between logical validity (also known as the formal properties of the inference process)
and truth. This also means that a true result may come from an invalid argument (see
below for the definition of an argument). For example, “all cats are cute; Fluffer is a cat;
therefore, Fluffer is cute,” is a valid inference; whereas, “all cats are cute; Fluffer is cute;
therefore, Fluffer is a cat,” is an invalid inference, even if Fluffer really is a cat.What is the historical basisfor mathematical logic?
Most mathematicians believe that systematic logic began with Aristotle’s collection of
works titled Organon(Tool), in which he introduced his ideas on logic. In particular,
Aristotle used general forms to describe logic, such as if all xare y; and all yare z;
then all xare z. He presented three laws basic to all valid thought: the law of identity,
or Ais A(for example, an acorn will always yield an oak tree and nothing else); the law
of contradiction, or Acannot be both Aand not A(for example, an honest woman can-
not be a thief); and the law of the excluded middle, or either or, in which Amust either
be Aor not A(for example, a dog can be brown or not brown). Interestingly, author
Ayn Rand divided her novel Atlas Shruggedinto three parts after these three princi-
ples as a tribute to Aristotle.Was mathematicsalways based on a logical foundation?
No, not all of mathematics was always based on a logical foundation, but some ancient
cultures did develop certain aspects of logic in their thought. The Greeks were proba-
bly one of the first cultures to understand logic’s role in mathematics and philosophy,
and they studied the subject extensively. For example, geometry, as presented by
Greek mathematician Euclid (c. 325–c. 270 BCE), had some foundations in logic.
Greek scientist and philosopher Aristotle’s (384–322 BCE) rules for syllogisms were
104 also based on logic, and he wrote the first systematic treatise on logic. But his logic
What is an argument?
I
n logic, an argument is not a “heated discussion,” although some mathemati-
cians may argue over the validity of certain mathematical arguments. In this
sense, an argument is a list of statements called premisesfollowed by a state-
ment called the conclusion. Generally, an argument is valid if the conjunction of
its premises implies its conclusion; stated differently, validity means that if all
the premises are true, then so is the conclusion. But remember: The validity of
an argument does not guarantee the truth of its premises, and thus it does not
guarantee the truth of its conclusion. It only guarantees that if the premises are
true, the conclusion will be true.