21 research outputs found
Categorical Abstract Algebraic Logic: Equivalential Ï-Institutions
The theory of equivalential deductive systems, as introduced by Prucnal and WroĆski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as Ï-Institutions. More precisely, the notion of an N-equivalence system for a given Ï-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a âtransfer theoremâ for N-equivalence systems is proven. For a Ï-Institutions I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmannâs Test, characterizing those N-protoalgebraic Ï-Institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of Ï-Institutions, as they relate to the existence of N-equivalence systems
Categorical Abstract Algebraic Logic: Equivalential Ï-Institutions
The theory of equivalential deductive systems, as introduced by Prucnal and WroĆski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as Ï-Institutions. More precisely, the notion of an N-equivalence system for a given Ï-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a âtransfer theoremâ for N-equivalence systems is proven. For a Ï-Institutions I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmannâs Test, characterizing those N-protoalgebraic Ï-Institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of Ï-Institutions, as they relate to the existence of N-equivalence systems
Order algebraizable logics
AbstractThis paper develops an order-theoretic generalization of Blok and PigozziÊŒs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)â(iv)
Admissible equivalence systems
Whenever a logic is the set of theorems of some deductive system, where the latter
has an equivalence system, the behavioral theorems of the logic can be determined
by means of that equivalence system. In general, this original equivalence system
may be too restrictive, because it su ces to check behavioral theorems by means
of any admissible equivalence system (that is an equivalence system of the small-
est deductive system associated with the given logic). In this paper, we present a
range of examples, which show that: 1) there is an admissible equivalence system
which is not an equivalence system for the initial deductive system, 2) there is
a non- nitely equivalential deductive system with a nite admissible equivalence
system, and 3) there is a deductive system with an admissible equivalence sys-
tems, such that this deductive system is not even protoalgebraic itself. We use
methods and results from algebraic and modal logic.FCT via UIMAFCT via KLog projec
Leibniz hierarchy
Mestrado em MatemåticaA Lógica Algébrica Abstracta estuda o processo pelo qual uma classe de ålgebras pode ser associada a uma lógica. Nesta dissertação, analisamos este processo agrupando lógicas partilhando certas propriedades em classes. O
conceito central neste estudo Ă© a congruĂȘncia de Leibniz que assume o papel desempenhado pela equivalĂȘncia no processo tradicional de Lindenbaum- Tarski.
Apresentamos uma hierarquia entre essas classes que Ă© designada por hierarquia de Leibniz, caracterizando as lĂłgicas de cada classe por
propriedades meta-lĂłgicas, por exemplo propriedades do operador de Leibniz.
Estudamos tambĂ©m a recente abordagem comportamental que usa lĂłgicas multigĂ©nero, lĂłgica equacional comportamental e, consequentemente, uma versĂŁo comportamental do operador de Leibniz. Neste contexto, apresentamos alguns exemplos, aos quais aplicamos esta nova teoria, capturando alguns fenĂłmenos de algebrização que nĂŁo era possĂvel formalizar com a abordagem
standard.
ABSTRACT: Abstract Algebraic logic studies the process by which a class of algebras can be associated with a logic. In this dissertation, we analyse this process by grouping logics sharing certain properties into classes. The central concept in this study is the Leibniz Congruence that assumes the role developed by the equivalence in the traditional Lindenbaum-Tarski process.
We show a hierarchy between these classes, designated by Leibniz hierarchy, by characterizing logics in each class by meta-logical properties, for example properties of the Leibniz operator.
We also study a recent behavioral approach which uses many-sorted logics, behavioral equational logic and, consequently, a behavioral version of the
Leibniz operator. In this context, we provide some examples, to which we apply this new theory, capturing some phenomena of algebraization that are not possible to formalize using the standard approach
Admissible rules and the Leibniz hierarchy
This paper provides a semantic analysis of admissible rules
and associated completeness conditions for arbitrary deductive systems,
using the framework of abstract algebraic logic. Algebraizability is not
assumed, so the meaning and signi cance of the principal notions vary
with the level of the Leibniz hierarchy at which they are presented. As
a case study of the resulting theory, the non-algebraizable fragments of
relevance logic are considered.This work is based on research supported in part by
the National Research Foundation of South Africa (UID 85407).https://www.dukeupress.edu/notre-dame-journal-of-formal-logichb2016Mathematics and Applied Mathematic
Logics of left variable inclusion and PĆonka sums of matrices
The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic âą. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic âą is related to the construction of PĆonka sums of the matrix models of âą. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy
Epimorphisms, definability and cardinalities
We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m+s+â”0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable âbridge theoremsâ, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic âą with suitable infinitary definability properties of âą, while not making the standard but awkward assumption that âą comes furnished with a proper class of variables.The European Unionâs Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project âSyntax Meets Semantics: Methods, Interactions, and Connections in Substructural logicsâ). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAÄR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.http://link.springer.com/journal/112252020-02-07hj2019Mathematics and Applied Mathematic