One way of changing an open formula into a closed formula, or sentence,
is by prefixing it with a quantifier-either the phrase "there exists a number
b such that ... ", or the phrase "for all numbers b". In the first instance, you
get the sentence
There exists a number b such that b plus 1 equals 2.Clearly this is true. In the second instance, you get the sentence
For all numbers b, b plus 1 equals 2.Clearly this is false. We now introduce symbols for both of these quantifiers.
These sentences are translated into TNT-notation as follows:
3b:(b+SO)=SSO
Vb:(b+SO) =SSO('3' stands for 'exists'.)
('V' stands for 'all'.)It is very important to note that these statements are no longer about
unspecified numbers; the first one is an assertion of existence, and the second
one is a universal assertion. They would mean the same thing, even if written
with c instead of b:
3c:(c+SO)=SSO
Vc:(c+SO)=SSO
A variable which is under the dominion of a quantifier is called a
quantified variable. The following two formulas illustrate the difference
between free variables and quantified variables:(b·b)=SSO
-3b:(b'b)=SSO(open)
(closed; a sentence of TNT)The first one expresses a property which might be possessed by some natural
number. Of course, no natural number has that property. And that is
precisely what is expressed by the second one. It is very crucial to under-
stand this difference between a string with afree variable, which expresses a
property, and a string where the variable is quantified, which expresses a truth
or falsity. The English translation of a formula with at least one free
variable-an open formula-is called a predicate. It is a sentence without a
subject (or a sentence whose subject is an out-of-context pronoun). For
instance,
"is a sentence without a subject"
"would be an anomaly"
"runs backwards and forwards simultaneously"
"improvised a six-part fugue on demand"are nonarithmetical predicates. They express properties which specific en-
tities might or might not possess. One could as well stick on a "dummy(^208) Typographical Number Theory