Declarative Sentences
As the Introduction said,
geometry consists of numerous declarative
sentences. A declarative sentence is a sentence that asserts the truth or
falsehood of something. For example, "That car is red" is a declarative
sentence. Other sentences can be interrogative, exclamatory, or imperative.
Examples are, respectively, "Is that car red?", "Wow, a red car!", and "Drive
that red car." Geometry most often concerns itself with declarative sentences.
Statements
More specifically, geometry and logic uses a precise kind of declarative
sentence that is either definitely true or false; such declarative sentences are
called statements. For example, "It is purple" is a declarative sentence,
but we don't know what "it" is, so we cannot argue its truth or falsehood.
"Fred is purple" is a declarative sentence that is definitely either true or
false; it is the kind of declarative sentence we can study under the rules of
logic. "An obtuse triangle is a triangle with one obtuse angle" is also a
declarative sentence that is either true or false (we know it is true, of
course) and so can be studied using the rules of logic. From this point forth,
we will define a statement as a declarative sentence that is either true or
false.
Every statement, by definition, has a truth value. Only two different truth
values exist: True or False. Any statement has either one truth value or
another. Either it is true, or it is false. These truth values are symbolized
by the capital letters T and F. In this way, entire statements can be
symbolized by a single letter. At the beginning of a problem, it might say, "p:
Brian runs barefoot." From that point on, "p" symbolizes this entire statement.
These symbols will become necessary when we look at more than one statement in
the same problem.
In the following lessons, we'll look at the different ways to classify and group
statements, and the different ways in which we can change them to learn more
about their subjects.
