Almost structural completeness; an algebraic approach

A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras

Paraconsistency properties in degree-preserving fuzzy logics

Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature. © 2014, Springer-Verlag Berlin Heidelberg.All the authors have been partially supported by the FP7 PIRSES-GA-2009-247584 project MaToMUVI. Besides, Ertola was supported by FAPESP LOGCONS Project, Esteva and Godo were supported by the Spanish project TIN2012-39348-C02-01, Flaminio was supported by the Italian project FIRB 2010 (RBFR10DGUA_02) and Noguera was suported by the grant P202/10/1826 of the Czech Science Foundation.Peer reviewe

On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics

The aim of this article is to explore the class of intermediate logics between the truth-preserving Łukasiewicz logic Ł and its degree-preserving companion Ł≤. From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in Ł≤ and derivable in L and we characterize the corresponding intermediate logics. From a semantical point of view, we first consider the family of logics characterized by matrices defined by lattice filters in [0,1], but we show there are intermediate logics falling outside this family. Finally, we study the case of finite-valued Lukasiewicz logics where we axiomatize a large family of intermediate logics defined by families of matrices (A,F) such that A is a finite MV-algebra and F is a lattice filter.The authors have been partially supported by the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Coniglio was also supported by FAPESP (Thematic Project LogCons 2010/51038-0), and by a research grant from CNPq (PQ 308524/2014-4). Esteva and Godo also acknowledge partial support by the MINECO project TIN2012-39348-C02-01.Peer Reviewe

On modal expansions of t-norm based logics with rational constants

[eng] According to Zadeh, the term “fuzzy logic” has two different meanings: wide and narrow. In a narrow sense it is a logical system which aims a formalization of approximate reasoning, and so it can be considered an extension of many-valued logic. However, Zadeh also says that the agenda of fuzzy logic is quite different from that of traditional many-valued logic, as it addresses concepts like linguistic variable, fuzzy if-then rule, linguistic quantifiers etc. Hájek, in the preface of his foundational book Metamathematics of Fuzzy Logic, agrees with Zadeh’s distinction, but stressing that formal calculi of many-valued logics are the kernel of the so-called Basic Fuzzy logic (BL), having continuous triangular norms (t-norm) and their residua as semantics for the conjunction and implication respectively, and of its most prominent extensions, namely Lukasiewicz, Gödel and Product fuzzy logics. Taking advantage of the fact that a t-norm has residuum if, and only if, it is left-continuous, the logic of the left-continuous t-norms, called MTL, was soon after introduced. On the other hand, classical modal logic is an active field of mathematical logic, originally introduced at the beginning of the XXth century for philosophical purposes, that more recently has shown to be very successful in many other areas, specially in computer science. That are the most well-known semantics for classical modal logics. Modal expansions of non-classical logics, in particular of many-valued logics, have also been studied in the literature. In this thesis we focus on the study of some modal logics over MTL, using natural generalizations of the classical Kripke relational structures where propositions at possible words can be many-valued, but keeping classical accessibility relations. In more detail, the main goal of this thesis has been to study modal expansions of the logic of a left-continuous t-norm, defined over the language of MTL expanded with rational truth-constants and the Monteiro-Baaz Delta-operator, whose intended (standard) semantics is given by Kripke models with crisp accessibility relations and taking the unit real interval [0, 1] as set of truth-values. To get complete axiomatizations, already known techniques based on the canonical model construction are uses, but this requires to ensure that the underlying (propositional) fuzzy logic is strongly standard complete. This constraint leads us to consider axiomatic systems with infinitary inference rules, already at the propositional level. A second goal of the thesis has been to also develop and automated reasoning software tool to solve satisfiability and logical consequence problems for some of the fuzzy logic modal logics considered. This dissertation is structured in four parts. After a gentle introduction, Part I contains the needed preliminaries for the thesis be as self-contained as possible. Most of the theoretical results are developed in Parts II and III. Part II focuses on solving some problems concerning the strong standard completeness of underlying non-modal expansions. We first present and axiomatic system for the non-nodal propositional logic of a left-continuous t-norm who makes use of a unique infinitary inference rule, the “density rule”, that solves several problems pointed out in the literature. We further expand this axiomatic system in order to also characterize arbitrary operations over [0, 1] satisfying certain regularity conditions. However, since this axiomatic system turn out to be not well-behaved for the modal expansion, we search for alternative axiomatizations with some particular kind of inference rules (that will be called conjunctive). Unfortunately, this kind of axiomatization does not necessarily exist for all left-continuous t-norms (in particular, it does not exist for the Gödel logic case), but we identify a wide class of t-norms for which it works. This “well-behaved” t-norms include all ordinal sums of Lukasiewiczand Product t-norms. Part III focuses on the modal expansion of the logics presented before. We propose axiomatic systems (which are, as expected, modal expansions of the ones given in the previous part) respectively strongly complete with respect to local and global Kripke semantics defined over frames with crisp accessibility relations and worlds evaluated over a “well-behaved” left-continuous t-norm. We also study some properties and extensions of these logics and also show how to use it for axiomatizing the possibilistic logic over the very same t-norm. Later on, we characterize the algebraic companion of these modal logics, provide some algebraic completeness results and study the relation between their Kripke and algebraic semantics. Finally, Part IV of the thesis is devoted to a software application, mNiB-LoS, who uses Satisfability Modulo Theories in order to build an automated reasoning system to reason over modal logics evaluated over BL algebras. The acronym of this applications stands for a modal Nice BL-logics Solver. The use of BL logics along this part is motivated by the fact that continuous t-norms can be represented as ordinal sums of three particular t-norms: Gödel, Lukasiewicz and Product ones. It is then possible to show that these t-norms have alternative characterizations that, although equivalent from the point of view of the logic, have strong differences for what concerns the design, implementation and efficiency of the application. For practical reasons, the modal structures included in the solver are limited to the finite ones (with no bound on the cardinality)

