PHIL 2020 Unit 4: Lecture 3 Truth Tables and Validity In addition to their ability to help visualize the truth conditions of our basic operators, we can use truth tables to evaluate arguments. Since we are no studying formal logic, we wont be able to use truth tables to evaluate the truth of premises, since their truth depends on what our simple statement abbreviations stand for. Nor can we use truth tables to evaluate inductive arguments, since their standard of relevance escapes binary categorization. It follows that truth tables are only of use in evaluating whether or not an argument is valid. A refresher on validity What does it mean for an argument to be valid? o Its premises entail its conclusion In terms of a truth table what would a valid argument form look like? o For every row in which every premise is true, the conclusion is also true. The truth table test So, we can test for an arguments validity by plotting out all of its simple sentence variables, and all of its premises, and its conclusion. The truth value of the simple sentence variables will determine the truth of the premises and the conclusion. Heres a simple example for the argument o (AB) (A v B): A B (A B) (A v B) T T T T T F F T F T F T F F F F Is the argument valid? o Yes, because in every case in which the premise is true, the conclusion is also true. In order to apply the test, we only need to look at the rows in which all the premises are true. This means a couple of interesting things. First, just like a conditional, the entailment claim associated with validity is false only if the antecedent (the premises) are true, and the consequent (the conclusion) is false. Second, because validity only talks about what is the case when the premises are true, an argument with a necessarily false premise is always valid. Limitations to the truth table test Omitting the trop row, where we name the relevant variables, premises and conclusion, hoe many rows would we need for an argument that made use of eighteen different simple statements? o 262, 144 Obviously, that would take a long time, so it would be nice if we could develop a more efficient method of testing for validity.