Rules of Inference for Predicate Calculus

7/25/2019 Rules of Inference for Predicate Calculus

1/8

Chapter

Propositions and Predicates

5 RULES OF INFERENCE FOR PREDICATE

CALCULUS

Before discussing the rules inference, we note that: i the proposition

formulas are also the predicate formulas; ii) the predicate formulas w here all

the variables are quantified) are the proposition formulas. Therefore, all the

rules of inference for the proposition formulas are also applicable to predicate

calculus wherever necessary.

For predicate formulas not involving connectives such as A x), P x, y). we

can get equivalences and rules inference similar t those given in

Tables 1.11 and 1.13. For Example, corresponding

to

1

6

in Table 1.11

we

get

P x v Q x P x

i \

-- ,

Q x)). Corresponding to RI

3

in Table 1.13

P

i \

Q

}

P,

we get

P x

i \

Q x

::::}

P x).

Thus we can replace propositional

variables by predicate variables in Tables 1.11 and 1.13.

Some necessary equivalences involving the two quantifiers and valid

implications are given in Table 1.14.

TABLE 4 Equivalences Involving Quantifiers

Distributivity of

j

over

3x CP x) O x)) = 3x PCI) 3x O x)

3x P O x))

=

P 3x O x))

Distributivity of ;I over

A

;Ix P x) O x))

x

P x) ;Ix O x)

;Ix

P

/\ O x)) = P .\ ;Ix O x))

3x P x)) = ;Ix . P x))

dx P x)) = 3x P x))

J ,7

3x P .\ O x)) = P /\ 3x O x))

- _

18

;Ix

P

O x)) = P ;Ix OCr))

RJ

l

;Ix P x)

3x

P x)

RJ

;Ix P x) ;Ix O x) ;Ix P x) O x))

RJ

3x P x)

O x)) 3x P x)

3x O x)

Sometimes when we wish to derive s J m e c o ~ l u s i o n from a given set

premises involving quantifiers. we may have to eliminate the quantifiers

before applying the rules inference for proposition formulas. Also, when

the conclusion involves quantifiers, we may have to introduce quantifiers. The

necessary rules

inference for addition and deletion

quantifiers are given

in Table 1.15.

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Theory o f

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TABLE

5 Rules of Inference for Addition and

Deletion of Quantifiers

RI

13

:

Universal instantiation

ix

P x

c

is

some element of the universe

RI

Existential instantiation

?:.xP x

. P c

c is some element for which P c is true

Rl

1s

: Universal generalization

P x

ix

P x

x should not

free in any of the given premises

RI

6

Existential generalization

P c

. =x P x

c is

some element of the universe

EXAMPLE 22

Discuss the validity

the following argument:

All graduates are educated.

Ram

is

a graduate.

Therefore. Ram is educated.

olut on

Let G x denote

x

is a graduate .

Let E x denote

x

is educated .

Let

R

denote Ram .

So the premises

are i)

If.r G x E x and ii) G R . The conclusion is E R .

l fx

G x E x Premise i)

G R E R Universal instantiation RI

G R)

Premise ii)

E R)

Modus ponens

RI

Thus the conclusion is valid.



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Chapter Propositions and Predicates ;J 25

EXAMPLE 23

Discuss the validity

of

the following argument:

All graduates can read and write.

Ram can read and write.

Therefore, Ram is a graduate.

olut on

Let G x) denote x is a graduate .

Let L x) denote x can read and write .

Let R denote Ram .

The premises are:

IIx G x)

=? L x)) and

L R).

The conclusion

is

G R).

G R)

=?

L R))

/\

L R))

=?

G R)

is not a tautology.

So we cannot derive

G R).

For example, a school boy can read and write

and he is not a graduate.

EXAMPLE 24

Discuss the validity of the following argument:

All educated persons are well behaved.

Ram is educated.

No well-behaved person is quarrelsome.

Therefore. Ram is not quarrelsome.

olut on

Let the universe of discourse be the set of all educated persons.

Let

PCx

denote x is well-behaved .

Let y

denote Ram .

Let

Q x)

denote

x

is

quarrelsome .

So the premises are:

i) II Y

PCx .

ii)

y

is a particular element of the universe

of

discourse.

iii) IIx P x) =?

Q x)).

To obtain the conclusion. we have the following arguments:

1. I Ix P x) Premise i)

2 PC\ Universal instantiation RI

3

3 IIx P x) =? Q x)) Premise iii)

4. PCy

=?

QC:y Universal instantiation

3

5 pry

Line 2

6 Q y) Modus ponens RIc;

Q y)

means that Ram is not quarrelsome . Thus the argument is valid.



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SUPPLEMENTARY EXAMPLES

EXAMPLE 25

Write the following sentences in symbolic form:

a) This book is interesting b ut the exercises are difficult.

b) This book is interesting but the subject is difficult.

c) This book is not interesting. the exercises are difficult but the subject

is not difficult.

d)

this

book

is interesting and the exercises are not difficult then the

subject is not difficult.

e) This book is interesting means that the subject is

not

difficult, and

conversely.

f) The subject is not difficult but this book is inter esting and the

exercises are difficult.

g) The subject is not difficult but the exercises are difficult.

h) Either the book is interesting

or

the subject is difficult.

olut on

Le t denote This book is interesting .

Le t

Q

denote T he exercises are difficult .

Let

denote Th e subject is difficult .

Then:

a /\

Q

b)

/\

c P /\ Q

d /\

- ,

Q)

=: }

R

e)

= >

-,

R

f

-,R

/\

/\ Q)

g R /\ Q

h) -,

v

EXAMPLE 26

Construct the truth table for

= -

= > - , Q) = > Q = >

olut on

The truth table is constructed as shown in Table 1.16.



