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A system of logic for partial functions under existence-dependent kleene equality

Published online by Cambridge University Press:  12 March 2014

H. Andréka ,
W. Craig  and
I. Németi
H. Andréka
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary
W. Craig
Affiliation:
Department of Philosophy, University of California, Berkeley, California 94720
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Extract

Ordinary equational logic is a connective-free fragment of first-order logic which is concerned with total functions under the relation of ordinary equality. In [AN] (see also [AN1]) and in [Cr] it has been extended in two equivalent ways into a near-equational system of logic for partial functions. The extension given in [Cr] deals with partial functions under two relationships: a relationship of existence-dependent existence and one of existence-dependent Kleene equality. For the language that involves both relationships a set of rules was given that is complete. Those rules in the set that involve only existence-dependent existence turned out to be complete for the sublanguage that involves this relationship only. In the present paper we give a set of rules that is complete for the other sublanguage, namely the language of partial functions under existence-dependent Kleene equality.

This language lacks a certain, often needed, power of expressing existence and fails, in particular, to be an extension of the language that underlies ordinary equational logic. That it possesses a fairly simple complete set of rules is therefore perhaps more of theoretical than of practical interest. The present paper is thus intended to serve as a supplement to [Cr] and, less directly, to [AN]. The subject is further rounded out, and some contrast is provided, by [Rob]. The systems of logic treated there are based on the weaker language in which partial functions are considered under the more basic relation of Kleene equality.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 834 - 839
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[AN]Andréka, H. and Németi, I., Generalization of the concept of variety and quasivariety to partial algebras through category theory, Dissertationes Mathematicae/Rozprawy Matematyczne, vol. 204 (1983).Google Scholar
[AN1]Andréka, H., Corrections of a calculus of equations for partial algebras (to appear).Google Scholar
[Bu]Burmeister, P., A model theoretic oriented approach to partial algebras. Part I, Akademie-Verlag, Berlin, 1986.Google Scholar
[Cr]Craig, William, Near-equational and equational systems of logic for partial functions, this Journal (to appear).Google Scholar
[Rob]Robinson, Anthony, Equational logic of partial functions under Kleene equality: a complete and an incomplete set of rules, this Journal (to appear).Google Scholar
[Ru]Rudak, L., A completeness theorem for weak equational logic, Algebra Universalis, vol. 16 (1983), pp. 331–337.CrossRefGoogle Scholar