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Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of NonClassical Logics I
 Studia Logica
, 1998
"... The main goal of this paper is to explain the link between the algebraic and the Kripkestyle models for certain classes of propositional logics. We start by presenting a Priestleytype duality for distributive lattices endowed with a general class of wellbehaved operators. We then show that fin ..."
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The main goal of this paper is to explain the link between the algebraic and the Kripkestyle models for certain classes of propositional logics. We start by presenting a Priestleytype duality for distributive lattices endowed with a general class of wellbehaved operators. We then show that finitelygenerated 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 nontopological Kripkestyle models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the abovementioned classes. Introduction In the study of nonclassical 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...
A Logic Of Vision
"... 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 JohnsonLaird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessari ..."
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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 JohnsonLaird, 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 multilayered 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.
Representation Theorems and the Semantics of NonClassical Logics , and Applications to Automated Theorem Proving
, 2002
"... We give a uniform presentation of representation and decidability results related to the Kripkestyle 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, d ..."
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Cited by 4 (2 self)
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We give a uniform presentation of representation and decidability results related to the Kripkestyle 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 Kripkestyle models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rstorder axiomatizable) for automated theorem proving by resolution for some nonclassical logics.
Algebraic Polymodal Logic: A Survey
 LOGIC JOURNAL OF THE IGPL
, 2000
"... 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. It begins with ..."
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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. It begins with
Conditional Quantification, Or Poor Man's Probability
, 2002
"... 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 t ..."
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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
Scopes and limits of modality in quantum mechanics
, 2008
"... 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 ..."
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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
Functional Monadic Bounded Algebras
, 2010
"... 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 isomor ..."
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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
FIXED POINT ALGEBRAS
"... Although selfreference in arithmetic was used to impressive effect by Gödel in 1930 (published in 1931) when he noted the sentence asserting its own unprovability to be unprovable, and although this use immediately appealed to philosophers and philosophical logicians, it has largely been ignored by ..."
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Although selfreference in arithmetic was used to impressive effect by Gödel in 1930 (published in 1931) when he noted the sentence asserting its own unprovability to be unprovable, and although this use immediately appealed to philosophers and philosophical logicians, it has largely been ignored by mathematical logicians. Indeed, it is only in the 1970s that arithmetic selfreference has begun to be systematically studied and applied. One aspect of this study is algebraic. In simplest terms one can distinguish two types of selfreference—extensional and nonextensional. Extensional selfreference lends itself quite readily to algebraic description and modelling, with some types of extensional selfreference even being amenable to algebraic study. The purpose of the present paper is to expound upon this algebraic modelling, touching briefly on its successes and delineating roughly the limits to this success. The central notion of this exposition is that of a fixed point algebra. This notion is a new one—it is untested and, hence, of only provisional interest. But it does appear useful: