first-order predicate calculus

Question

We are given some functions, what they mean, and some statements and told to write the English sentence that describes the statement. I am having some trouble with this. Here is what we are given.

"Let $B(x)$ mean $x$ is a bird, let $W(x)$ mean $x$ is a worm, and let $E(x, y)$ mean $x$ eats $y$."

We are told to write the English sentence of $\forall x \forall y(E(x, y) \rightarrow B(x) \land W(y))$

This is where I get stuck. We were given an example where $\forall x \forall y(B(x) \land W(y) \rightarrow E(x, y))$ which says, "Every bird eats every worm," which makes sense since B(x) and W(y) are on the left side of the implication arrow. I'm just not sure how to describe it with the E(x, y) on the left. Are these two statements the same thing? Any help would be appreciated.

Thanks

Answer 1

$E(x,y)→B(x)∧W(y)$ reads :

"if $x$ eats $y$, then $x$ is a bird and $y$ is a worm".

Thus we can try with :

"if something eats something, the first one is a bird and the second one is a worm"

that looks quite "weird".

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The two statements have not the same meaning.

To show it, consider an interpretation in the set $\mathbb Z$ of integers; $B(x)$ is "$x$ is negative", $W(x)$ is "$x$ is positive" and $E(x,y)$ is "$x$ is less than $y$".

With this interpretation, we have that :

"if $x$ is negative and $y$ is positive, then $x$ is less than $y$"

is true, while :

"if $x$ is less than $y$, then $x$ is negative and $y$ is positive"

is false.