[Math] Why can we use implication with the universal quantifier but not with the existential quantifier

first-order-logiclogicquantifiers

I've been looking at this question amongst others and am beginning to get a better understanding of quantifiers when used with implication and conjunction, however the accepted answer posted by Dustan Levenstein is raising a question for me. I understand the reasons of why using implication for an existential quantification is a bad idea as the following example describes:

Using implication with existential quantification:

Using the statement, There is at least one piece of fruit in the bowl that is an apple that is delicious then using implication gives the following:

$$\exists{x} \in F, A(x) \implies D(x)$$

Is not best practice because as Dustan Levenstein states in his answer, it doesn't quite capture the exact sentiments of the statement as we are also capturing fruit which is not an apple – regardless of whether it is delicious or not, as per the truth table for implication states:

Implication truth table

Using implication with universal quantification:

But by the same token, why can we then use implication with the universal quantification and not have the same problem?

$$\forall{x} \in F, A(x) \implies D(x)$$

Doesn't this then say using natural language:

  • Every piece of fruit in the bowl is an apple and is delicious (First row of the implication truth table) – which is our intended statement.
  • Every piece of fruit in the bowl is not an apple and is delicious (Third row of the implication truth table).
  • Every piece of fruit in the bowl is not an apple and is not delicious (Fourth row of the implication truth table).

So, it would seem that using implication with universal quantification leads to the original problem of using implication with existential quantification – i.e. that we aren't capturing the true intention. Yet according to all explainations I've found, this is not the case. Why?

Further Explaination

This isn't to take anything away from the answer provided by Mees De Vries but to explain my faulty thinking which brought about this problem.

If we use the universal quantification and implication, we have:

$$\forall{x} \in F, A(x) \implies D(x)$$

Which as Mees De Vries says:

For every piece of fruit in the fruit bowl, one of the following is true:

  • It is an apple and it is delicious.
  • It is not an apple and it is delicious.
  • It is not an apple and it is not delious.

Now it's not a problem to have three truth results for this. Because the truth table still lines up with the English statement – every piece of fruit we pick up could be a delicious apple, or to put another way, every apple we pick up is delicious.

The key issue here is that using the universal quantifier with implication is not about saying every piece of fruit you pick up will be an apple and delicious. We are just saying picking up a delicious apple is one of three possibilities (the other two being that it's not an apple, delicious or otherwise).

If we wanted to ignore the last two possiblities, we are then saying you pick a piece of fruit up, and it has to be an apple and it has to be delicious, because there are no other possibilities. In this case using implication no longer gives the correct truth table, so we need to use conjunction:

$$\forall{x} \in F, A(x) \land D(x)$$

Which has a truth table:

Conjunction truth table

This now matches the English statement that every piece of fruit you pick up is an apple and delicious.

Best Answer

Your universal statement $\forall x \in F(A(x) \to D(x))$ states, in natural language, Every apple in the fruit bowl is delicious, or in other words, Every piece of fruit in the fruit bowl which is an apple, is delicious. If you intend to say Every fruit in the bowl is a delicious apple, or equivalently Every fruit in the bowl is delicious and is an apple, or The fruit bowl contains only delicious apples, then indeed the translation into formal language should be $\forall x \in F(A(x) \land D(x))$.

At the end of your question, you split your implication into three cases, but you put the universal quantifier in front of each case. This might be what is causing confusion. A better way of putting it would be:

For every piece of fruit in the fruit bowl, one of the following is true:

  • It is an apple and it is delicious.
  • It is not an apple and it is delicious.
  • It is not an apple and it is not delious.

In other words, the only case that is not possible is that there is a non-delicious apple. But, again, that is another way of saying that, disregarding all other fruits, every apple in the fruit bowl is delicious. So it corresponds to the correct translation of $\forall x\in F(A(x) \to D(x))$, as it should.

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