Mixed Quantification
 

  We just saw how quantifiers can be introduced when we're dealing with relations. Our examples contained names. But we can quantify over relations where there are no names. Consider the sentence:

Now this is admittedly an awkward sentence, but it is a sentence nonetheless. How would we translate it? Our propositional function is "Txy". So we need a quantifier for each of the two variables. So we have:

which reads: "There is an x, there is a y, such that x is taller than y." Similarly, to say "Everyone is taller than everyone" we would use universal quantifiers:

The real fun, however, comes with what we call "mixed quantification." Consider the sentence:

This sentence clearly uses both universal and existential quantification. But what order do we put the quantifiers? Which comes first, the universal or the existential quantifier? Let's look at both possibilities:

This says: "For every x, there is some y, such that x is taller than y." In other words, Everyone is taller than someone." That's clearly not the same thing as our original sentence. Let's look at the other possibility:

This says "there is an x, for every y, such that x is taller than y."  So someone (some x) is taller than everyone (every y). 6 is clearly the correct translation.

However, there are two other possibilities for the order of the quantifiers. Instead of 5 we could have:

and instead of 6 we could have:

Notice how 7 differs from 5 and 8 from 6. The quantifiers are the same, just the order has been changed. But the order actually changes the meaning of the proposition.

5 says "Everyone is taller than someone (or other)."  7 says "someone (e.g. Fred) is such that everyone is taller than (him/her)."

6 says "Someone (e.g. Fred) is taller than everyone." 8 says "Everyone has  someone (or other) taller than them." The difference is subtle, but very important.

The next wrinkle is dealing with negation. Negations can go in front of quantifiers, between quantifiers, and after the quantifiers. The quantifier negation rules apply as before. Let's look at some examples.

9 says that it's not the case that for all y, there is an x, such that x is taller than y. This is just the negation of 8. So it says that it is not the case that everyone is taller than someone or other.  Note that this is equivalent to:

We get 10 by moving the negation to the other side of the quantifier and changing the quantifier from universal to existential, using the quantifier negation rules. 10 says "There is someone that someone is not taller than." 

Of course, there is an exercise to practice translating sentences of this sort.