Results

**1 - 1**of**1**### 1 Quasi-Identity Logic

, 2001

"... symbols function symbols constant symbols variables In Chapter III of LMCS we looked at Birkhoff’s study of equational logic. The next larger interesting fragment of first-order logic with equality is the logic of quasi-identities. Quasi-identities are universal Horn formulas of the form ∀x [p1 ≈ q1 ..."

Abstract
- Add to MetaCart

symbols function symbols constant symbols variables In Chapter III of LMCS we looked at Birkhoff’s study of equational logic. The next larger interesting fragment of first-order logic with equality is the logic of quasi-identities. Quasi-identities are universal Horn formulas of the form ∀x [p1 ≈ q1 ∧ · · · pn ≈ qn = ⇒ p ≈ q], including the possibility that n = 0 and we simply have an equation. As with identities, we usually omit writing the universal quantifiers. The study of quasi-identities, and the corresponding model classes called quasi-varieties, has been pursued mainly in Eastern Europe, following the lead of Mal’cev. In Western Europe and North America the focus has been on identities and varieties, a direction initiated by Birkhoff. The rules for working with quasi-identities are not as simple, or standardized, as the rules of Birkhoff. Of course the usual rules of first-order logic suffice to derive all the quasiidentity consequences of a set of quasi-identities, but one might prefer to have a logical system which only produces quasi-identities from quasi-identities. One such was given by Selman [5] in 1972 with four axiom schemes and six rules of inference. By considering a conjunction p1 ≈ q1 ∧ · · · ∧ pn ≈ qn as a set of equations, rather than an abbreviation for some particular way of inserting parentheses, we can omit his fifth rule (which handles rearranging the parentheses and repeat copies of equations): 1 AXIOMS: (a) p ≈ q ∧ r ≈ s = ⇒ r ≈ s (b) p ≈ p (c) p ≈ q ∧ q ≈ r = ⇒ r ≈ p (d) p1 ≈ q1 ∧ · · · ∧ pn ≈ qn = ⇒ f(p1,..., pn) ≈ f(q1,..., qn).