For Friday, read Chapter 3, section 4. Nongraded Homework: Problems at the end of section 4, set I only; Power of Logic web tutor, 7.4, A, B, and C. Graded homework #3 is due at the beginning of class on Friday.

A B A → [~ (A & B) → ~ B] T T T   T T  T  T T  T T  T T   T * Answer: Tautology

F G ~ (F → G) & G T T  T  T    T  T     * Answer: Contradiction

Using truth-tables to test for logical equivalence: Make a truth-table that includes both formulae. If their truth-values match on each row, they are logically equivalent; if there is even one line where the truth-values of the two statements (as a whole) don’t match, the statements are not logically equivalent.

D G ~ D v ~ G ~ (D & G) T T     T T   T T T   T T T  T    T T T T  * * * * On each line, the value of the m.o.’s match each other. So, these two statements are logically equivalent.

Exercises on p

Using Truth-tables to test arguments for validity. Place the entire argument on a truth-table: list the premises from left to right, separated by commas; put the conclusion on the far right after a ‘  ’  Fill out the table. If there is at least one row where all premises are true and the conclusion false, then the argument is invalid; if there is no row with all true premises and a false conclusion, the argument is valid.

Z S GZ → (S v G), Z & G  S T T T T T T T T T  T T  T T  T T T T  ***** T        T T T T  T  T  T T  T   T T T      T    * * * * * * Answer: Invalid

Why does it work? Each row on the truth-table represents a relevant possibility (an interpretation), and taken together the rows represent all of the relevant possibilities (all possible interpretations). So, if there is a row on the truth-table with all true premises and a false conclusion, then it is possible to have all true premises and a false conclusion—thus the argument is invalid; and if there is no row with all true premises and a false conclusion, it is impossible to have all true premises and a false conclusion, and by definition, the argument is valid.

P Q SP → Q, S → Q, ~ Q  ~ P & ~ S T T T T T     T T  T T    T T  T   T    T    T T   T  T T T T  T    T  T T  T T T   T T  T T      T T T T T T

Answer? Valid. There is no line showing all true premises and a false conclusion. Thus, it is impossible for the argument to have all true premises and a false conclusion. So, the reasoning can’t go wrong. Whenever the premises are all true, the conclusion is as well. Remember, when an argument is valid, there is no particular line that proves validity.

A B GA → (B & G), ~ B  A  ~ B T T T T  T T    T  T   T      T T T   T  T    T T T   T F F F Answer: Invalid, proven by line seven