DHARMPROPOSITIONAL LOGIC 137In order to determine the truth value of (one-place predicate) statement E(x) : where x is
a even number will be true, because
E = {2, 4}.
The truth value for the statement M(x): where x is greater than 5, will be false because
set M find no element from given domain set D.
The truth value for 2-place predicate statement G (x, y) : where x > y will also be true
where, set G = {(2, 1), (3, 1) (4, 1), (3, 2), (4, 2), (4, 3)}.
If G(x, y) i.e. x is greater or equal to y then its truth value also be true where, set G
contains all above elements including (1, 1), (2, 2), (3, 3) and (4, 4).5.7.2 Variables and Quantifiers..................................................................................
Consider the statement discussed earlier,
“Rhodes is a good boy” : G (r), where G be the predicate “good boy”
and r is the name “Rhodes”
Consider another statement,
“Stephen is a good boy” : G(s), where predicate G “good boy” is same with
different name “Stephen” symbolizes by s.
Consider one more similar statement,
“George is a good boy” : G(g) with same predicate G and different name
“George” symbolizes by g.
These statements G(r), G(s), G(g) and possibly several other statements shared the prop-
erty in common that is predicate G “good boy” but subject is varies from one statement to the
other statement. If we write G(x) in general that states “x is G” or “x is a good boy” then the
statements G(r), G(s), G(g) and infinite many statements of same property can be obtained by
replacing x by the corresponding name. So, the role of x is a substitute called variable and G(x)
is a simple (atomic) statement function.
G( )xPredicate Symbol Variable
We can obtain the statement from statement function G(x), when variable x is replaced
by the name of the object.
A compound statement function can be obtain from combining one/more atomic state-
ment function using connectives viz, ∧, ∨, ~, → etc. for example,
G(x) ∧ M(x); G(x) ∨ M(x); ~ G(x); G(x) → M(x); etc.
The idea of statement function of two/more variables is straightforward.
Quantifier
Consider the statement,
“Everyone is good boy”
The translation of the statement can be written by G(x) s.t. “x is a good boy”. To symbol-
ize the expression “every x” or “all x” or “for any x” we use the symbol “(∀x)” that is called
universal quantifier. So, given statement can be expressed by an equivalent statement ex-
pression,
(∀x) G(x) : read as “for all x, x is G” (where G stands for good boy)