Quantifiers and vs if then connective

Question

Consider the statement: Jane saw a police officer and Roger saw one too. The answer given is: (re-edited) $\exists x (P(x) \land S(j,x)) \land\exists y (P(y) \land S(r,y))$ where P is police, S(w,y) is w saw y. Is this synonymous with the statement: There exists an x such that if x is a police officer, then Jane saw the police officer and there exists a y such that if y is a police officer, then Roger saw the police officer: $\exists x (P(x) \implies S(j,x)) \land \exists y(P(y)\implies S(r,y))$

It makes sense to me through reasoning but I haven't learned truth tables for quantifiers and if you rewrite the if then statement into $\neg P \vee Q$ it seems like it won't work.

What about: Jane saw a police officer, and Roger saw him too. The correct answer is: $\exists x (P(x) \land S(j,x) \land S(r,x))$ Can I rewrite this as: There exists an x such that if x is a police officer then Jane saw the police officer and Roger saw the police officer: $\exists x (P(x) \implies S(j,x) \land S(r,x))$

Thanks a bunch.

Answer 1

The problem with your proposed statements is as follows. Consider: “There exists an $x$ such that if $x$ is a police officer, then Jane saw [$x$] and there exists a $y$ such that if $y$ is a police officer, then Roger saw [$y$].” Let $x$ and $y$ be any two things that are not police officers. Such things obviously exist. Then, this statement is vacuously true: remember, a false premise implies anything. (The implication “if P, then Q” is true whenever P is false). In particular, this statement holds true irrespective of whether Jane and Roger saw police officers or not. Therefore, this statement and the original one are not synonymous.

Answer 2

First of all, there are no truth tables for predicates. But you are on the right track about rewriting your statement.

As you point out, the statement $\exists x (P(x) \implies S(j,x))$ is indeed equivalent to $\exists x \neg(P(x) \land \neg S(j,x)) $.

I think you can see that this is quite different from $\exists x (P(x) \land S(j,x)) $.

EDIT: If $\forall x \neg P(x)$, then $\exists x (P(x) \land S(j,x)) $ will be false and $\exists x \neg(P(x) \land \neg S(j,x)) $ will be true (assuming a non-empty domain of discussion). So, in general, the two statements are not identical.

Answer 3

If by "synonymous" you mean materially equivalent ($\leftrightarrow$), then the answer is "no".

1. $\exists x (P(x) \land S(j,x)) \land\exists y (P(y) \land S(r,y))$

2. $\exists x (P(x) \implies S(j,x)) \land \exists y(P(y)\implies S(r,y))$

3. $\exists x (P(x) \land S(j,x) \land S(r,x))$

4. $\exists x (P(x) \implies S(j,x) \land S(r,x))$

Consider a world with only one inhabitant in it who is not a police, say: Robinson Crusoe on his island. The island is a counterexample to the material equivalence of (1&2 and 3&4) because: (1) & (3) are false due to there being no police on it, while (2) & (3) are true because there is someone on it who's not a police, namely, Mr. Crusoe.

Answer 4

Your question is whether statements $\exists x (P(x) \land Q(x))$
and $ \exists x (P(x) \to Q(x)) $ both work as equally good renditions of vernacular "Some P are Q." or if you like "There exists atleast one P that is Q."

It becomes clear that the second rendition won't do once you realize that in the case where no object in the domain is actually in the extension of the predicate P (ie, there are no Ps), the antecedent of your conditional is false, and that makes the conditional true not only for at least one x, but for every x. This clearly shows that it can't be used as a translation of a given vernacular statement, which clearly implies the existence of at least one P.

To put it in another way:

The statement "There exists at least one object that if it is a table, then it is wooden" is true if there is at least one object that is not a table (because this would make the conditional statement where the name of that object is substituted for the variable true, as the antecedent would be false).

Notice, by the way, that this only applies when the vernacular "if... then ..." is taken to be properly translated by using material conditional connective. I mention this because, in fact, the intuition in cases like this one might point to counterfactual conditional as a better option (if it was the case that..., than it would be the case that ...) But as we are in classical logic, we standardly understand it as the former.

Counterfactual conditionals are not truth-functional as the value of propositions formed by using them is not a function of the actual truth values of the constituent propositions.

On the other hand, the statement: "There exists at least one object that is a table and is wooden." is true only if there exists at least one object that is a table.

For the statement "There is at least one x such that if it is a table, then it is wooden." rendered to formal language by using material conditional to be true, it is enough that there is one object that is not a table.

If this puzzles you, have in mind that this is due to the way material conditional behaves.

It is an idealization and as such, in many cases, it seems that it is not an adequate translation for everyday "if... then...".

It is simply the only plausible candidate amongst truth-functional two-place connectives that can be constructed in classical logic for rendering everyday statements using "if... then..." to formal language.