much farther, but it is weaker than certain branches of mathematics, such as arith-
metic and set theory.
What is a quantifierin predicate calculus?
A sentence or many sentences containing a variable (such as x) can be made into true
or false propositions simply by using a quantifier. The quantifier actually assigns a
truth value to the sentence, depending on the set of values allowed for that variable.
There are two major quantifiers: the existential and universal quantifiers, which are
represented by the logical operator symbols of and , respectively, although there
are also more exotic types of logic that use different quantifiers.
How is predicate calculus interpreted?
Predicate calculus may be a general system of logic, but it accurately expresses a large
variety of assertions and provides many types of reasoning. It is definitely more flexible
than Aristotle’s syllogisms and more useful (in many cases) than propositional calculus.
Predicate calculus makes heavy use of symbolic notation, using lowercase letters
a, b, c, ..., x, y, zto denote the subject (in predicate calculus, often referred to as
“individuals”), and uppercase letters M, N, P, Q, R, ...to denote predicates. The sim-
plest of assertions are formed by moving the predicate with the subject.
For example, using the “all” quantifier means that when you have an arbitrary vari-
able you must prove something true about that variable, and then prove that it does not 111
FOUNDATIONS OF MATHEMATICS
Who was responsible for expanding the ideas of predicate calculus?
T
he German philosopher and mathematician Friedrich Ludwig Gottlob Frege
(1848–1925) presented a way to rearrange sentences to make their logic clear-
er and to show how the sentences relate in various ways in his 1879 treatise,
Begriffsschrift(German for “Concept Script”). Before Frege began his work, for-
mal logic (in the form of propositional or sentential calculus; see above) used such
words as “and,” “or,” and so on, but the method could not break the sentences
down into smaller parts. For example, formal logic could not show how the sen-
tence “Cats are animals” actually entails “parts of cats are parts of animals.”
Frege added words such as “all,” “some,” and “none,” using variables and
quantifiers to rearrange the sentences, therefore making them more precise in
their meaning. He also developed two of the major qualifier symbols for predi-
cate calculus, the upside-down A () and the backward E (). Frege’s work was
the foundation for modern logical theory, even though his work was defective in
several respects and was considered awkward to use. By the 1910s and 1920s,
Frege’s system was modified and streamlined into today’s predicate calculus.