6.3 Truth Tables for Propositions Propositional Logic 6.3 Truth Tables for Propositions

Truth Tables for Propositions The truth tables for the tilde, dot, wedge, etc., are all “truth tables for propositions.” Now we will apply a few new rules to show we can make truth tables for more complex propositions.

Truth Tables for Propositions Let’s take this example: (A v ~B)  B Since there are 2 different simple propositions, we need 4 lines to assess all the possible combinations of truth and falsity…

Truth Tables for Propositions (A v ~B)  B __________ Draw 4 lines T T T Enter TTFF, and TFTF T F F Since B appears again, we put TFTF in again F T T F F F

Truth Tables for Propositions (A v ~B)  B __________ Begin with the most enclosed operators and move out to the least enclosed Here, then, is the truth table for (A v ~B)  B T T T F T T F T T T F F F F T F T T F F T T F F

Classifying Statements If the main column has all Ts, the statement is said to be Logically true Tautologous Trivially true (true under all interpretations). [(G  H) • G]  H T

Classifying Statements If the main column has all Fs, the statement is said to be Logically false Self-contradictory Trivially false (false under all interpretations). (G v H) Ξ (~G • ~H) F

Classifying Statements If the main column has any mix of Ts and Fs, the statement is said to be Logically undetermined Contingent Empirical (true or false under various interpretations). (A v ~B)  B F T

Comparing Statements If two statements have the same truth values on each line of their truth tables, the statements are Logically Equivalent T F T F

Comparing Statements Opposite truth values on each line: Logically Contradictory T F F T

Comparing Statements At least one line has both true: Logically Consistent T F F T

Comparing Statements No line has both true: Logically Inconsistent T F