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A paraconsistent higher order logic
 INTERNATIONAL WORKSHOP ON PARACONSISTENT COMPUTATIONAL LOGIC, VOLUME 95 OF ROSKILDE UNIVERSITY, COMPUTER SCIENCE, TECHNICAL REPORTS
, 2004
"... Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paracons ..."
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Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledgebased systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional manyvalued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.
Multidimensional Type Theory: Rules, Categories, and Combinators for Syntax and Semantics
, 2004
"... Abstract. We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multidimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantic ..."
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Abstract. We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multidimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantics, but the general perspective is important. For example, issues of pragmatics could be handled as additional dimensions. One of the main problems addressed is the rather complicated repertoire of operations that exists besides the notion of categories in traditional Montague grammar. For the syntax we use a categorial grammar along the lines of Lambek. For the semantics we use socalled lexical and logical combinators inspired by work in natural logic. Nabla provides a concise interpretation and a sequent calculus as the basis for implementations.... Lambek originally presented his type logic as a calculus of syntactic types. Semantic interpretation of categorial deductions along the lines of the CurryHoward correspondence was put on the categorial agenda in J. van Benthem (1983) The semantics of variety in categorial grammar, Report 8329*, Simon Fraser University, Canada. This contribution made it clear how the categorial type logics realize Montagues Universal Grammar program — in fact, how they improve on Montagues own execution of that program in offering an integrated account of the composition of linguistic meaning and form. Montagues adoption of a categorial syntax does not go far beyond notation: he was not interested in offering a principled theory of allowable ‘syntactic operations’
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, 2005
"... Conference on Logic Programming, also colocated with CP’05, 11th ..."
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