Chapter 1: Propositions and Predicates ,\;! 21
The logical connectives involving predicates can be used for declarative
sentences involving predicates. The following example illustrates the use of
connectives.
EXAM PLE 1.21
Express the following sentences involving predicates in symbolic form:
1. All students are clever.
2. Some students are not successful.
3. Every clever student is successful.
4. There are some successful students who are not clever.
5. Some students are clever and successful.
Solution
As quantifiers are involved. we have to specify the universe of discourse. We
can take the universe of discourse as the set of all students.
Let C(x) denote 'x is clever'.
Let Sex) denote 'x is successful'.
Then the sentence 1 can be written as 'IIx C(x). The sentences 2-5 can be
written as
::Jx (, Sex»~,
::Jx (S(x) /\ ,C(x»,
'IIx (cex) :::::} Sex»~,
::Jx (C(x) , Sex»~
1.4.2 WELL-FORMED FORMULAS OF PREDICATE CALCULUS
A well-formed formula (wff) of predicate calculus is a string of variables such
as Xl, x2, •.• , X/1' connectives. parentheses and quantifiers defined recursively
by the following rules:
(i) PCYl, ... , x,J is a wff. where P is a predicate involving n variables
Xl, X20 ••. , J.-11'
(ii) If a is a wff. then , a is a wff.
(iii) If a and [3 are wffs, then a v [3, a i\ [3, a :::::} [3, a¢:::}[3 are also
wffs.
(iv) If a is a wff and x is any v~e; then 'IIx (a), ::Jx (a) are wffs.
(v) A string is a wff if and only if it is obtained by a finite number of
applications of rules (i)-(iv).
Note: A proposition can be viewed as a sentence involving a predicate with 0
Variables. So the propositions are wffs of predicate calculus by rule (i).
We call wffs of predicate calculus as predicate formulas for convenience.
The well-formed formulas introduced in Section 1.1 can be called proposition
formulas (or statement formulas) to distinguish them from predicate formulas.
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The logical connectives involving predicates can be used for declarative
sentences involving predicates. The following example illustrates the use of
connectives.
EXAM PLE 1.21
Express the following sentences involving predicates in symbolic form:
1. All students are clever.
2. Some students are not successful.
3. Every clever student is successful.
4. There are some successful students who are not clever.
5. Some students are clever and successful.
Solution
As quantifiers are involved. we have to specify the universe of discourse. We
can take the universe of discourse as the set of all students.
Let C(x) denote 'x is clever'.
Let Sex) denote 'x is successful'.
Then the sentence 1 can be written as 'IIx C(x). The sentences 2-5 can be
written as
::Jx (, Sex»~,
::Jx (S(x) /\ ,C(x»,
'IIx (cex) :::::} Sex»~,
::Jx (C(x) , Sex»~
1.4.2 WELL-FORMED FORMULAS OF PREDICATE CALCULUS
A well-formed formula (wff) of predicate calculus is a string of variables such
as Xl, x2, •.• , X/1' connectives. parentheses and quantifiers defined recursively
by the following rules:
(i) PCYl, ... , x,J is a wff. where P is a predicate involving n variables
Xl, X20 ••. , J.-11'
(ii) If a is a wff. then , a is a wff.
(iii) If a and [3 are wffs, then a v [3, a i\ [3, a :::::} [3, a¢:::}[3 are also
wffs.
(iv) If a is a wff and x is any v~e; then 'IIx (a), ::Jx (a) are wffs.
(v) A string is a wff if and only if it is obtained by a finite number of
applications of rules (i)-(iv).
Note: A proposition can be viewed as a sentence involving a predicate with 0
Variables. So the propositions are wffs of predicate calculus by rule (i).
We call wffs of predicate calculus as predicate formulas for convenience.
The well-formed formulas introduced in Section 1.1 can be called proposition
formulas (or statement formulas) to distinguish them from predicate formulas.
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