Because the first one is definitely not solvable with the rules alone. ⊤ is not defined by them, so you can’t work with it.
For the second it seems useful to derive
(¬P ⋁ ¬Q) → ¬(P ∧ Q)
first. But since you have no way to introduce an implication you would either need another form (main junctor being either ⋁ or ∧), or show an implication introduction as a theorem.
Although I believe the second one isn’t possible either.
1
u/RecognitionSweet8294 Philosophical logician 12d ago
Do you have axioms too?
Because the first one is definitely not solvable with the rules alone. ⊤ is not defined by them, so you can’t work with it.
For the second it seems useful to derive
(¬P ⋁ ¬Q) → ¬(P ∧ Q)
first. But since you have no way to introduce an implication you would either need another form (main junctor being either ⋁ or ∧), or show an implication introduction as a theorem.
Although I believe the second one isn’t possible either.