Metaphilosophy 2: A Brief Introduction to Logic

Logic is the fundamental tool of philosophy. While philosophers puzzle over inductive logic, deductive arguments form the vast bulk of philosophical argumentation. Consequently, I will be discussing deduction at this point in time.

There are three broad types of basic logic: propositional, predicate, and meta-predicate. Propositional logic deals with what are expressed by whole sentences, namely, propositions. It studies the effects of the operations of the so-called logical operators on the truth-values of various propositions. The logical operators are: "... and ...", "... or ...", "if ... then ...", and "not ...". Each operator has an introduction and elimination rule associated with it that is designed to capture the standard English meaning of the operator; also, a good number of philosophers associate each operator with a truth-table, which governs how it affects truth-values in complex propositions.

An important type of logical argument in philosophy is called reductio ad absurdum or reductio for short. The name is Latin for "reduction to absurdity" and signifies a general type of argument where you suppose some proposition, prove a contradiction (an absurdity) from it, and subsequently conclude that the proposition is false. Because of the nature of truth-tables, if you suppose for reductio some proposition that begins "not ...", you end up concluding "not not ..." and can conclude "..." from there. That is, if you suppose not-P, you may be able to prove, via reductio, P.

Predicate logic breaks propositions down into a predicate-subject format; usually, this is signified by something of the form "Ps" (read "s is P"). Predicate logic allows generalization over subjects, but not predicates, by introducing two new logical operators "for all ..." and "there exists ..." (a subject goes in the "..."); these operators are called the universal and existential respectively. Introduction and elimination rules exist for these operators as well; truth-table advocates utilize so-called truth-trees to establish the effects of these new operators on the truth-values of propositions.

Finally, meta-predicate logic doesn't introduce any new operators, but allows the universal and existential operators to generalize over predicates as well as subjects. Meta-predicate logic is important because certain important things, like "finite, but arbitrarily large", cannot be expressed in any merely predicate logic.