Syllogisms rewritten in first order logic

Question

When given the syllogism "Some x are y" (e.g. "Some red things are fruits"), is this syllogism best rewritten in first order logic as $(\exists x (R(x) \rightarrow F(x))$ or $(\exists x (R(x) \wedge F(x))$? Furthermore, if the first one, how might one then prove that $\exists x (F(x) \rightarrow R(x))$? I'm having a bit of a hard time wrapping my mind around proofs with the existential quantifier.

Thanks!

Answer 1

You want the statement to be true only when you can find a specific example of an object in the domain that is a red fruit. In the first case, $\exists x (R(x) \rightarrow F(x))$ is true whenever there are no red things, so it definitely does not carry the meaning of "Some red things are fruits." The second one, $\exists x (R(x) \wedge F(x))$ is true if and only if you can find an object in the domain that is both red and a fruit (that is, a red fruit).

Answer 2

Don't worry about proofs until you clearly understand translation - once you understand what the existential quantifier means, proofs will be easy.

$(\exists x)(R(x) \rightarrow F(x))$ says "there is a thing so that if that thing is red, then that thing is a fruit". That's practically the weakest statement possible - remember, $P \rightarrow Q$ is always true if $P$ is false. So all that's needed to make $(\exists x)(R(x) \rightarrow F(x))$ true is to have one thing that is not red. That's clearly not what "some red things are fruits" is supposed to say.

"Some red things are fruits" means "there are some things which are red and are also fruits". This translates directly into $(\exists x)(R(x) \wedge F(x))$: "there is a thing which is red and is a fruit".

As a general rule of thumb, $\rightarrow$ and $\exists$ do not play nicely together - using both of them in the same sentence almost always results in something wildly weaker than what you wanted. $\rightarrow$ and $\forall$ can combine to make useful statements - for example, $(\forall x)(R(x) \rightarrow F(x))$ means "for each thing, if that thing is red then it is a fruit" or (put more naturally) "everything red is a fruit". On the flip side, $\wedge$ and $\exists$ work well together, but $\wedge$ and $\forall$ usually produce something much more powerful than intended - see $(\forall x)(R(x) \wedge F(x))$ "everything is a red fruit".