Table of Contents - 1. CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
Logic can be defined as the "study of the principles of correct reasoning"
(Merrilee H. Salmon's 1991 "Informal Reasoning and Informal Logic" in
Informal Reasoning and Education) or as "a formal system using symbolic
techniques and mathematical methods to establish truth-values" (the Oxford
English Dictionary).
This section formally defines the logic system we will use. In particular,
it defines symbols for declaring truthful statements, along with rules for
deriving truthful statements from other truthful statements. The system
defined here is classical first-order logic with equality (the most common
logic system used by mathematicians).
We begin with a few housekeeping items in pre-logic, and then introduce
propositional calculus (both its axioms and important theorems that can be
derived from them). Propositional calculus deals with general truths about
well-formed formulas (wffs) regardless of how they are constructed. This is
followed by proofs that other axiomatizations of classical propositional
calculus can be derived from the axioms we have chosen to use.
We then define predicate calculus, which adds additional symbols and rules
useful for discussing objects (beyond simply true or false). In particular,
it introduces the symbols ("equals"), ("is a member of"), and
("for all"). The first two are called "predicates". A predicate
specifies a true or false relationship between its two arguments.