6
Formal Logic
We can string statement letters, connectives, and parentheses (or brackets)
together to form new expressions, as in
(A S B) ` (B S A)
Of course, just as in a computer programming language, certain syntax rules
(rules on which strings are legitimate) prevail; for example,
A )) `` S BC
would not be considered a legitimate string. An expression that is a legitimate
string is called a well-formed formula, or wff. To reduce the number of parentheses required in a wff, we stipulate an order in which connectives are applied. This
order of precedence is
1. connectives within parentheses, innermost parentheses first
2. ′
3. `, ~
4. S
5. 4
This means that the expression A ~ B′ stands for A ~ (B′), not (A ~ B)′. Similarly,
A ~ B S C means (A ~ B) S C, not A ~ (B S C ). However, we often use parentheses anyway, just to be sure that there is no confusion.
In a wff with a number of connectives, the connective to be applied last is the
main connective. In
A ` (B S C )′
the main connective is `. In
((A ~ B) ` C ) S (B ~ C′)
the main connective is S. Capital letters near the end of the alphabet, such as P, Q,
R, and S, are used to represent wffs. Thus P could represent a single statement letter, which is the simplest kind of wff, or a more complex wff. We might represent
((A ~ B) ` C ) S (B ~ C′)
as
P S Q
if we want to hide some of the details for the moment and only concentrate on the
main connective.
Wffs composed of statement letters and connectives have truth values that
depend on the truth values assigned to their statement letters. We write the
truth table for any wff by building up the component parts, just as we did for
(A S B) ` (B S A). The main connective is addressed in the last column of the
table.