Appendix A
Logic and Proofs
A.1 The Logic of Propositions
The theory of deductive logic, as it applies to mathematics,
divides naturally into two main areas:- the calculus of propositions, and
- the calculus of propositional functions.
Mathematics is expressed in language; indeed, some would say that it is a
language. Mathematical truth is expressed in sentences. Even though these
sentences may be abstract or symbolic, they must be clear and unambiguous.
Moreover, some sentences taken together imply other sentences. Logic enables
us to clarify these relationships and to distinguish valid implications from in-
valid ones. It should not be surprising, in view of the role of proofs in mathe-
matics , that the principles of logic play a significant role in this book.
There are two types of sentences that occur frequently in mathematics: (1)
propositions, and (2) propositional functions. We discuss (1) here, and defer
(2) to Section A.2.Definition A.1.1 A proposition is a declarative sentence that is either true
or false, but not both.
(We say that a proposition has a definite "truth-value," either Tor F , but not
both.)
For example, "3 + 5 = 10" is a proposition; it has truth-value F, but it is
still a proposition. On the other hand, "x + y = 10" is not a proposition; it is
neither true nor false. Indeed, there is no way of knowing whether it is true or
false because x and y are unspecified.
The actual words used in uttering a proposition are not important in this
context; it is the meaning of the sentence that is important. Thus, two different
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