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412 Mastering Technical Mathematics
There are several ways in which negation, also called the logical NOT operation, can
be symbolized. In propositional logic, a common symbol is a drooping minus sign (¬).
Let’s use it here. Some texts use a tilde () to represent negation. Others use a minus
sign or em dash (–). Some put a line over the letter representing the sentence; still others
use an accent symbol. In our system, the sentence “It’s not cold outside” can be denoted
as ¬C.
Suppose someone comes along and says, “You are correct to say ¬C. In fact, I’d say it’s
hot outside!” Suppose this is symbolized H. Does H mean the same thing as ¬C? No!
You’ve seen days that were neither cold nor hot. There can be in-between states such as
“cool” (K), “mild” (M), and “warm” (W). But there is no in-between condition when it
comes to C and ¬C. In propositional logic, either it is cold, or else it is not cold. Either
it’s hot, or else it is not hot. A proposition is either true, or else it is false (not true). Of
course, temperature opinions like this depend on who you ask, so maybe this is not such
a good example. But you should get the general idea!
There are logical systems in which in-between states exist. These go by names such as
trinary logic or fuzzy logic. But discussions of those types of logic belong in a different
book. In this chapter, you can assume that any given proposition is either true or
false; there is no “neutral” or “don’t know” truth state, nor any “continuum” of truth
values.
CONJUNCTION (AND)
Propositional logic doesn’t bother with how the phrases inside a sentence affect one
another, but it is concerned with the ways in which complete sentences interact.
Sentences can be combined to make bigger ones, called compound sentences. The truth
or falsity of a compound sentence depends on the truth or falsity of its components, and
on the ways in which those components are connected.
Suppose someone says, “It’s cold outside, and it’s raining outside.” Using the symbols
above, you can write this as
C AND R
In logic, it’s customary to use a symbol in place of the word AND. There are several
symbols in common use, including the ampersand (&), the inverted wedge (^), the
asterisk (*), the period (.), the multiplication sign (), and the raised dot (·). Let’s use
the ampersand. Then the above compound sentence becomes
C & R
The formal term for the AND operation is logical conjunction. A compound sentence
containing one or more conjunctions is true if (but only if) both or all of its components