Compiled Labelled Deductive Systems: A Uniform Presentation of Non-classical Logics
Research Studies Press, 2004 - 343 páginas
This book introduces a general framework, called Compiled Labelled Deductive System (CLDS), for a unified presentation of logics. It shows how this technique is applied to various families of non-classical logics widely used in Computer Science and Artificial Intelligence. It includes complete natural deduction proof systems, together with proofs and examples, for the families of propositional and predicate modal logics, propositional conditional logics of normality, multiplicative linear logic and for Lukasiewicz fuzzy logic. The CLDS framework generalises the notion of a logical theory to a structured theory consisting of labelled formulas and relations between the labels. This formalisation makes the logics more amenable to applications. Moreover, the framework provides a semantic approach based on a first-order translation, which can be uniformly applied to any logic whose semantics is first-order axiomatisable. Such a semantic approach also facilitates the development of automated theorem provers for non-classical logics.
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Soundness Completeness and Correspondence Results
Derivability and Semantic Entailment
Formal Results for Predicate MLDS
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accessibility additional allows applied approach arbitrary associated assumptions axiom schemas axiomatic system called Chapter classical CLDS Cmcc Cmce completeness conditional conditional logic connectives considered constant contradiction corresponding declarative term declarative unit deduction system defined Definition denote derivation described domain equal example exists expressed extended extended algebra Figure finite first-order formalisation formula FOT(C frames given ground term Hence holds includes inconsistent inference rules initial interpretation introduced Kripke labelled deductive labelling algebra labelling language Lemma maximal consistent configuration MLDS modal language modal logic natural deduction normal notation notion operators particular possible world predicate proof proof system properties propositional proved quantified quantified modal R-literals reasoning refer relation represented respect satisfiability semantic semantic entailment shown sometimes soundness specific standard step strict syntactic Theorem theory tion translation true truth variable wffs written