I'm studying for a logic exam I've come across the following L-formula
$$ \forall i (\operatorname{in}(i,xs) \Rightarrow \exists j (i = 2 \cdot j)) $$
Where the $\operatorname{in}(x,xs)$ predicate means "$x$ is in $xs$".
To my knowledge, I translated this to English as "Everything in $xs$ is twice as big as some number"
But it does not say that $j$ is in the list. Is my thinking correct to say that an example of a list with 4 elements that satifies this sentence would be something like
$$[2,4,8,16]$$
And a list that violates the formula would be something like
$$[0,0,0,0]$$
The empty list does satisfy this sentence. I am quite unsure about an example of a list that violates the sentence.
I am unsure if I have translated the sentence correctly. Does anyone see a problem here?
The interpretation depends on your universe of discourse (i.e., what kind of object all varables stand for). I shall assume that it is $\mathbb Z$. Then $\exists j(i=2j)$ is true for $i=42$ because $i=2j$ with $j=21$. More generally, $\exists j(i=2j)$ is true iff $i$ is an eve number. So your original statement transalates as: All members of $xs$ are even. (Or: $xs$ is a subset of the set of even numbers). Note that indeed $j$ may or may not be in $xs$, it can be any number (here: integer). Hence $[0,0,0,0]$ would satisfy the formula! On the other hand $[2,4,6,8,10,12,1001]$ violates ist - just one odd member suffices.