We develop a new logic for deriving consequences of multialgebraic theories (specifications). Multilagebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set valued operations. Atomic formulae are set inclusion t ≺ t ′ –the interpretation of t is included in the interpretation of t ′ , and element equality t. = t ′ – t and t ′ denote the same element of the carrier. We introduce the Rasiowa-Sikorski logic R-S for proving multilagebraic tautologies and show its soundness and completeness. We then extend this system for proving consequences of specifications based on translation of theories into logical formulae. Finally, we show how such a translation may be avoided –introduction of specific cut rules leads to a sound and complete Gentzen system for proving directly consequences of specifications.

We recall basic facts about the institution of multialgebras, and introduce a new, quantifier-free reasoning system for deriving consequences of multialgebraic specifications. We then show how can be used for combining specifications developed in other algebraic frameworks. We spell out the definitions of embeddings of institution of partial algebras, and membership algebras, MA.

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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the so-caled ‘institution theory ’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system ’ and a mathematical concept of ‘homomorphism ’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here. 1.