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  1. ~∀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.

أحمد الدسوقي
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    Universal Instantiation is: "from $\forall x Px$, derive $Pa$ for $a$ whatever". The universal quantifier must be the outer part of the formula. – Mauro ALLEGRANZA Sep 23 '22 at 15:28
  • Thus, the rule is: "we can't instantiate a negated universally quantified statement." – Mauro ALLEGRANZA Sep 23 '22 at 15:29
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    From $\sim\forall x(A(x))$ alone, it is not certain that there are objects that are apples (or satisfy $A(x)$). – peterwhy Sep 23 '22 at 15:36
  • @peterwhy So, are you saying that it is possible for all objects not to have the property A? – أحمد الدسوقي Sep 23 '22 at 15:39
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    All @peterwhy is saying is that even if the statement $\sim!\forall x(A(x))$ is known to be true, although you may conclude the existence of an object $x$ that does NOT satisfy $A(x)$, without knowing anything more about the nature of $A(x)$ you may not draw any general conclusions about the existence of an object $x$ that DOES satisfy $A(x)$. – Lee Mosher Sep 23 '22 at 17:34

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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.

whoisit
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  • You are saying that statement 1 allows the possibility of all the objects in the universe not having the property of being an apple? And therefore, we don't have 100% certainty that will allow us to instantiate statement 1. – أحمد الدسوقي Sep 23 '22 at 17:42