We are given a sentence and told to write the quantifier statement that corresponds with it. I had a similar question like this the other day, but this one goes the opposite way.
Let $B(x)$ mean $x$ is a bird, $W(x)$ mean $x$ is a worm, and $E(x, y)$ mean $x$ eats $y$.
We are told to give the statement for the sentence "Not all birds eat worms." Here is what I am thinking so far:
- $\exists x(B(x) \rightarrow \forall y(W(y) \land \lnot E(x, y)))$
I am not sure if this is the correct statement or not. What I think my answer says is, "There exists a bird such that for all worms, the bird does not eat worms." Since there exists a bird that does not eat worms, doesn't that satisfy the sentence "Not all birds eat worms"?
I also came up with another idea but I'm not sure which one is right:
- $\exists x \forall y(B(x) \land W(y) \rightarrow \lnot E(x, y))$
Which one is correct? (If any of them are) Both make a lot of sense to me which is why I don't know which one is it. The second one to me says, "There is a bird as for all worms such that the bird does not eat worms." Any help on going about solving these types of problems?
Your first answer isn't quite correct. It says "There is a thing such that if it is a bird, then for all objects $y$, $y$ is a worm and $x$ doesn't eat $y$". That isn't what you meant to say: I, for instance, satisfy that predicate.
The second attempt has the same problem: let $x$ be me. Then the predicate is satisfied because $B(x)$ is never true.
"Not all birds eat worms" is "there is an object which is a bird and which does not eat worms", or "there is an object which is a bird, and such that for all objects which are worms, the bird-object doesn't eat the worm-object".