Analytic proof systems for classical and modal logics of restricted quantification

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Abstract

This thesis is a study of the relationship between proof systems for propositional logic and for logics of restricted quantification incorporating restriction theories. Such logics are suitable for the study of special purpose reasoning as part of a larger system, an important research topic in automated reasoning. Also, modal and sorted logics can be expressed in this way. Thus, results on restricted quantification apply to a wide range of useful logics. D'Agostino's "expansion systems" are used to generalise results to apply to a variety of tableau-like propositional proof systems.

A certain class of propositional expansion systems is defined, and extended for restricted quantification in two different ways. The less general, but more useful, extension is proved sound and complete provided that the restriction theory can be expressed as a set of definite Horn clauses.

In the definite clauses case, the result is used to present a generalisation of Wallen's matrix characterisations of validity for modal logics. The use of restricted quantification enables more logics to be covered than Wallen did, and the use of expansion systems allows analogues of matrices to be defined for proof systems other than tableaux.

To derive the results on matrices, the calculi for restricted quantification are made weaker, and so can be unsound for some restriction theories. However, much greater order independence of rule applications is obtained, and the weakening is sound if one of two new conditions introduced here hold, namely "alphabetical monotonicity" or "non-vacuity". Alphabetical monotonicity or non-vacuity are shown to hold for a range of interesting restriction theories associated with order sorted logics and some modal logics.

I also show that if non-vacuity holds, then instantiation in restricted quantification can be completely separated from propositional reasoning. The major problem left open by this thesis is whether analogues of the previous matrix characterisations can be produced based on the proof systems introduced for nondefinite clause restriction theories.

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