General Logic: The Predicate Calculus
There are many different types of arguments which cannot be accounted for by the propositional calculus so as to include arguments whose validity depends not only on the external relationship of propositions but also on the internal construction of the propositions themselves. Simple illustrations of such arguments would include Aristotelian syllogisms. We also want our new logic to have a notation which fits in with the propositional calculus and, further, we want to keep the algebraic feature that manipulation of symbols according to certain rules leads to the expression of new logical relationships.
Our
new expanded logic is sometimes called the
first-order (or restricted) calculus of propositional functions,
sometimes
the first-order
(or restricted) functional calculus, and
sometimes the first-order
(or restricted) predicate calculus. The
central notion to be explained is that of a propositional
function or predicate. Let there be fixed some nonempty domain of
discourse, that is, some set of objects which our logic will be
about. Examples might include the set of physical objects, the set of
living animals, or the set of natural numbers. The members of the
domain of discourse will be called individuals.
An
n-place
propositional function (or n-place
predicate) is a function of n
individual
variables, where the domain of definition is the domain of discourse
and the domain of values is a set of propositions. Hence, when
each variable in a propositional function (predicate) has assigned to
it an individual, the result is a proposition. A one-place
propositional function or predicate is a property,
a
two-place propositional function or predicate - a binary
relation, a
three-place propositional function or predicate - a
ternary relation and
so on. Sometimes we wish to express particular or universal
propositions (as in the case of the syllogism). We may do this by
making use of quantifiers,
that
is, operators which make particular or universal propositions out of
predicates. We shall use two operators: "
" which is calledthe
existential quantifier, and
"
" which is calledthe
universal quantifier. The
formula "
(
x)A{x)"
is read "there exists an x,
A of
x"
or
"for some x,
A of
x".
It
means that there is at least one individual in the domain of
discourse which has the property represented by A.
The
formula
"
(
/x)A
(x)
"
is read "for all x,
A of
x"
or
"given any x,
A of
x".
It means that every individual in the domain of discourse has the
property represented by "A".
The
order in which the quantifiers occur is important.
As the first step toward defining validity for the predicate calculus, we might raise the question whether the technique of a truth table might be extended from the propositional calculus to the predicate calculus. The answer is "yes". But the technique, while possible, does not give us a mechanical test of validity as was in the case in the propositional calculus. Nevertheless, truth tables will enable us to define what we mean by validity in the predicate calculus.
Set-Theoretic Logic: Higher-Order Predicate Calculi
Once the notion of quantification is understood, it is natural to introduce predicate variables and to extend its use to them. A logic which contains quantification over only one domain — the individuals — is called a first-order (or restricted) predicate calculus. A logic which contains quantification with two types of variables - individual variables and predicate variables for individuals — is called a second-order predicate calculus. The higher-order predicate calculi are each isomorphic with various fragments of set theory. In fact, it seems likely that the theory as originally proposed by Cantor was meant to embody somewhat the same theoretical structure as is found in w-order logic. In the latter, of course, there are individuals, properties of individuals, properties of properties of individuals.