### Due Friday January 17.

PLACE ASSIGNMENTS IN THE ASSIGNMENT BOXES (by section) ON ROSS 5th FLOOR.
1. Create truth tables for the following sentences.(This can be done by hand or on SYMLOG.
(a) ([(A -> C) & (B -> C)] -> [((B v A) -> C)]
(b) ([¬ (A v B) & C]-> [( ¬ A & C) -> (B & C)])
(c) ([¬ A -> A] -> A)

2. Give a 'parsing tree' for each sentence in 1 to demonstrate that the sentence is a well-formed formula of SL.

3. Two formulae which have the same final column in a truth table are logically equivalent ( truth-functionally equivalent ).
(a) How many distinct truth tables are possible, using exactly one letter (D) and logical connectives.
(b) Give a well-formed formula for each of the possible truth tables in (a).
(c) Give an informal argument that every well-formed formulae, with one letter (D) and logical connectives, is logically equivalent to one of the formulae you gave in (b).
(d) Can two well formed formulae, one with the letter D, one with the letter A, be logically equivalent?
4. A sentence is truth-functionally true (or a tautology) if every line of its truth table makes it true.
A sentence is truth-functionally false (or a contradiction) if every line of its truth table makes it false.
A sentence is truth-functionally indeterminate (or contingent ) if some line of its truth table makes it false and some line makes it true.
(a) For the sentences in 1, state which are truth-functionally true, truth- functionally false and truth-functionally indeterminate.

(b) Consider two wffs of SL: \alpha formed using only the atomic letter A, and \beta formed using only the atomic letter B. Give informal arguments to show that:
(i) If \alpha and \beta are tautologies, then they are logically equivalent.
(ii) If \alpha and \beta are contingent, then they are not logically equivalent.
(iii) If \alpha and \beta are contradictions, then they are logically equivalent.

NOTE: Future assignments will require you to do exercises from the text and to include printouts from the program SYMLOG. Be sure you have access to these by next week.