Biconditional—Biconditional (double implication or bi-implication) is true only if
the two statements have the same value, either true or false; it refers to “if and only
if....” in natural language.
What is a truth table?
A truth table is a two-dimensional array of truth values derived by determining the
validity of arguments through assigning all possible combinations of truth values to
the statements. This simple form of logic depends on a combination of certain state-
ments, using terms such as “not” or “and,” along with the input values.
The first columns correspond to the possible input values and the last columns to
the operations being performed; the rows list all possible combinations of true (T) or
false (F) inputs, together with the corresponding outputs. The following is a truth
table for the three most common binary operations of logic (“if ... then,” “or,” “and”),
using sand tas the statements:
stif s, then tsor tsand t
TT T T T
TF F T F
FT T T F
FF T F FWhat are logical operatorsin truth tables?
Logical operators in truth tables include such words as “and” or “or,” which are all
represented by certain symbols (for more about logical operators in predicate calcu-
lus, see below). For example, “and” (also called the conjunction operator) is also
referred to as a binary operator. It is one of the most useful logical operators, as in “p 109
FOUNDATIONS OF MATHEMATICS
What is propositional calculus?
P
ropositional calculus is not the calculus most of us hear about, but it is con-
sidered by many to be the foundation of symbolic logic. (Actually, the term
“calculus” is a generic name for any area of mathematics that deals with calcu-
lating; thus, arithmetic could be called the “calculus of numbers.”) Also known
as truth-functional analysis, sentential calculus, or the calculus of propositions
(or, as seen above, any declarative sentence that is either true or false), proposi-
tional calculus deals with statements that can be assigned truth values. In gen-
eral, it uses symbols to denote logical operators (such as andand or), and paren-
theses for grouping formulas.