In each case, justify your answer: if the answer is yes, give a metalogic proof; if the answer is no, give a counterexample. 4. Do the formulas P and P Q in... → Q together semantically imply the formula (a) ...Łukasiewicz's system? (b) ... Kleene's? (c) ...the logic of paradox? (d) ...the system of supervaluations?

Translate into predicate logic and show the logical computation of the negation of the statement: No duck talks or all ducks squawk. Give a key for your predicate. Additionally, give the negation of the statement in simple English.

Translate the given statement into symbolic form. Be sure to define all pred- icates used. Use the variable a with domain D = {a : a is an animal } as your main variable. Then, negate the symbolic statement. In your answer, negations should appear only in front of predicates. Finally, rewrite the negated statement clearly in plain English. "Any animal that is wild can be tamed." 2.

Let p, q, and r represent the following simple statements. p: I eat bananas. q: I like ice cream. r I like to eat. Write the following compound statement in its symbolic form. If I do not eat bananas then: I do not like ice cream and I do not like to eat. The symbolic form is