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An abstract logic 〈A, C〉 consists of a finitary algebraA and a closure systemC onA. C induces two other closure systems onA, CP andCI, by projective and inductive generation respectively. The various relations amongC, CP andCI are determined. The special case thatC is the standard equational closure system on monadic terms is studied in detail. The behavior of Boolean logics with respect to projective and inductive generation is determined.

Two characterizations are given of those structural consequence operations on a propositional language which can be defined via proofs from a finite number of polynomial rules.

A signed ∑-equation is an expression of the form t ≈ t′ or t ≉t′, where t and t′ are ∑-terms (for some ranked set ∑). We characterize those classes of ∑-algebras which are models of a set of signed ∑-equations. Further we consider the problem of finding a complete deductive system analogous to equational logic for the logical consequence operation restricted to signed equations.

1. The SCI (Sentential Calculus with Identity) is obtained from the classical sentential calculus by the addition of (1) a new binary connective, the identity con? nective (denoted by"=") and (2) axioms which 'say5 that "p == q" means ccp is identical to q" (also: "the situation p is the same as the situation q"). The new axioms are the weakest possible; no presuppositious about the meaning of "is identical to35 are in? cluded (other than "p = p"). We do not attempt to say what the range of the sentential variables p,q,r, ... is. (In the classical propositional calculus, they are intended to range over a two element set.) In this paper, a number of results about the semantics of the SCI are given without proof. The proofs of these and other results are contained in the much longer "In? vestigations into the Sentential Calculus with Identity53 [1].