Chapter Three Truth Tables

1. Computing Truth-Values We can use truth tables to determine the truth-value of any compound sentence containing one of the five truth- functional sentence connectives. This method can also be used to determine the truth-value of more complicated sentences. This procedure is called truth table analysis.

2. Logical Form The assignment of a truth-value to a compound sentence from the truth-values of its atomic constituents is called a valuation. Expressions that contain only sentence variables and sentence connectives are called sentence forms. If we replace sentence variables with sentence constants we end up with a substitution instance of the original sentence form. Logical form is not the same as logical equivalence.

3. Tautologies, Contradictions, and Contingent Sentences A sentence that is true in virtue of its logical form is a tautology. Contradictions are sentences that cannot possibly be true. The form of a contradiction is a contradictory sentence form.

Tautologies, Contradictions, and Contingent Sentences, continued A contingent statement is one whose form has at least one T and one F in its truth table.

4. Logical Equivalences When two statements are logically equivalent, the truth-value of one determines the truth-value of the other. That is, each has the same truth-value under the same truth conditions.

5. Truth Table Test of Validity We can use truth tables to determine if any argument in sentential logic is valid. Recall: An argument is valid if and only if it is not possible for its premises to all be true while its conclusion is false. So, if an argument is valid there will be no line in a truth table in which all the premises are true and the conclusion false.

Truth Table Test of Validity, continued We can test an argument for validity by conjoining the premises into the antecedent of a conditional, putting the conclusion as the consequent, and testing its form to see if it is a tautological form. If it is, the argument is valid.

Truth Table Test of Validity, continued If the corresponding conditional, or test statement form, of an argument is a tautology, then premises are said to logically imply or entail the conclusion of the argument.

Truth Table Test of Validity, continued A logical implication is a tautology who main connective is a horseshoe. A counterexample is an assignment of truth-values that will yield true premises and a false conclusion.

6. Truth Table Test of Consistency We can use a truth table to check for consistency by constructing the truth table for the conjunction of the forms and looking for a line on which all the substitutions are true. The set is consistent if and only if there is such a line.

7. Validity and Consistency The counterexample set of an argument consists of the premises of the argument together with the denial of the conclusion. If the counterexample set is consistent then the argument is invalid. All arguments with inconsistent premises are valid.

8. The Short Truth Table Test for Invalidity All it takes to show that an argument is invalid is a single counterexample—a single line of a truth table on which the premises are all true and the conclusion false. It is often possible to produce such a counterexample by assigning a truth-value to the entire sentence and then working to find the appropriate assignment of truth-values to the atomic constituents.

The Short Truth Table Test for Invalidity, continued We find the logical form of the argument, then assign an F to the conclusion and the try to assign a T to each premise. If we can do this the argument is invalid.

9. The Short Truth Table Test for Consistency All it takes to show that a set of sentences is consistent is to produce a single line in a truth table that makes them all true.

10. A Method of Justification for the Truth Tables We can build the truth table for any connective given the following information: 1) A set of intuitively valid and invalid arguments 2) In a valid argument, if the premises are true the conclusion must be true 3) In an invalid argument, there must be the possibility that the premises are true and the conclusion false

A Method of Justification for the Truth Tables, continued To justify a truth table, find valid and invalid argument forms, using the connective in question, that force the lines of the truth table.

Key Terms Contingent statement Contingent sentence form Contradiction Contradictory sentence form Corresponding conditional of an argument Counterexample of an argument Counterexample set Logical equivalence Logically equivalent Logical implication

Key Terms, continued Logically implies Sentence form Substitution instance Tautologous sentence form Tautology Test statement form Truth table analysis Valuation

Key Terms, continued Truth-function Truth-functional operator Truth table Truth-value Wedge Well-formed