Question Title may seems similar but Its not a Duplicate Question.
Question:-
Suppose I want to formulate a statement "All Apples are Delicious.
Let F be the domain of fruits and
A(x) : is an apple
D(x) : is delicious
Let's say:
∀x∈F,
A(x)⟹D(x)
Is Correct and means all apples are delicious.
And
∃x∈F,A(x)⟹D(x)
It means there is some fruit that if it is an apple, it is delicious.
It is Incorrect because if x is not an apple then also it is true i.e it also includes other fruits
My Problem:-
Why we can't apply the same logic (that is if x is not an apple then also it is true.) for Universal Quantification (∀x∈F, A(x)⟹D(x))?
Why Only for Existential Quantifier we don't want it to hold true in a universe that doesn't contain any apples why not for Universal Quantifier?
P.S. :- Sorry in advance because my English is not upto that mark. Edits are welcome 🙂
Best Answer
I'm going to answer the question I think you're trying to ask, so correct me if I'm wrong.
First, you want to express the sentence "All apples are delicious," and you correctly say $$\forall x \in F,A(x) \Rightarrow D(x)$$ Note that if all apples are delicious, this statement is true of every fruit. If the fruit is an apple, it's true because the apple is delicious. If the fruit is not an apple, it's true because the hypothesis is false, and the statement is true.
Now I guess you wanted to express the statement there is some apple that is not delicious, which you should have written as $$\exists x \in F,A(x) \wedge D(x),$$ or something equivalent. Instead you wrote $$\exists x \in F,A(x) \Rightarrow D(x)$$ Now, this statement can fail in two ways. First it may be that no apple is delicious. Second, there may be some fruit that isn't an apple. Once again, if the hypothesis is false, the statement is true. That was very convenient in the first case. I want to make a statement that formally applies to every fruit, although I'm really just talking about apples. The convention that an implication with a false hypothesis is always true is just what we need. However, trying to use it in the second case backfires. Remember, to prove an existential statement, it's enough to produce one example. We don't want that to be an example of something were not even really talking about.
I think beginners often find that technicalities obscure what is really quite straightforward. Universal quantifiers at the beginning of a sentence are actually sort of redundant. All you really want to say is "All apples are delicious." With only the predicates $A$ and $D$ at your disposal, what can you say but "if it's an apple, then it's delicious?" The other case, is similar, all you can is, "There is some thing that is both an apple and delicious."