[Math] Why can’t we use implication for the existential quantifier

logicquantifiers

I'm not quite sure that I really understand WHY I need to use implication for universal quantification, and conjunction for existential quantification.

Let $F$ be the domain of fruits and

$$A(x) : \text{is an apple}$$

$$D(x) : \text{is delicious}$$

Let's say:
$$\forall{x} \in F, A(x) \implies D(x)$$
Is correct and means all apples are delicous.

Whereas,
$$\forall{x} \in F, A(x) \land D(x)$$
is incorrect because this would be saying that all fruits are apples and delicious which is wrong.

But when it comes to the existential quantifier:
$$\exists{x} \in F, A(x) \land D(x)$$
Is correct and means there is some apple that is delicious.

Also,
$$\exists{x} \in F, A(x) \implies D(x)$$
Is incorrect, but I cannot tell why. To me it says there is some fruit that if it is an apple, it is delicious.

I cannot tell the difference in this case, and why the second case is incorrect?

Best Answer

To me it says there is some fruit that if it is an apple, it is delicious.

This is absolutely correct. There exists a fruit such that if it is an apple, then it is delicious. Let $x$ be such a fruit. We have two cases for what $x$ may be here:

  • $x$ is an apple. Then $x$ is delicious. This is the $x$ you are searching for.
  • $x$ is not an apple. Now the statement "if $x$ is an apple, then $x$ is delicious" automatically holds true. Since $x$ is not an apple, the conclusion doesn't matter. The statement is vacuously true.

So the statement $\exists{x} \in F, A(x) \implies D(x)$ fails to capture precisely your desired values of $x$, i.e., apples which are delicious, because it also includes other fruits.

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