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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.
Paraconsistent assertions
 In Second German Conference on Multiagent System Technologies
, 2004
"... Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion. We argue that paraconsistent logics are especially advantageous in order to deal with assertions made ..."
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Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion. We argue that paraconsistent logics are especially advantageous in order to deal with assertions made by intelligent agents. Other propositional attitudes like knowledge and beliefs can in principle be treated along the same lines. We propose a manyvalued paraconsistent logic based on a simple notion of indeterminacy. The proposed paraconsistent logic has a semantics that extends the one of classical logic and it is described using key equalities for the logical operators. A case study is included. We briefly compare with logics based on bilattices. We finally investigate how to translate the paraconsistent logic into classical predicate logic thereby allowing us to make use of automated deduction of classical logic in the future. We base our initial translation on recent work by Muskens. Our final translation is polynomial in the size of the translated formula and follows the semantics for the paraconsistent logic directly. The major motivation behind paraconsistent logic has always been the thought that in certain circumstances we may be in a situation where our information or theory is inconsistent, and yet where we are required to draw inferences in a sensible fashion... Numerous examples of inconsistent information/theories from which one might want to draw inferences in a controlled way have been offered by paraconsistent logicians. For example: 1. information in a computer data base; 2. various scientific theories; 3. constitutions and other legal documents; 4. descriptions of fictional (and other nonexistent) objects; 5. descriptions of counterfactual situations. The first of these is fairly obvious...
Paraconsistent Query Answering Over DLLite Ontologies
"... Consistent query answering over description logicbased ontologies is an important topic in ontology engineering as it can provide meaningful answers to queries posed over inconsistent ontologies. Current approaches for dealing with this problem usually consist of two steps: the first step is extrac ..."
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Consistent query answering over description logicbased ontologies is an important topic in ontology engineering as it can provide meaningful answers to queries posed over inconsistent ontologies. Current approaches for dealing with this problem usually consist of two steps: the first step is extracting some consistent subontologies of an inconsistent ontology, then posing the query over these subontologies. In this paper, we propose an alternative approach for consistent query answering over DLLite ontologies based on fourvalued semantics, where DLLite is a family of tractable DLs. We give an algorithm to compute answers to a query over inconsistent DLLite ontologies and show that it is tractable. In particular, it is PTime in the size of TBox, and LOGSPACE in the size of ABox. 1
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’
Supralogic: Using transfinite type theory with type variables for paraconsistency
 In III World Congress on Paraconsistency
, 2003
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Multiagent System Technologies — MATES 2004 c ○ SpringerVerlag LNCS Paraconsistent Assertions
"... Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion. We argue that paraconsistent logics are especially advantageous in order to deal with assertions made ..."
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Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion. We argue that paraconsistent logics are especially advantageous in order to deal with assertions made by intelligent agents. Other propositional attitudes like knowledge and beliefs can in principle be treated along the same lines. We propose a manyvalued paraconsistent logic based on a simple notion of indeterminacy. The proposed paraconsistent logic has a semantics that extends the one of classical logic and it is described using key equalities for the logical operators. A case study is included. We briefly compare with logics based on bilattices. We finally investigate how to translate the paraconsistent logic into classical predicate logic thereby allowing us to make use of automated deduction of classical logic in the future. We base our initial translation on recent work by Muskens. Our final translation is polynomial in the size of the translated formula and follows the semantics for the paraconsistent logic directly. The major motivation behind paraconsistent logic has always been the thought that in certain circumstances we may be in a situation where our information or theory is inconsistent, and yet where we are required to draw inferences in a sensible fashion... Numerous examples of inconsistent information/theories from which one might want to draw inferences in a controlled way have been offered by paraconsistent logicians. For example: 1. information in a computer data base; 2. various scientific theories; 3. constitutions and other legal documents; 4. descriptions of fictional (and other nonexistent) objects; 5. descriptions of counterfactual situations. The first of these is fairly obvious...