Valid Arguments

The tautologies in the preceding section are not all that there are. If you want to make a deduction based on a sentence, check its truth table. If it is a tautology, use it. Tautologies provide lots of reasoning theorems before we ever start deduction within a mathematical system.

There are actually two branches of formal logic: the statement calculus, involving statements and reasoning by tautology; and the predicate calculus, involving quantified sentences. All we are doing is taking a quick guided tour through informal logic, and so we will not study these areas in great detail. However, from the predicate calculus we get another collection of reasoning sentences, some of which are listed below. These cannot be verified by tautology.

Logic Axiom 2: Let U be a universal set. Each of the following is a rule of reasoning.

1. ,
2. ,
3. .

An argument is an assertion that from a certain set of sentences (called premises or assumptions) one can deduce another sentence Q (called an inference or conclusion). Such an argument can be denoted by

Arguments are either valid (correct) or invalid (incorrect).

Logic Axiom 3: [Rule of Substitution] Suppose . Then P and Q may be substituted for one another in any sentence. Logic Axiom 4: Every sentence of the type

is true. Logic Axiom 5: Every sentence of the type

is true.

To prove a sentence of the type false, one could try to prove true. This is referred to as providing a counterexample.