thanks in advance for your help, I feel like I am losing my mind over this. I can't understand the difference between two formulas:
- $\exists x (Px \rightarrow \forall y Py)$
- $(\exists x Px \rightarrow \forall y Py)$
Again, I don't understand why it makes a difference, whether the existential quantifier is outside the brackets or not. 1. is a tautology and the second is not. I can think of a counterexample for 2. but I feel like it should be possible to get a counterexample for the first. Sorry for my studipity.
The difference is that in the first statement the proposition is that there exists an $x$ such that the statement $(Px\rightarrow \forall yPy)$ is true. This is indeed tautological since, if $\forall yPy$ the implication is true for each $x$. If not $\forall yPy$ there exists an $x$ such that $\neg Px$. implying that for given $x$ the statement $Px\rightarrow\forall yPy$ holds true.
In the second proposition, this is an implication. The implication that $\exists xPx$ implies $\forall yPy$. You could write it with the following parentheses $(\exists xPx)\rightarrow(\forall yPy)$. A counterexample to this being tautological, is for a statement $P$ that is true for some but not all $x$. (For example $x^2 = 1$ in the theory of integer arithmetic.) In this case the premise is true $\exists xPx$, but the implication does not follow $\forall xPx$.