Consider the FOL formula: ((∃y.(P(y))) ∧ (∃y.(Q(y)))) →(∃y.(P(y) ∧ Q(y))). (a) Is this formula valid? (b) Is this formula satisfiable? (c) Is this formula contingent? Justify your answers.

(a) The formula is not valid. Validity in first-order logic (FOL) means that the formula is true in all possible interpretations or models. However, this formula is not always true. For example, consider an interpretation where P(y) is true for some object 'a' and Q(y) is true for some different object 'b'. In this case, the antecedent of the implication is true (since there exists an 'a' such that P(a) is true and there exists a 'b' such that Q(b) is true), but the consequent is false (since there is no single 'y' for which both P(y) and Q(y) are true). Therefore, the whole implication is false in this interpretation, so the formula is not valid.

(b) The formula is satisfiable. Satisfiability in FOL means that there is some interpretation or model in which the formula is true. Consider an interpretation where P(y) and Q(y) are both true for the same object 'c'. In this case, both the antecedent and the consequent of the implication are true, so the whole implication (and thus the whole formula) is true. Therefore, the formula is satisfiable.

(c) The formula is contingent. Contingency in FOL means that the formula is neither valid (true in all interpretations) nor unsatisfiable (false in all interpretations). As we've seen in parts (a) and (b), this formula is not valid, but it is satisfiable. Therefore, it is contingent.