Thinking Mathematically Arguments and Truth Tables

Definition of a Valid Argument An argument is valid if the conclusion is true whenever the premises are assumed to be true. An argument that is not valid is said to be an invalid argument, also called a fallacy.

An Example of an Argument p  q If I get an A on the final I will pass the course. pI got an A on the final.  qI will pass the course The “argument” is “If I get an A on the final I will pass the course and I got an A on the final therefore I will pass the course.” [(p  q)/\p]  q

Valid Arguments Valid arguments are tautologies. That is they are always true. p  q If I get an A on the final I will pass the course. pI got an A on the final.  qI will pass the course

Testing the Validity of an Argument with a Truth Table 1.Use a letter to represent each simple statement in the argument. 2.Express the premises and the conclusion symbolically. 3.If the argument contains n premises, write the symbolic conditional statement of the form [(premise 1)/\(premise 2)/\.../\(premise n)]→conclusion.

Testing the Validity of an Argument with a Truth Table 4.Construct a truth table for the conditional statement in step 3. 5.If the final column of the truth table has all trues, the conditional statement is a tautology, and the argument is valid. If the final column does not have all trues, the conditional statement is not a tautology, and the argument is invalid.

Discuss the Standard Forms of Arguments and some Fallacies (page 144) Show Transitive Reasoning is valid. This uses a larger truth table.