures) of $�A$ satisfies Hrushovski's extension property, then is it true that the automorphism group $���Aut��\left(�A�\right)�$ of $�A$ contains a dense, locally finite subgroup? In order to investigate this question, in the previous decades a coherent variant of Hrushovski's extension property has been introduced and studied. Among other results, we provide equivalent conditions for the existence of a dense, locally finite subgroup of $���Aut��\left(�A�\right)�$ in terms of a (new) variant of the coherent extension property. We also compare our notion with other coherent extension properties.

Läuchli showed in the absence of the Axiom of Choice ($�\mathrm{AC}$) that $��{�\left(�2^{�f�i�n�\left(�m�\right)�}�\right)�}^{{\aleph }_0}�=�2^{�f�i�n�\left(�m�\right)�}�$ and, consequently, $��2^{2^m}�+�2^{2^m}�=�2^{2^m}�$ for all infinite cardinals $�m$, where $��f�i�n�\left(�m�\right)�$

Assessors:

Hanne Andersen, Copenhagen, Denmark. Rachel Ankeny, Adelaide, Australia. Valeria de Paiva, Mountain View CA, U.S.A. Gerhard Heinzmann, Nancy, France. Concha Martinez Vidal, Santiago de Compostela, Spain. Tomáš Marvan, Prague, Czech Republic. Dhruv Raina, Delhi, India. Cheng Sumei, Shanghai, China. Alasdair Urquhart, Toronto, Canada. Andrés Villaveces, Bogotá, Colombia.

Former Presidents: († = deceased)

Menachem Magidor (Israel), Elliott Sober (United States of America), Wilfrid Hodges (United Kingdom), Adolf Grünbaum† (United States of America), Michael Rabin (United States of America), Wesley Salmon† (United States of America), Jens-Erik Fenstad† (Norway), Lawrence J. Cohen† (United Kingdom), Dana S. Scott (United States of America), Jerzy Łoś† (Poland). Patrick Suppes† (United States of America), Jaakko Hintikka† (Finland & United States of America), Andrzej Mostowski† (Poland), Stephan Körner† (United Kingdom), Yehoshua Bar-Hillel† (Israel), Georg Henrik von Wright† (Finland), Stephen C. Kleene† (United States of America).

Former president Jens Erik Fenstad passed away on 14 April 2020.

The Executive Committee of the Division is composed of the President, the Vice-Presidents, the Secretary-General, the Treasurer, and the immediate Past President. The Council consists of the Executive Committee plus the Assessors.

Ordinary Members Present (number of votes and names of delegates in parentheses; total votes: 68 before Agenda item 5; 69 after Agenda item 5). Argentina (2; 1 after Agenda item 5; Víctor Rodríguez), Australia (3; Pamela Robinson), Austria (1; Georg Schiemer), Belgium (1; Peter Verdée), Brazil (2; Elaine Pimentel), Canada (3; Zeyad El Nabolsy), P. R. China (3; Chen Bo), Czech Republic (2; Libor Behounek, Robin Kopecký), Denmark (2; Magdalena Malecka), Estonia (1; Peeter Müürsepp), Finland (2; Uskali Müki), France (4; Andrew Arana, Paola Cantu, Karine Chemla, Gerhard Heinzmann), Germany (4; Benedikt Löwe), Iran (1; Benedikt Löwe), Italy (4; Daniele Molinini), Japan (4; Mitsuhiro Okada), Republic of Korea (2; Insok Ko), Mexico (2; Atocha Aliseda, Ambrosio Velasco-Gómez), The Netherlands (2; Benedikt Löwe), Poland (2; Piotr Błaszczyk), Romania (1; Sorin Costreie), Russian Federation (3; Ilya Kasavin, Andrei Rodin, Lada Shipovalova), South Africa (1; Sean Muller), Spain (2; José A. Díez Calzada), Sweden (3; Valentin Goranko), Taiwan (2; Husan-Chih Lin, Kok-Yong Lee), United Kingdom (4; Robin Hendry), and United States of America (5; Hasok Chang, Philip Kircher, Helen Longino). After Agenda item 5, the number of votes of Argentina was reduced to 1 and the new Ordinary Member Portugal (2; João Luis Cordovil, Joan Bertran-San-Millán) was admitted; therefore, the number of votes of ordinary members increased by one.

Ordinary Members Absent. Eire (cf. Agenda item 5), Hungary (1), India (1), Israel (1), Moldov

We apply the mereological concept of parthood to the coding of finite sequences. We propose a first-order theory in which coding finite sequences is intuitive and transparent. We compare this theory with Robinson arithmetic, adjunctive set theory and weak theories of finite strings and finite trees using interpretability.

We provide two proofs of the compactness theorem for extensions of first-order logic based on team semantics. First, we build upon Lück's [16] ultraproduct construction for team semantics and prove a suitable version of Łoś' Theorem. Second, we show that by working with suitably saturated models, we can generalize the proof of Kontinen and Yang [13] to sets of formulas with arbitrarily many variables.

Starting from the generalized continuum hypothesis ($�\mathrm{GCH}$), we build a cardinal and $�\mathrm{GCH}$ preserving generic extension of the universe, in which there exists a set $��A�\subseteq �{\omega }_2�$ of size $�{\aleph }_2$ so that every countably infinite subset of $�A$ or $��{\omega }_2�\setminus �A�$ is Cohen generic over the ground model.

We study the 2-categories BIon, of (generalized) bounded ionads, and $�{\text{Acc}}_{\omega }$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex) geometric sketches, which serve as categorical specifications of geometric theories. We provide reconstruction and completeness-like results. We relate abstract elementary classes to locally decidable topoi. We introduce the notion of categories of saturated objects and relate it to atomic topoi.