- ~∀x(A(x))
Let A represent the category of things that are apples. Then, statement 1 is saying: it is not the case that everything is an apple. This means that we can have things that are apples and things that aren't apples. For example, 80% percent of the universe can be apples whereas the other 20% can be anything else. So, why can't we instantiate statement 1 by saying that we have an object that is an apple. Specifically, if I made the inference that Sara is an apple as an example of universally instantiating statement 1, why would this be incorrect? Would it be incorrect because we don't have the certainty as which object is an apple and which is not? In other words, we know that are objects that are apples but we don't have the 100% certainty that will allow us to talk about a certain instance?
The book I am reading says that we can't instantiate a negated universally quantified statement without providing any explanation. In my opinion I think the book would be correct if statement 1 is open to the interpretation that all items in the universe aren't apples. According to such interpretation we can't have anything that is an apple, thus, instantiating the first statement would logically invalid.
Counter Example:
Consider a universe $S = \{pear_1, pear_2\}$
For both $s \in S$, $A(s)$ is false.
That is, $\forall sA(s)$ is a false statement. That is, $\lnot \forall sA(s)$ is a true statement.
If we allow what you say, we'd be able to say that there exists an apple. But there is no apple in $S$, yet $\lnot \forall sA(s)$ is a true statement.