Logic, as a distinct field of study, dates approximately to Aristotle. Its fundamental charge is to distinguish sound reasoning from unsound reasoning. In an effort to do this systematically, rather than in an ad hoc manner, logicians usually start by abstracting away all the particulars of the argument and studying only its form: for example, if you know that at least one of the statements P and Q is true, and you know that P is false, you can conclude that Q must be true, regardless of what P and Q actually mean.
Since logic, like other branches of mathematics, deals only with unambiguous statements, and since English is a notoriously ambiguous language, logicians (and other mathematicians) usually work in a language of their own, in which every term and symbol has a clear, well-defined meaning, or (like P and Q in the above example) can be manipulated without regard to any specific meaning. We call this approach ``symbolic logic''.
We'll study two fundamentally different flavors of symbolic logic: propositional logic, in which the variables refer to statements that can be either true or false, and predicate or first-order logic, in which most of the variables refer to objects in some world, and we can ask what properties (or ``predicates'') are true of which objects. After learning the language used to talk about such logics, and a number of techniques for rigorously proving logical statements, we'll turn our attention to specific applications of logic, such as set theory, arithmetic, and computer expert systems. I hope to finish the semester with a discussion of the limits of logical reasoning, and the startling result (due to Kurt Gödel) that in any ``reasonably powerful'' logical system, there will be statements that are true and can be expressed in logical language, but cannot be proven.