Page 59 - Discrete Mathematics and Its Applications

38 1 / The Foundations: Logic and Proofs

EXAMPLE 2 Let A(x) denote the statement “Computer x is under attack by an intruder.” Suppose that of the
computers on campus, only CS2 and MATH1 are currently under attack by intruders. What are
truth values of A(CS1), A(CS2), and A(MATH1)?

Solution: We obtain the statement A(CS1) by setting x = CS1 in the statement “Computer x
is under attack by an intruder.” Because CS1 is not on the list of computers currently under
attack, we conclude that A(CS1) is false. Similarly, because CS2 and MATH1 are on the list of
computers under attack, we know that A(CS2) and A(MATH1) are true. ▲

We can also have statements that involve more than one variable. For instance, consider the
statement “x = y + 3.” We can denote this statement by Q(x, y), where x and y are variables
and Q is the predicate. When values are assigned to the variables x and y, the statement Q(x, y)
has a truth value.

EXAMPLE 3 Let Q(x, y) denote the statement “x = y + 3.” What are the truth values of the propositions
Q(1, 2) and Q(3, 0)?

Solution: To obtain Q(1, 2), set x = 1 and y = 2 in the statement Q(x, y). Hence, Q(1, 2) is
the statement “1 = 2 + 3,” which is false. The statement Q(3, 0) is the proposition “3 = 0 + 3,”
which is true. ▲

CHARLES SANDERS PEIRCE (1839–1914) Many consider Charles Peirce, born in Cambridge, Mas-
sachusetts, to be the most original and versatile American intellect. He made important contributions to an
amazing number of disciplines, including mathematics, astronomy, chemistry, geodesy, metrology, engineer-
ing, psychology, philology, the history of science, and economics. Peirce was also an inventor, a lifelong student
of medicine, a book reviewer, a dramatist and an actor, a short story writer, a phenomenologist, a logician, and a
metaphysician. He is noted as the preeminent system-building philosopher competent and productive in logic,
mathematics, and a wide range of sciences. He was encouraged by his father, Benjamin Peirce, a professor of
mathematics and natural philosophy at Harvard, to pursue a career in science. Instead, he decided to study logic
and scientific methodology. Peirce attended Harvard (1855–1859) and received a Harvard master of arts degree
(1862) and an advanced degree in chemistry from the Lawrence Scientific School (1863).
In 1861, Peirce became an aide in the U.S. Coast Survey, with the goal of better understanding scientific methodology. His service
for the Survey exempted him from military service during the Civil War. While working for the Survey, Peirce did astronomical and
geodesic work. He made fundamental contributions to the design of pendulums and to map projections, applying new mathematical
developments in the theory of elliptic functions. He was the first person to use the wavelength of light as a unit of measurement.
Peirce rose to the position of Assistant for the Survey, a position he held until forced to resign in 1891 when he disagreed with the
direction taken by the Survey’s new administration.
While making his living from work in the physical sciences, Peirce developed a hierarchy of sciences, with mathematics at the
top rung, in which the methods of one science could be adapted for use by those sciences under it in the hierarchy. During this time,
he also founded the American philosophical theory of pragmatism.
The only academic position Peirce ever held was lecturer in logic at Johns Hopkins University in Baltimore (1879–1884). His
mathematical work during this time included contributions to logic, set theory, abstract algebra, and the philosophy of mathematics.
His work is still relevant today, with recent applications of this work on logic to artificial intelligence. Peirce believed that the study
of mathematics could develop the mind’s powers of imagination, abstraction, and generalization. His diverse activities after retiring
from the Survey included writing for periodicals, contributing to scholarly dictionaries, translating scientific papers, guest lecturing,
and textbook writing. Unfortunately, his income from these pursuits was insufficient to protect him and his second wife from abject
poverty. He was supported in his later years by a fund created by his many admirers and administered by the philosopher William
James, his lifelong friend. Although Peirce wrote and published voluminously in a vast range of subjects, he left more than 100,000
pages of unpublished manuscripts. Because of the difficulty of studying his unpublished writings, scholars have only recently started
to understand some of his varied contributions. A group of people is devoted to making his work available over the Internet to bring
a better appreciation of Peirce’s accomplishments to the world.