Uses for Truth Tables Determine the truth conditions for any compound statementDetermine the truth conditions for any compound statement Determine whether a statement is a tautology, a contradiction or neither,Determine whether a statement is a tautology, a contradiction or neither, Determine whether two formulae are equivalent.Determine whether two formulae are equivalent. Determine whether an argument is valid or not.Determine whether an argument is valid or not.

Logical Implication n One statement logically implies another if, but only if, whenever the first is true, the second is true as well n If a statement, S 1, implies S 2 then the conditional (S 1  S 2 ) will be a tautology. Implication is the validity of the conditional. n Truth tables allow one to test for logical implication

Validity and Logical Implication n An argument is valid if, but only if, its premises logically imply its conclusion.

Deductive Validity n A characteristic of arguments in which the truth of the premises guarantees the truth of the conclusion. n A characteristic of arguments in which the premises logically imply the conclusion.

Truth Table Tests for Validity n n Construct a column for each premise in the argument n n Construct a column for the conclusion n n Examine each row of the truth table. Is there a row in which all the premises are true and the conclusion is false. If so, the argument is non- valid. If not, then the argument is valid.

Truth-Table Test for Validity for R  S, R  S Modus Ponens (MP) R  S R  T  T S  4  3 T2 T1  S S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T  T   T T  T T T VALID NO

Truth-Table Test for Validity for R  S, S  R Fallacy of Affirming the Consequent R  S S  T  T S  4  3 T2 T1 RR Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T YES NON-VALID T  T  T  T   T T

Truth-Table Test for Validity for R  S, ~S  ~R Modus Tollens (MT) R  S ~S  T  T S  4  3 T2 T1  ~R Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T  T T  T  T T   VALID NO

Truth-Table Test for Validity for R  S, ~R  ~S Evil Twin of Modus Toll ens R  S ~R~R  T  T S  4  3 T2 T1  ~S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid? T T  T T T   T  T  NON-VALID YES

Truth-Table Test for Validity for R  S, ~R  S Disjunctive Syllogism (DS) R  S  ~R  T  T S  4  3 T2 T1  S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid?  T T T T T   T  T VALID NO

Truth-Table Test for Validity for R  S, R  ~S R  SR  T  T S  4  3 T2 T1 ~S~S Rcase Is there a case in which the premises are all true but the conclusion is false? Is the argument form valid or non-valid?  YES NON-VALID T T T   T T T  T 

Key Ideas n Grouping and Meaning n Paraphrasing Inward n Truth Functional Operators n Truth Tables Using Truth TablesUsing Truth Tables Testing for tautologies and contradictionsTesting for tautologies and contradictions Testing for equivalenceTesting for equivalence n Testing for validity