Negations of Simple Quantified Propositions
In Propositional Logic, negations were easy. There the tilde is the only prefix operator, that is, the only operator that goes in front of a well-formed formula. In Quantificational Logic, we have two more prefix operators: the existential quantifier and the universal quantifier. When we have a negation and a quantifier, the key questions will be: What's the order? Where does the tilde go?
Suppose we want to take the negation of "Everyone is tall." We saw in the last section that the QL translation of this sentence is:
(x)Tx
So for the negation, where does the tilde go? There are two possibilities:
| first possibility: | ~(x)Tx |
| second possibility: | (x)~Tx |
Let's read the two wffs carefully. The first says "It's not the case that everything has the property T." The second says everything has the property of being not-T. Now let's compare this to the English sentence "It's not the case that everything is tall." If not everything is tall, does it follow, as the second possibility holds, that nothing is tall? Clearly not! Suppose that Ignat is not tall. If Ignat is not tall, then not everything is tall (first possibility). But it doesn't follow that everyone else is not tall (second possibility). Ignat's brother, Fred, might very well be tall.
So there's a big difference between putting the tilde to the left of the quantifier and putting it to the right. We get wffs with completely different meanings. Your exercise for this section is to show that the difference we've just noted is also a difference for existentially quantified wffs.
In English it's not always clear whether a sentence expresses a negation of a quantified wff. It's not uncommon for someone to say "Everyone is not here." Now does that mean "Not everyone is here" or does it really mean "everyone is not here," that is, that no one is here? Note that if the person uttering "Everyone is not here" counts as someone, then one cannot truthfully utter something that means "No one is here." Maybe that accounts for the folksy reply to the question "Is anyone here?" with "No one except us mice!"
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