Truth Tables, Continued 6.3 and 6.4 March 14th

6.3 Truth tables for propositions Remember: a truth table gives the truth value of a compound proposition for every possible truth value of its simple components. Each possible arrangement of truth-values gets a line in the truth table. As a result, the total number of lines is equal to the number of possible combinations of truth values for simple propositions.

6.3 Truth tables for propositions L = # of lines n = # of different simple propositions L = 2 n EXAMPLE: consider the statement, (A ⋅ B) ⊃ C A, B, C are three simple statements 2 3 L = 8 L =

6.3 Truth Table - demonstration EXAMPLE: (A V ~B) ⊃ B Two different propositions 2 2 = 4, so four lines

6.3 Truth Table - demonstration EXAMPLE: (A V ~B) ⊃ B Two different propositions 2 2 = 4, so four lines (A V ~ B) ⊃ B T T F T T T T T T F F F F F F T T T F T T F T F

6.3 Classifying statements Tautologies Tautologous statements, also known as logically true statements, are true regardless of the truth values of their components. Examples: - A V ~A - (A ⊃ B) ⊃ (B ⊃ A)

6.3 Classifying statements Self-Contradictory Statements Self-contradictory statements, also known as logically false statements, are false regardless of the truth values of their components. Examples: - A ⋅ ~A - (B V C) ≡ (~B ⋅ ~C)

6.3 Classifying statements Contingent Statements Contingent statements are true or false depending on the truth values of their components. Examples: - A V B - B ⊃ C

6.3 Your Turn Tautologous, self-contradictory, or contingent? 1. N ⊃ (N ⊃ N) 2. (A ⊃ B) ⋅ (A ⋅ ~B) 3. [C ⊃ (D ⊃ E)] ≡ [(C ⊃ D) ⊃ E]

6.3 Your Turn Tautologous, self-contradictory, or contingent? 1. N ⊃ (N ⊃ N) tautologous 2. (A ⊃ B) ⋅ (A ⋅ ~B) self-contradictory 3. [C ⊃ (D ⊃ E)] ≡ [(C ⊃ D) ⊃ E] contingent

6.3 COMPARING STATEMENTS We can also use truth tables to see if statements have a particular sort of logical relationship...

6.3 COMPARING STATEMENTS Logically equivalent statements Logically equivalent statements have the same truth value on each line under their main operators.

6.3 COMPARING STATEMENTS Logically equivalent statements Example: B ⊃ C ~ C ⊃ ~ B T T T F T T F T T F F T F F F T F T T F T T T F F T F T F T T F

6.3 COMPARING STATEMENTS Logically equivalent statements Example: B ⊃ C ~ C ⊃ ~ B T T T F T T F T T F F T F F F T F T T F T T T F F T F T F T T F Note: to compare the truth values for the main connectives, you must use the same combination of truth-value possibilities for both statements, as you go down the rows...

6.3 COMPARING STATEMENTS Logically contradictory statements Logically contradictory statements have opposite truth values on each line under their main operators.

6.3 COMPARING STATEMENTS Logically contradictory statements Example: B ⊃ C B ⋅ ~ C T T T T F F T T F F T T T F F T T F F F T F T F F F T F

6.3 COMPARING STATEMENTS Logically contradictory statements Example: B ⊃ C B ⋅ ~ C T T T T F F T T F F T T T F F T T F F F T F T F F F T F Note: to compare the truth values for the main connectives, you must use the same combination of truth-value possibilities for both statements, as you go down the rows...

6.3 COMPARING STATEMENTS Logically consistent statements Two pairs of statements are logically consistent if there is at least one line on which the truth values for the main operators are both true.

6.3 COMPARING STATEMENTS Logically consistent statements Example: B V C B ⋅ C T T T T T F T F F F T T F F F F F F F F T

6.3 COMPARING STATEMENTS Logically consistent statements Example: B V C B ⋅ C T T T T T F T F F F T T F F F F F F F F T There is at least one line where both statements are true at the same time.

6.3 COMPARING STATEMENTS Logically inconsistent statements Two pairs of statements are logically inconsistent if there is no line on which the truth values for the main operators are both true.

6.3 COMPARING STATEMENTS Logically inconsistent statements Example: A ≡ BA ⋅ ~ B T T T T F F T T F F T T T F F F T F F F T F T F F F T F There are no lines in which both statements are true (where both primary operators have true values).

6.3 RELATIONSHIPS BETWEEN STATEMENTS, THROUGH TRUTH-TABLES While only some pairs of statements are equivalent or contradictory with regard to one another, all pairs of statements will fall into the category of consistent or inconsistent.

6.3 RELATIONSHIPS BETWEEN STATEMENTS, THROUGH TRUTH-TABLES Consistency and inconsistency can be evaluated for a large group of propositions - the group will be consistent as long as there is some combination of truth values for the basic statements that exists under which all of the propositions are true. That is, as long as there’s at least one line where all would have “T” under their main connective.

6.3 Regarding Consistency We can use consistency and inconsistency in real life to evaluate how rational a person’s stated position is. If their stated position is inconsistent, their position can’t be said to make any logical sense: If you were to conjoin the statements, their position would be a self-contradictory compound statement.

6.4 Truth Tables For Arguments If you’ve been paying attention, you may have already figured out how truth tables could be useful for evaluating arguments… They provide the standard technique, in fact, for testing the validity of arguments in propositional logic.

6.4 Truth Tables For Arguments Constructing a Truth Table for an Argument: 1. Translate into symbols w/ letters for simple propositions. 2. Put the translated argument on top of the table. Use a single slash between premises and a double-slash between the last premises and the conclusion. 3. Draw a truth table for the symbolized argument as if it were a proposition broken into parts. 4. Look for a line where all the premises are true and the conclusion false. If there is no such line, the argument is valid.

6.4 Truth Tables For Arguments P1) If children who kill people are as responsible for their crimes as adult killers are, then execution is a justifiable punishment for them. P2) Juvenile killers are not as responsible for their crimes as adults are. C) Therefore, execution is not a justifiable punishment for them.

6.4 Truth Tables For Arguments P1) If children who kill people are as responsible for their crimes as adult killers are, then execution is a justifiable punishment for them. P2) Juvenile killers are not as responsible for their crimes as adults are. C) Therefore, execution is not a justifiable punishment for them. R = child killers are as responsible for their crime as adult killers are E = execution is a justifiable punishment for child killers

6.4 Truth Tables For Arguments P1) If R, then E P2) Not R C) Not E R = child killers are as responsible for their crime as adult killers are E = execution is a justifiable punishment for child killers P1) R ⊃ E P2) ~ R C) ~ E

6.4 Truth Tables For Arguments P1) R ⊃ E P2) ~ R C) ~ E R ⊃ E / ~ R //~ E T T TF T T F FF TT F F T TT FF T F T FT F

6.4 Truth Tables For Arguments P1) R ⊃ E P2) ~ R C) ~ E R ⊃ E / ~ R //~ E T T TF T T F FF TT F F T TT FF T F T FT F The argument is invalid!

6.4 Truth Tables For Arguments This strategy uses the definition of validity to check arguments. Remember, valid arguments are those in which it is impossible for the premises to be true and the conclusion false at the same time. If the premises are true, the conclusion must be true.