5/8

Chapter

Propositions and Predicates ml

TABLE 6 Truth Table of Example 1.26

P

Q

R

Q R

T

T T

F T T F

T T

T

T F

F T

T

F F T

T

F T

T T T

T

T

T

T

F F

T F F F F T

F

T T

F T

T F T T

F T F F

T

T F T

T

F

F T T T T T T T

F F

F

T F T T T

T

EXAMPLE 28

State the converse, opposite and contrapositive

to

the following statements:

a

If

a triangle is isoceles, then two of its sides are equal.

b If there is no unemployment in India, then the Indians won t go to

the USA for employment.

olut on

If P

:::::} Q is a statement, then its converse, opposite and contrapositive

s t ~ t n t s are, Q :::::}

P P

:::::} Q and Q :::::}

P

respectively.

a

Converse If

two

of

the sides of a triangle are equal, then the triangle

s

isoceles.



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Opposite-If

the triangle is not isoceles, then two of its sides are not

equal.

Contrapositive-If

two of the sides of a triangle are not equal, then

.the triangle is

no t

isoceles.

b

Converse-If

the Indians

won t

go to the

USA

for employment, then

there is

no

unemployment in India.

Opposite-If

there is unemployment in India. then the Indians will

go

to the

USA

for employment.

c

Contrapositive-If

the Indians go to the

USA

for employment , then

there is unemployment in India.

EXAMPLE 1 29

Show that:

-

P

/\ -

Q /\

R v

Q /\ R

v

P /\

R

:::> R

olut on

-

P

/\ -

Q

/\

R

v Q

/\ R

v

P

/\

R

:::> P /\

-

Q /\

R

v Q /\ R v P

/\

R by using the associative

la w

:::>

-, P v Q /\ R v Q /\ R v P /\ R by using the

DeMorgan s la w

:::>

h P v Q /\ R v Q v P /\ R by using the distributive law

:::> -,

P v Q v P v Q /\ R

by

using

the commutative

an d distributive laws

by

using s

by using 1

X MPL

1 3

Using identit ies, prove that:

Q v

P

/\

-

Q

V -

P

/\

-

Q is a tautology

olut on

Q

v P /\ -

Q

V -

P

/\ -

Q

:::>

Q v P

/\

Q V --- v

-,

P

1\

Q

by

using the distributive law

:::>

Q v

P

\

T)

v

-, /\

-, Q

by

using

s

:::> Q

v

P

v

P

v Q

by

using the

DeMorgan slaw

and

1

9

:::> P V

Q

V P

v Q

by using

the

commutative

law

>T

Hence the given

fonnula

is a tautology.

by

using

s



7/8

Chapter

Propositions and Predicates

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EXAMPLE 3

Test the validity of the following argument:

If

I get the notes and study well, then I will get first class.

I didn t get first class.

So either I didn t get the notes or I didn t study well.

olut on

Let P denote get the notes .

Let Q denote I study well .

Let R denote 1 will get first class.

Let S denote I

didn t get first class.

The given premises are:

i) P

;\

Q

}

R

ii) -

R

The conclusion is - P v - , Q

l . P ; \Q=: }R

2.

- R

3. - P

;\

Q)

P

v Q

Thus the argument

is

valid.

EXAMPLE 32

Premise i)

Premise ii)

Lines I, 2 and modus tollens.

DeMorgan s law

Explain

a

the conditional proof rule and b) the indirect proof.

olut on

a)

I f

we want to prove

A

}

then we take

A

as

a premise and construct

a proof of B This is called the conditional proof rule. It

is

denoted

by

CPo

b) To prove a formula

0:

we construct a proof of

-

0:

}

F. In

particular.

to

prove

A

}

B

we construct a proof

of A ;\ -, B

}

F.

EXAMPLE

33

Test the validity of the following argument:

Babies are illogical.

Nobody is despised who can manage a crocodile.

Illogical persons are despised.

Therefore babies cannot manage crocodiles.



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olut on

Let B x denote x is a baby .

Let lex denote x is illogical .

Let

D x

denote

x

is despised .

Let C x denote x can manage crocodiles .

Then the premises are:

i)

Vx

B x ::::} I x

ii) Vx C x ::::} , D x ) )

iii)

Vx

l x ::::} D x

The conclusion is Vx B x ::::}

,

C x .

1.

Vx

B x

::::}

I x

Premise i

2

Vx C x ::::} ,D x Premise ii)

Vx l x

::::}

D x Premise iii)

B x

::::}

I x) 1

Universal instantiation

5

C x ::::}

,D x

2 Universal instantiation

6 I x ::::} D x) 3 Universal instantiation

7

B x

Premise of conclusion

8

I x)

4,7 Modus pollens

9

D x

6,8 Modus pollens

10. ,C x 5,9 Modus tollens

11. B x ::::} , C x 7,10 Conditional proof

2

Vx

B x ::::} , C x ) ) 11, Universal generalization.

Hence the conclusion is valid.

EXAMPLE 34

Give an indirect proof

Q, P Q, P v S S

olut on

We have to prove S. So we include iv) S as a premise.

P v S Premise iii)

2. S Premise iv)

P 1,2, Disjunctive syllogism

4. P

Q

Premise ii)

5. Q 3,4, Modus ponens

6.

Q Premise i)

7.

Q /\

Q 5.6, Conjuction

8. F 1

8

We get a contradiction. Hence Q, P

::::}

Q, P v S

S.

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Rules of Inference for Predicate Calculus