Logics of formal inconsistency

Orientadores: Walter Alexandre Carnielli, Carlos M. C. L. CaleiroTexto em ingles e portuguesTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias HumanasTese (doutorado) - Universidade Tecnica de Lisboa, Instituto Superior TecnicoResumo: Segundo a pressuposição de consistência clássica, as contradições têm um cará[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocínio sensato pode então ter lugar. Uma lógica é paraconsistente se ela rejeita uma tal pressuposição, e aceita ao invés que algumas teorias inconsistentes conquanto não-triviais façam perfeito sentido. A? Lógicas da Inconsistência Formal, LIFs, formam uma classe de lógicas paraconsistentes particularmente expressivas nas quais a noção meta-teónca de consistência pode ser internalizada ao nível da linguagem obje[c]to. Como consequência, as LIFs são capazes de recapturar o raciocínio consistente pelo acréscimo de assunções de consistência apropriadas. Assim, por exemplo, enquanto regras clássicas tais como o silogismo disjuntivo (de A e {não-,4)-ou-13, infira B) estão fadadas a falhar numa lógica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presunção de que estamos raciocinando em um ambiente consistente (neste caso, pelo acréscimo de (consistente-.A) como uma hipótese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustrações destas lógicas e de suas propriedades, mostrando que tais lógicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocínio consistente dentro de tais sistemas inconsistentes são também ilustradas Em cada caso, interpretações em termos de semânticas polivalentes, de traduções possíveis ou modais são fornecidas, e os problemas relacionados à provisão de contrapartidas algébricas para tais lógicas são examinados. Uma abordagem formal abstra[cjta é proposta para todas as definições relacionadas e uma extensa investigação é feita sobre os princípios lógicos e as propriedades positivas e negativas da negação.Abstract: According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and (not-A)-or-B, infer B) are bound to fail in a paraconsistent logic (because A and (not-.4) could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of (consistent-/!) as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature. Several ways of performing the recapture of consistent reasoning inside such inconsistent systems are also illustrated. In each case, interpretations in terms of many-valued, possible-translations, or modal semantics are provided, and the problems related to providing algebraic counterparts to such logics are surveyed. A formal abstract approach is proposed to all related definitions and an extended investigation is carried out into the logical principles and the positive and negative properties of negation.DoutoradoFilosofiaDoutor em Filosofia e Matemátic

Dualities in modal logic

Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics