AND q,” represented by the symbols or &. The “or” (also called the disjunction oper-
ator) is also a binary operator, as in “p OR q”, and represented by the symbols and |.
The “not” (also called the negation or inversion) operator is known as a unary opera-
tor,and is represented by the symbols ~ or (in computer programming, NOT is
often represented by the !). The “implies” (or implication operator) is also a binary
operator; its symbols include , , and Ã.
But note: Not all logical operators seem to represent words the way we are accus-
tomed to using them, and many times they seem to contradict their proper defini-
tions. But in a truth table, the logical operator means what it means—without the
usual nuances of our English language.What is a formula?
In mathematics, a formula is generally a rule, principle, or fact that is displayed in
terms of mathematical symbols. (Although the Latin plural form of formula is “formu-
lae,” “formulas” is the accepted common use in mathematics.) These equations
express a definite, fixed relationship between certain quantities (usually expressed by
letters), with their relationship indicated by algebraic symbols. For example, scientist
Albert Einstein’s famous E mc^2 is a formula representing energy (E) equal to mass
(m) times the speed of light (c) squared. The word “formula” is also used in logic; it is
written as a propositional or sentential formula, or “a formula in propositional calcu-
lus is one that uses ‘and,’ ‘or,’ and so on.”What is predicate calculus?
Predicate calculus (also called first-order logic, functional calculus, or quantification
theory) is a theory in symbolic logic that uses statements such as “there exists an
object such that ...” or “for all objects, it is the case that....” It is a much more solid
110 theory than propositional calculus, taking the interrelationship between sentences
Why are truth tables important to computers?
I
n many ways, truth tables are directly connected to digital logic circuits. In the
case of computers, the terms would be AND, OR, NAND, NOR, NOT, XOR, or the
“gates” that open and close in response to such terms. In such a circuit, values at
each point can take on values of only true (1) or false (0); this is also known as the
computer binary system. In general, there is also a three-valued logic,in which
possible values are true, false, and “undecided.” A further generalization called
fuzzy logicexamines the “truth” as a continuous quantity ranging from 0 to 1.
(For more about fuzzy logic and computers, see “Math in Computing.”)