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Some characterization theorems for infinitary universal Horn logic without equality

Published online by Cambridge University Press:  12 March 2014

Pilar Dellunde  and
Ramon Jansana
Pilar Dellunde
Affiliation:
Area of Logic and Philosophy of Science, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain, E-mail: dellunde@cc.uab.es
Ramon Jansana
Affiliation:
Department of Logic, History and Philosophy of Science, Universitat de Barcelona, 08028 Barcelona, Spain, E-mail: jansana@cerber.ub.es
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Extract

In this paper we mainly study preservation theorems for two fragments of the infinitary languages Lκκ, with κ regular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, when κ is ω, we obtain the corresponding theorems for the first-order case.

The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only one κ-ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 4 , December 1996 , pp. 1242 - 1260
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

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