**7.1 Semantics Demonstration
Multiple Quantifiers**

Assume the following interpretation:

universe of discourse = the counting numbers: 1,2,3, etc.

Gxy: x is greater than y

Now, consider the English sentence "For every number w there is a greater number z" and its PL symbolization:

(*) (^w)(%z)Gzw

Is (*) true? How do we tell? (Start the Demo and see!)

Think
first about the definition of __truth__ for a universally quantified sentence
like (*).

(*) is true if *every* substitution instance of (*),

(%z)Gz_

is true.

That is, (*) is true just in case *any* number n makes the following
true:

(%z)Gzn

(using 'n' as PL

and English name.)

But, using our definition of truth for an *existential* quantifier,
'(%z)Gzn' is true iff there is at least *one*
true substitution instance. That is, one number whose name makes

G_n

true.

*Is there such
a number???*

Yes! All we need is a name for a number greater than n (whatever n might be).

One example of such a number is n+1.^{†}

So, we've just seen that '(%z)Gzn' *is*
true because there is a number greater than n.

But we could say the same for *whatever* number is named in place
of n.

That is, *every* substitution instance

'(%z)Gz_'

of (*) is true. That makes (*) true.