Speaking about Transistive Frames in Propositional Languages

This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication

Frontal Operators in Weak Heyting Algebras

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation τ(a) ≤ b ∨ (b → a), for all a, b ∈ A. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces hX, ≤, T, Ri where hX, ≤, T i is a WH - space [6], and R is an additional binary relation used to interpret the modal operator. We will also study the WH -algebras with successor and the WH -algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH -spaces.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; ArgentinaFil: San Martín, Hernán Javier. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Interpolation Property on Visser's Formal Propositional Logic

 In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property

Frontal operators in weak Heyting algebras

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving nite meets which also satis es the equation τ (a) ≤ b ∨ (b → a), for all a; b ∈ A. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces ⟨X;≤; T;R⟩ where ⟨X;≤; T⟩ is a WH- space [6], and R is an additional binary relation used to interpret the modal operator. We will also study the WH-algebras with successor and the WH-algebras with gamma. For these varieties we will give two topological dualities. The rst one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH-spaces.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

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)

Estudio de álgebras de Heyting débiles con operadores modales

En esta tesis de doctorado se estudian diferentes variedades de álgebras de Heyting débiles dotadas de operadores unarios que generalizan diferentes variedades de álgebras de Heyting modales.Doctor en Ciencias Exactas, área MatemáticaUniversidad Nacional de La PlataFacultad de Ciencias Exacta