Note that NOT is a unary operator; it can
relate to one statement at a time. The other four—AND, OR, IF,
and IFF (short for IF AND ONLY IF)—are binary operators; they
always express relationships between pairs of statements. Note
also that each of the operands
(statements upon which an operator operates) has a specific functional
designation. In the case of conjunction, disjunction, and
bi-condition, the two statements p and q have the same
designation; but in the case of condition the statement associated with
IF has a different function from the statement associated with THEN.
It's common practice, in what's called statement logic or
proposition logic, to symbolize
statements as lower-case letters. This avoids confusing a letter
representing a statement with the capital "T's" and "F's" we use to
denote true and false values. Although for no apparent reason
logicians seem to favor p and q (and r, s,
t, and u when they're feeling really frisky), we're free
to use any letters of the alphabet that we like—with the exception of
v, which could be too easily confused with the
˅ OR
symbol. If we
prefer to start with a and b and work our way through the
whole alphabet, that's perfectly okay.
Or if we want to keep track of which plain-language statement is
represented by which letter in a complicated problem, we might use
whatever letters are suitable for the job (e.g., b for "Brian
goes bowling" and s for "Sue goes skiing"), as long as we're
careful to use a
different letter for each statement.
One minor point before we move on is that
logicians haven't yet agreed on a single standard
notation
for logical operators. Although the
˅ for OR
now seems fairly universal, there are alternative symbols for each of
the other operators:
NOT is sometimes represented by an overline (p)
instead of a preceding tilde (~p);
IF...THEN by
p É q
instead of an arrow;
IF-AND-ONLY-IF...THEN by
p ≡ q
instead of a double arrow;
AND may be represented variously as
p & q,
p ∙ q,
or simply pq,
as well as p * q.
So, if you happen to read a paper on
symbolic logic or take a college course in the subject, don't let it fry
your brain if you find these same operations represented by different
symbols or syntax. They might look different, but the underlying
concepts are exactly the same.
The
reader might well wonder how these five operations could possibly
represent the broad array of plain-language relationships from
although to whenever. We'll deal with that issue later
in Simplifying Plain Language.
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