The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. Introduction In the study of non-classical propositional logics (and especially of modal logics) there are two main ways of defining interpretations or models. One possibility is to use algebras -- usually lattices with operators -- as models. Propositional variables are interpreted over elements of these algebraic models, an...

This essay attempts to develop a psychologically informed semantics of perception reports, whose predictions match with the linguistic data. As suggested by the quotation from Miller and Johnson-Laird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessarily differs from situation semantics. On the psychological side, our main inspiration is Marr's (1982) theory of vision, which can easily accomodate fallible perception. In Marr's theory, vision is a multi-layered process. The different layers have filters of different gradation, wkich makes vision at each of them approximate. On the logical side, our task is therefore twofold to fomalise the layers and the ways in which they may refine each other, and to develop logical means to let description vary with such degrees of refinement.

We give a uniform presentation of representation and decidability results related to the Kripke-style semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripke-style models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rst-order axiomatizable) for automated theorem proving by resolution for some non-classical logics.

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.

This paper was begun in Amsterdam and finished at HCRC in Edinburgh. I thank Keith Stenning for his friendship and hospitality, and the EPSRC for financial support. can only be determined up to some interval, the truthvalue of X A cannot be ascertained unambiguously. In probability theory, this type of incomplete information is treated by means of conditional expectation. Still referring to our example, let # be the #-algebra generated by the intervals from the partition of IR. Then = X -1 # is strictly contained in B

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of “relatively functional ” algebra, and show that every member of MBA is isomorphic to a relatively functional one. In [1], an equationally defined class MBA of monadic bounded algebras was introduced. Each of these algebras comprises a Boolean algebra B with a distinguished element E, thought of as an existence predicate, and an operator ∃ on B reflecting the properties of the existential quantifier in logic without existence assumptions. MBA was shown to be generated by a certain proper

Algebraic Logic. In preparation. Manuscript.

representation theorems of three-valued structures

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems. 1