Order algebraizable logics

AbstractThis paper develops an order-theoretic generalization of Blok and Pigozziʼs 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)–(iv)

Structural and universal completeness in algebra and logic

In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.Comment: This is a preprin

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+ℵ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⊢ with suitable infinitary definability properties of ⊢, while not making the standard but awkward assumption that ⊢ comes furnished with a proper class of variables.The European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAČR). 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

An algebraic study of residuated ordered monoids and logics without exchange and contraction.

Thesis (Ph.D.)-University of Natal, Durban, 1998.Please refer to the thesis for the abstract

On the logics of algebra.

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.We present and consider a number of logics that arise naturally from universal algebraic considerations, but which are ‘inherently unalgebraizable’ in the sense of [BP89a], essentially because they have no theo- rems. Of particular interest is the membership logic of a quasivariety, which is determined by its theorems, which are the relative congruence classes of the term algebra together with the empty-set in the case that the quasivariety is non-trivial. The membership logic arises by a more general technique developed in this text, for inducing deductive systems from closed systems on the free algebras of quasivarieties. In order to formalize this technique, we develop a theory of logics over constructs, where constructs are concrete categories. With this theory in place, we are able to view a closed system over an algebra as a logic, and in particular a structural logic, structural with respect to a suitable construct, typically the construct con- sisting of all algebras in a quasivariety and all algebra homomorphisms between these algebras. Of course, in such a case, none of these logics are generally sentential (i.e., structural and finitary deductive systems in the sense of [BP89a]), since the formulae of sentential logics arise from the terms of the absolutely free term algebra, which is generally not a member of the quasivariety under interest. In such cases, where the term algebra is not a member of a quasivariety, the free algebra of the quasivariety on denumerably countable free generators takes on the role played by the term algebra in sentential logics. Many of the logics that we encounter in this text arise most naturally as finitary logics on this free algebra of the quasivariety and generally are structural with respect to the quasivariety. We call such logics canons, and show how such structural canons induce sentential calculi, which we call the induced ideal ; the filters of the ideal on the free algebra are precisely the theories of the canon. The membership logic is the ideal of the cannon whose theories are the relative congruence classes on the free algebra. The primary aim of this thesis is to provide a unifying framework for logics of this type which extends the Blok-Pigozzi theory of elementarily algebraizable (and protoalgebraic) deductive systems. In this extension there are two parameters: a set of formulae and a variable. When the former is empty or consists of theorems, the Blok-Pigozzi theory is recovered, and the variable is redundant. For the membership logic, the appropriate variant of equivalent algebraic semantics encompasses the relatively congruence regular quasivarieties. These results have appeared in [BR03]. The secondary aim of this thesis is to analyse our theory of parameterized algebraization from a non- parameterized perspective. To this end, we develop a theory of protoalgebraic logics over constructs and equivalence between logics from different constructs, which we then use to explain the results we obtained in our parameterized theories of protoalgebraicity, algebraic semantics and equivalent algebraic semantics. We relate this theory to the theory of deductively equivalent -institutions [Vou03], and as a consequence obtain a number of improved and new results in the field of categorical abstract algebraic logic. We also use our theory of protoalgebraic logics over constructs to obtain a new and simpler characterization of structural finitary n-deductive systems, which we then use to close the program begun in [BR99], by extending those results for 1-deductive systems to n-deductive systems, and in particular characterizing the protoalgebraicity of the sentential n-deductive system Sn(K,N), which is the natural extension of the 1-deductive system S(K, ) introduce in [BR99], in terms of the quasivariety K having hK,Ni-coherent N-classes (we cannot see how to obtain this result from the standard characterization of protoalgebraic n- deductive systems of [Pal03], which is very complex). With respect to this program of completing [BR99], we also show that a quasivariety K is an equivalent algebraic semantics for a n-deductive system with defining equations N iff K is hK,Ni-regular; a notion of regularity that we introduce and characterize by a quasi-Mal’cev condition. The third aim of this text is to unify as many disparate arguments and notions in algebraic logic under the banner of continuous translations between closed systems, where our use of the term continuous is in the topological sense rather than in the order-theoretic sense, and, where possible, to give elementary, i.e. first order, definitions and proofs. To this end, we show that closed systems, closure operators and conse- quence relations can all be characterized elementarily over orders, and put into one-to-one correspondence that reflects exactly, the standard correspondences between the well-known concrete notions with the same name. We show that when the order is the complete power order over a set, then these elementary structures coincide with their well-known counterparts with the same name. We also introduce two other elementary structures over orders, namely the closed equivalence relation and something we term the proto-Leibniz relation; these elementary structures are also in one-to-one correspondence with the earlier mentioned structures; we have not seen concrete versions of these structures. We then characterize the structure homomorphisms between these structures, as well as considering galois relations between them; galois relations are pairs of order-preserving function in opposite directions; we call these translations, and they are elementary notions. We demonstrate how notions as disparate as structurality, semantics, algebraic semantics, the filter correspondence property, filters, models, semantic consequence, protoalge- braicity and even the logic S(K, ) of [BR99] and our logic Sn(K,N), all fall within this framework, as does much of our parameterized theory and much of the theory of -institutions. A brief summary of the standard theory of deductive systems and their algebraization is provided for the reader unfamiliar with algebraic logics, as well as the necessary background material, including construct and category theory, the theory of structures and algebras, and the model theory of structures with and without equality

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