Abstract
There are different ways of constructing a logic. One possibility is to define a Hilbert calculus, i.e. a kind of grammar that produces all formulae to be considered true. A logic can also be defined by a model theoretic semantics for the logical connectives in the language. In this paper a general theory is presented for the transition from a Hilbert calculus to its model theoretic semantics such that soundness and completeness are automatically guaranteed.
For a given Hilbert calculus we start with a general neighbourhood semantics for n-place connectives. This semantics does not impose any built-in properties. A quantifier elimination algorithm is used to translate Hilbert axioms and rules into corresponding semantic properties. By proving certain key lemmas from these semantic properties, neighbourhood semantics can be systematically strengthened up to a version of the semantics which has as many Hilbert axioms built in as possible. The work is still incomplete and will be continued.
This work was supported by the ESPRIT project 3125 MEDLAR, by the “Sonder-forschungsbereich” 314, “Künstliche Intelligenz und wissensbasierte Systeme” of the German Research Council (DFG) and by the MFT funded project ‘LOGO’. The first author is a SERC/UK Senior Research Fellow.
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© 1993 British Computer Society
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Gabbay, D., Ohlbach, H.J. (1993). From a Hilbert Calculus to its Model Theoretic Semantics. In: Broda, K. (eds) ALPUK92. Workshops in Computing. Springer, London. https://doi.org/10.1007/978-1-4471-3421-3_13
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DOI: https://doi.org/10.1007/978-1-4471-3421-3_13
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