Quantified Negations
We have just seen that it really matters whether the tilde goes before
or after the quantifier. We showed that the negation of a quantified
statement requires that the tilde appear at the far left of the wff. A more
elegant way of describing this is to say that in the negation of a quantified
wff, as in the negation of any wff, the tilde is the main
operator.
When the tilde is "inside" the wff, that is, between the quantifier and the propositional function, we have a quantified wff, since the quantifier is the main operator. We can describe such wffs as quantified negations. Here are some examples of quantified negations and their translations into QL:
| English Sentence | Propositional Function | Translation |
| Nothing is easy. | Ex: E is easy | (x)~Ex |
| Everything is not funny. | Fx: x is funny | (x)~Fx |
| Something is not here. | Hx: x is here | (∃x)~Hx |
There's an important wrinkle: We translated "Nothing is easy" as "For every x, x is not easy." But we could have also translated it as "It's not the case that there is an x, such that x is easy" or more simply "Something is not easy." In general:
| ~(x)Fx | is logically equivalent to | (∃x)~Fx |
| ~(∃x)Fx | is logically equivalent to | (x)~Fx |
Verify this for yourself with some examples. A good way to remember this equivalence is as follows: If you have a negation of a quantified sentence, you can turn it into a quantified negation by moving the negation from the left of the quantifier to the right, and also changing the quantifier. This last step is crucial. If the quantifier is universal, change it to existential. If it is existential to begin with, change it to the universal quantifier. These equivalences are quite important, and we will make extensive use of them shortly!
Now you should practice the skill of distinguishing negations of quantified wffs from quantified negations. Guess what? Here's an exercise for doing just that!
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