From Total Equational to Partial First-Order Logic

Abstract

The focus of this chapter is the incremental presentation of partial first-order logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are fully described. Alternatives to partiality, like (variants of) error algebras and order-sortedness, are also discussed, emphasizing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragments are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Finally some more powerful algebraic frameworks are sketched.