Linear Spaces, Boolean Algebras, and First Order Logic

Abstract

After a brief introduction to algebras, in 2.1 we begin with linear spaces (LSs) and Boolean algebras (BAs). In 2.2 we cover the basic notions of universal algebra, building up to, but not including, Birkhoff’s Theorem. Ultrafilters are introduced in 2.3. In 2.4 we turn to the propositional calculus, and the (lower) predicate calculus is discussed in 2.5. The elements of proof theory are sketched in 2.6, and some preliminaries from multivariate statistics (which are, for the most part, just linear algebra in new terminological garb) are discussed in 2.7. The material is selected for its utility in later chapters, rather than for internal cohesion, and is largely orthogonal to the standard logic prerequisites to compositional semantics covered in for example the two volumes of Gamut (1991). A first course in linear algebra (but not in multivariate statistics) is assumed.