Consider the following two statements -
- "All apples are delicious"
- "Some apples are delicious"
and, consider the following four domain sets -
- {delicious apple 1, delicious apple 2, delicious orange, unpleasant mango}
- {delicious apple, unpleasant apple, delicious orange, unpleasant mango}
- {unpleasant apple 1, unpleasant apple 2, delicious orange, unpleasant mango}
- {delicious orange, unpleasant orange, delicious mango, unpleasant mango}
Also, let P(x) be "x is an apple", and Q(x) be "x is delicious".
Now, let's look at the truth values of the following with respect to each set -
$\forall x, P(x) \implies Q(x)$ is True in Set 1, False in Set 2, False in Set 3, and True in Set 4.
$\forall x, P(x) \land Q(x)$ is False in Set 1, False in Set 2, False in Set 3, and False in Set 4.
$\exists x, P(x) \implies Q(x)$ is True in Set 1, True in Set 2, True in Set 3, and True in Set 4.
$\exists x, P(x) \land Q(x)$ is True in Set 1, True in Set 2, False in Set 3, and False in Set 4.
Now, "All apples are delicious" is True in Set 1, False in Set 2, and False in Set 3. However, I'm not sure about its truth value in Set 4 according to our natural language usage.
Similarly, "Some apples are delicious" is True in Set 1, True in Set 2, and False in Set 3. Again, I'm not sure about its truth value in Set 4.
If we go with the accepted rule of associating conjunction with the existential quantifier and implication with the universal quantifier, then "All apples are delicious" must be True in Set 4, and "Some apples are delicious" must be False in Set 4.
However, I'm not sure whether I'm entirely convinced by this, as there are no apples in Set 4 at all.
So, does natural language actually correspond to the aforementioned accepted rule, or are we forced to accept it just because we don't have other logical connectives and quantifiers, and we are thus forced to make do with what we have?
