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)

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

On the injectivity of the Leibniz operator

The class of weakly algebrizable logics is defined as the class of logics having monotonic and injective Leibniz operator. We show that \monotonicity" can- not be discarded on this definition, by presenting an example of a system with injective and non monotonic Leibniz operator. We also show that the non injectivity of the non protoalgebraic inf-sup fragment of the Classic Propositional Calculus, CPC_{inf,sup}, holds only from the fact that the empty set is a CPC_{inf,sup}-filter.FCT via UIM

Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

The ubiquity of conservative translations

We study the notion of conservative translation between logics introduced by Feitosa and D'Ottaviano. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.Comment: 15 pages; to appear in Review of Symbolic Logi

Closure properties for the class of behavioral models

Hidden k-logics can be considered as the underlying logics of program specification. They constitute natural generalizations of k-deductive systems and encompass deductive systems as well as hidden equational logics and inequational logics. In our abstract algebraic approach, the data structures are sorted algebras endowed with a designated subset of their visible parts, called filter, which represents a set of truth values. We present a hierarchy of classes of hidden k-logics. The hidden k-logics in each class are characterized by three different kinds of conditions, namely, properties of their Leibniz operators, closure properties of the class of their behavioral models, and properties of their equivalence systems. Using equivalence systems, we obtain a new and more complete analysis of the axiomatization of the behavioral models. This is achieved by means of the Leibniz operator and its combinatorial properties. © 2007 Elsevier Ltd. All rights reserved.FCT via UIM

Equivalential Algebras with Conjunction on Dense Elements

We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety

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