What is a fallacy?
A fallacy is an incorrect result—in this case, one arrived at through misleading reason-
ing when examining a logical argument. One of the more common fallacies in logic is
believing incorrectly that if “p implies q” is true, “then q implies p” is also true. The
idea of such invalid arguments was well known in the past: With Greek mathematician
Aristotle’s syllogisms, an argument was valid if it adhered to all the laws; to be false, it
only needed to break one law. Euclid, another Greek mathematician, was known to
have written an entire book on fallacies in geometry, but the book has since been lost.What is a paradox?
In logic, paradoxes are statements that seem to be self-contradictory or contrary to
one’s expectations. These arguments imply both a proposition and its opposite. One of
the most famous paradoxes was stated by English logician Bertrand Russell
(1872–1970) in 1901 and deals with sets: “If sets that are not members of themselves
are normal, is the set of normal sets itself normal?” (For more information about Rus-
sell’s paradoxes in set theory, see below.)What are some paradoxesthat deal with space and time?
There are numerous paradoxes that deal with the counterintuitive aspects of continuous
120 space and time. One of the most well known is the dichotomy (or racetrack) paradox.
What are some examples of paradoxes throughout history?
T
he oldest paradoxes may be from the Greek Epimenides the Cretan (lived
sometime during the 6th century BCE), who stated, “All Cretans are liars.” If
this statement is true—and any other culture you would care to put in the Cre-
tans’ place—then the implication is that the statement is a lie. This is also called
the “Liar’s paradox.”The number of paradoxes has continued almost ad infinitumsince then.
Some of the more popular ones include those listed as Zeno’s paradoxes. They
are named after Greek philosopher Zeno of Elea (c. 490–c. 425 BCE), a disciple of
the philosopher Parmenides, who believed that reality was an absolute,
unchanging whole—and, thus, that many things we take for granted, such as
motion, were simply illusions. In order to defend his master’s highly debated
philosophy, Zeno developed his paradoxes. Most of Zeno’s paradoxes are still
highly debated by modern mathematicians and philosophers, thus proving
another paradox: Nothing truly changes throughout history—or does it?