Consider the quantified statements L Forall xPx doubleheadar

Consider the quantified statements L: Forall x(P(x) doubleheadarrow Q(x)) and R: Forall xP(x) doubleheadarrow Forall xQ(x). Establish that L and R are not logically equivalent by specifying an appropriate universe for x and meanings for P(x) and Q(x) and showing that L and R have different truth values for this choice.

Solution

The above mentioned statements L and R are not logically equivalent. Before proving the logical unequivalence between the two using truth table let us have a look on some explanation that will prove that they are not logically equivalent.

Let us take two elements a and b in any domain, such that P(a) is true and Q(a) is false and P(b) is false and Q(b) is true.

Now for statement L: The propositions for all x P(x) is false, since P(b) is. Also, since Q(a) is false then for all x Q(x) is also. So because both propositions are false, the biconditional for all x P(x) implies for all x Q(x) is true. However, the biconditional P(a) implies Q(a) is false since the two prositions have different truth values.The same can be said for P(b) implies Q(b).Consequently, the statement for all x (P(x) implies Q(x)) is false. Since for all x (P(x) implies Q(x)) is false for this domain while for all x P(x) implies for all x Q(x) is true, thus the two statements are not logically equivalent