Abstract
Two derivation operations are defined which are sound and complete for interpretations of first-order theories with equality in classes of partial algebraic structures. Łoś' theorem on ultraproducts has been generalized for such interpretations.
The following generalized form of Herbrand's theorem is proved: For an arbitrary first-order theory & and an arbitrary formula α may be constructed an enumerable set of open formulas Hα such that & ↣ α holds iff there exists β ε Hα with & ↣ β. whereby ↣ denotes either the usual or one of the mentioned above derivation operators. The enumeration of Hα is determined by a procedure which is closely related to the connection method proof procedure.
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Petermann, U. (1989). An extended Herbrand theorem for first-order theories with equality interpreted in partial algebras. In: Kreczmar, A., Mirkowska, G. (eds) Mathematical Foundations of Computer Science 1989. MFCS 1989. Lecture Notes in Computer Science, vol 379. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51486-4_88
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DOI: https://doi.org/10.1007/3-540-51486-4_88
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