First-Order Logic: Syntax
FOL; Predicate calculus; Predicate logic
First-order logic (FOL) is a formalization of the most common kind of mathematical reasoning. It is characterized by the quantification of variables that range over a “universe of discourse” (a set of values). Less complex reasoning is captured by propositional (a.k.a. Boolean or sentential) logic. More complex reasoning is captured by second-order or even higher-order logic.
The syntactic aspects of FOL comprise a vocabulary (a.k.a. language or signature), formulae and, in particular, sentences (a.k.a. assertions), and a proof system (one of many equivalent ones!), structures (a.k.a. models or interpretations), and the satisfaction (a.k.a. truth or validity or “holds in”) relationship between sentences and structures. All are detailed below.
FOL is the source of the relational paradigm that was introduced by E. F. Codd in 1970 and has been dominating database technology for 30+ years.
A first-order vocabulary...