HSS 2016 Solutions to Problem Set II-1

# HSS 2016 Solutions to Problem Set II-1 - Deductive Logic Dr...

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Deductive Logic Dr. James Pearson Solutions to Problem Set II 1. a) p q –p –q –p . –q p q –(p q) Since the last column is different to the fifth column on both the second and fourth rows, these schemata are not logically equivalent. [Note, you could have determined this after completing just two rows of the table.] b) p q –p –q p q – (p q) p . –q –p . q (p . –q) (–p . q) Since the fifth column and the last column are identical, these schemata are logically equivalent. c) p q r p . q q r (p . q) r p (q r) Since the final two columns are identical, these schemata are logically equivalent.

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2. (a) p q r –r p . –r q . –r (p . –r) (q . –r) p q ß !!! ß !!! In the third and fifth rows, the first schema is true while the second schema is false, so the first schema does not imply the second. A truth-value assignment establishing the failure is where “p” and “r” are both true, and “q” is false. Another is where p” is false, and both “q” and “r” are true. [Note, you could have determined this after completing just three rows of the table.] (b) p q r –p q r q . r p (q r) –p (q . r) ß !!! Since in the last row, the first schema is true while the second schema is false, the first schema does not imply the second. A truth-value assignment exhibiting this failure is where “p”, “q”, and “r” are all false. (c) p q r p q –(p q) r . q – (p q) (r .q) q r ß !!! ß !!!
In the second and sixth rows, the first schema is true while the second schema is false, so the first schema does not imply the second.

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