Page 137 - Intelligent Communication Systems

1 I 6 INTELLIGENT COMMUNICATION SYSTEMS
10.4.1 Definitions and Operations for Predicate Logic

An individual constant represents a specific object and is notated a. b<
c,,...
An individual variable represents any object and notatedx, y, z,...,
A functional symbol represents a relation between or among objects and is
notated fix, y), g(z, w),.... Here the functional symbol g shows the
relationship between z and w.
A predicate symbol represents a predicate for objects and is notated P(x, y),
Q(z),..., where P and Q are predicate symbols.
A logical symbol represents an operation on predicate symbols and is
notated <-», ~, —>, v, or A.
A term can contain individual constants, individual variables, and/or
functions.
Quantifiers come in two forms: existential quantifier (3) and universal quantifier
(V). For example, our earlier statement (3) means that if x is a human, then x
dies for all x. And our earlier statement (4) means that a red flower x exists if x
is a flower.
In predicate logic a logical expression is defined as follows:
(1) If t {,t 2,...,t nare terms and P is a predicate with n parameters, then
P(t {, t 2,..,, t n) is an atomic formula and a logical expression.
(2) If P(ti, t 2,..., ?„) and Q(s^ s 2,..., s m) are logical expressions, then
-/>(/„ %..., ?„), P(f,, f 2 ,..., *„) A Q( Sl, S 2,..., S m), />(*„ ? 2 ,..., t n) V Q(S,,
s 2,..., sj, ?(/„ ? 2,..., /„) -> 0(5!, %..., 5j, 00i, %••-, *«). andP(/,,
? 2 ,.-., ?„) <-> fi(^i, ^2»- • •> ^w) are logical expressions.
(3) If P(XI, x 2,...,x n) is a logical expression, then Vjc t, Vjc 2,..., Vjc n P(x^
x 2,..., x n) and BJCJ, 3jc 2,..., 3x n P(x {, x 2,,.., x n) are logical expressions.
A variable qualified by V or 3 is a bound variable. A variable that is not qual-
ified by V or 3 is a free variable. A logical expression operation of predicate logic
is shown in Table 10.3.
An area that a bound variable influences is called a scope of the variable,
When a quantifier q is given, the scope of qx[...] is [...] for x, and the scope of
qxl[...qx2[..,]] is [...qx2[...]] forxl and [...] forx2. Interpretations of \fx3yP(x,y)
and Vx 3 yP(x, y) are different and shown in Figures 10.7 and 10.8, respectively,
where x is one of the elements (a, b, c}and y is one of the elements {d, e, /}.

TABLE 10.3 Logical Operation of Predicate Logic

PAQ PvQ P-»Q P<-»Q ~p ~Q

T T T T F F
F T F F F T
F T T F T F
F F T T T T