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!)
or (%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.