3.6 Analyzing Arguments with Truth Tables

Truth Tables (Two Premises) Determine whether the argument is valid or invalid. If the floor is dirty, then I must mop it. The floor is dirty. _ I must mop it. Testing the Validity of an Argument with a Truth Table Assign letters to represent each component in the argument. Express each premise and the conclusion symbolically. Form the symbolic statement of the entire argument (conjunction of ALL the premises = antecedent of a conditional, conclusion = consequent). Complete a truth table for the conditional formed in Step 3. If it is a tautology, the argument is valid (otherwise, it’s invalid).

p q p  q (p  q)  p [(p  q)  p]  q The pattern of the argument shown here, p  q p _ q is a common one, known as modus ponens, or the law of detachment.

Using a Truth Table to Determine Validity Determine whether the argument is valid or invalid. If my check arrives in time, I’ll register for fall semester. I’ve registered for fall semester _ My check arrived in time. If a conditional and its converse were logically equivalent, then this argument would be valid. Therefore, an argument in the form p  q q _ p is an example of what is called the fallacy of the converse. p q p  q (p  q)  q [(p  q)  q]  p

Using a Truth Table to Determine Validity Determine whether the argument is valid or invalid. If a man could be in two places at once, I’d be with you. I am not with you. _ A man can’t be in two places at once. An argument in the form of p  q ~q _ ~p is called modus tollens.

Using a Truth Table to Determine Validity Determine whether the argument is valid or invalid. If it rains today, then I’ll stay home. It didn’t rain today. I didn’t stay home. An argument in the form p  q ~p _ ~q is called the fallacy of the inverse.

Using a Truth Table to Determine Validity Determine whether the argument is valid or invalid. I‘ll buy a car or I’ll take a vacation. I won’t buy a car. _ I’ll take a vacation If an argument in the form p  q ~p _ q then it is valid by the law of disjunctive syllogism.

Using a Truth Table to Determine Validity Determine whether the argument is valid or invalid. If it squeaks, then I use WD-40. If I use WD-40, then I must go to the hardware store. If it squeaks, then I must go to the hardware store. An argument in the form p  q q  r p  r is called reasoning by transitivity (or the law of transitivity).

Summary of Argument Forms Valid Argument Forms Modus Ponens Modus Tollens Disjunctive Syllogism Law of Transitivity p  q p  q p ~q ~p q  r q p  r Invalid Argument Forms (Fallacies) Fallacy of the Converse Fallacy of the Inverse p  q q ~p p ~q