\exists quantifier and being explicit about quantity

Question

I have a question regarding the use of the $ \exists $ quantifier when trying to signify explicit quantities.

Consider

$F:$ the set of all fruits

$A(x):$ x is an apple

$R(x):$ x is rotten

If I wanted to express 'One apple is rotten' would this be logically sound?

$\exists x \in F, \forall y \in F, x \neq y \wedge R(x) \wedge \neg R(y)$

Also how would the above be different from:

$\exists x \in F, \forall y \in F, x \neq y \Rightarrow R(x) \wedge \neg R(y)$

Thanks!

Answer 1

As Asaf pointed out, there are a couple of flaws with how you've formulated your expressions. The following should help:

"At least one apple is rotten." translates to "There is a fruit (call it $x$) that is both an apple and rotten.": $$ \exists x \in F~~~[A(x) \land R(x)] $$

"Exactly one apple is rotten." translates to "There is a particular type of fruit (call it $x$) such that for any other fruit (call it $y$), the other fruit can only be an apple and be rotten if and only if the other fruit was actually the particular type of fruit.": $$ \exists x \in F~~~\forall y \in F,~~[A(y) \land R(y) \iff x=y] $$

Answer 2

In the first form, you say that every member of $F$ is not $x$, and thus you allow $x$ itself. So essentially you are stating that amongst other things, $x\neq x$. This is impossible, of course.

On the other hand, in the second form you say that if $x\neq y$, then $y$ is not rotten. This means that it is possible that $x$ itself is rotten.

Answer 3

Here is a slightly different approach that can be easily generalized to assert the existence of exactly $n$ objects with a given property $P$. (For example, $P(x)$ could be defined as $A(x) \wedge R(x)$, meaning that $x$ is a rotten apple.)

Define the sentence $\psi_n$ saying $$\exists x_1 \cdots \exists x_n\, \left(\bigwedge_{1 \le i < j \le n} x_i \ne x_j \wedge \bigwedge_{1 \le i \le n} P(x_i)\right).$$ This says that there are $n$ objects, all different (no two are the same) and all with property $P$. So it says that at least $n$ objects have property $P$. Then the sentence $\psi_n \wedge \neg \psi_{n+1}$ says that exactly $n$ objects have property $P$.

For example, the sentence $$\left(\exists x_1\,P(x_1)\right) \wedge \neg\left(\exists x_1\exists x_2\, \left(x_1 \ne x_2 \wedge P(x_1) \wedge P(x_2)\right)\right)$$ says that exactly one object has property $P$.