Book contents
19 - Axiomatisation
Published online by Cambridge University Press: 06 July 2010
Summary
Finite axiomatisation
Although in principle a calculus may be axiomatised by supplying recursive rules of inference for it, in practice we expect the number of rules to be finite. For example the classical propositional calculus, characterised by the classical matrix M, can in principle be axiomatised by the rule ‘from A1,…,Am infer B if A1,…,Am, ⊢M B’. As an exercise in axiomatisation this is not trivial, since it calls for a demonstration that ⊢M, is compact; and if it is unsatisfactory it is because the rule is felt to be a portmanteau containing infinitely many distinct rules. But to make this feeling coherent we must be able to say what is to count as a single rule. For propositional calculi, and for other calculi too if a suitable definition of substitution can be devised, the appropriate idea is that of a scheme. The familiar example is the axiom scheme, a set of axioms comprising every substitution instance s(A) of a given formula A. By analogy we define a rule scheme to be a set of rules comprising ‘from s(X) infer s(Y)’ for every substitution instance of given X and Y. A single-conclusion rule scheme will similarly comprise ‘from s(X) infer s(B)’ for every substitution instance of given X and B. Axiom schemes are covered by this definition as rule schemes with zero premisses. Most of the familiar rules - modus ponens, adjunction, etc. - are rule schemes in this sense.
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- Multiple-Conclusion Logic , pp. 325 - 358Publisher: Cambridge University PressPrint publication year: 1978