∀x:X.¬(P^Q) ¬∃x:X. ¬(¬P v ¬Q)
I want to prove that the left hand side entails () the right hand side using propositional and predicate logic. Thank you
Do P and Q depend on x? Also, see this sticky thread.
Hmm, I am not sure I understand "p and q are of type x." In terms of types, I would say, "P and Q are of type X -> Prop" where Prop is the type of propositions. But I understand that P and Q are unary predicates that accept arguments of type X.
A derivation depends on the axioms you have. For example, if you have an axiom (¬∃x. ¬Ax) <-> ∀x. Ax, then you can immediately use it to derive ¬∃x:X. ¬¬(Px ^ Qx). Similarly, if you have an axiom A ^ B <-> ¬(¬A v ¬B), then you can derive ¬∃x:X. ¬(¬Px v ¬Qx). Otherwise, it is significantly more tricky.
Also, do you use Fitch-style natural deduction (also called flag notation) or tree-like natural deduction?
P ^ ¬P is not an axiom since it is contradictory.
I'll describe the derivation in words. Assume $\displaystyle \exists x\,\neg(\neg Px\lor\neg Qx)$. We need to derive a contradiction. Apply existential elimination to get an x and $\displaystyle \neg(\neg Px\lor\neg Qx)$. Use universal elimination with that x on $\displaystyle \forall x\,\neg(Px\land Qx)$ to get $\displaystyle \neg(Px\land Qx)$. We are going to make two assumptions and show that they are contradictory.
Assume $\displaystyle Px$ and assume $\displaystyle Qx$. Then $\displaystyle Px\land Qx$, which contradicts $\displaystyle \neg(Px\land Qx)$. Therefore, we close $\displaystyle Px$ and derive $\displaystyle \neg Px$. Using disjunction introduction, this implies $\displaystyle \neg Px\lor \neg Qx$, which contradicts $\displaystyle \neg(\neg Px\lor\neg Qx)$. Therefore, we close $\displaystyle Qx$ and derive $\displaystyle \neg Qx$. As before, this implies $\displaystyle \neg Px\lor \neg Qx$, which contradicts $\displaystyle \neg(\neg Px\lor\neg Qx)$. Thus we closed both $\displaystyle Px$ and $\displaystyle Qx$ and derived a contradiction.
Note that I have not used the law of excluded middle $\displaystyle A\lor\neg A$. Using it, it may be possible to make the derivation a little clearer. Basically, once you have $\displaystyle \neg(\neg Px\lor\neg Qx)$ and $\displaystyle \neg(Px\land Qx)$, you consider (using disjunction elimination) four cases where either $\displaystyle Px$ or $\displaystyle \neg Px$ and either $\displaystyle Qx$ or $\displaystyle \neg Qx$ hold. In each case, it is possible to derive a contradiction using the two premises above. This is similar to constructing a truth table.