rules of inference

1. RULES OF INFERENCES.
guided By :-
Dr.Ashok Mishra
PRESENTED BY :-
Rajnikant pandey
(220101120115)

2. Rules Of Inference
 The rules of inference (also known as inference rules) are
a logical form or guide consisting of premises (or
hypotheses) and draws a conclusion.
 An argument is valid when the conclusion logically follows
from the truth values of all the premises.

3. Quantifier
These are the words or phase which refers to quantity.
There are 2 types
Universal quantifier
For every, for each, for all, for any etc.
Existential Quantifier
For some, there exists, atleast etc.

4. Rule of inferences for
predicate Logic:-
Universal specification:- Us.
Us: ∀ x P(x)->P(a).
Existential specification:- Es.
Es: ∃ x P(x)-> P(a).
Universal Generalization:- Ug.
Ug: P(a)-> ∀ x P(x).
Existential Generalization:- Eg.
Eg: P(a)-> ∃ x P(x).

5. There are two ways to form Logical
Arguments

6. Show that the following sentences imply the conclusion “It rained:”
“If it does not rain or if it is not foggy, then the sailing race will held
and the life-saving demonstration will go on.”
“If the sailing race is held, then the trophy will be awarded.”
“The trophy was not awarded.”
Let,
p = “It rains”
q = “It is foggy”
r = “The sailing race will be held”
s = “Life-saving demonstrations will go on”
t = “The trophy will be awarded”
Now, the premises and conclusion is
converted into following logical form.
(¬p ∨ ¬q) → (r ∧ s), r → t and ¬t
Conclusion: ‘p’
Step No. of Rule of inferences
1. ¬t Premise
2. r → t Premise
3. ¬r Modus tollens
(1) ∧(2) →(3)
4. ¬r ∨ ¬s Addition, (3) →(4)
5. ¬(r ∧ s) Demorgan law
(4) →(5)
6. (¬p∨¬q)→(r∧s) Premise
7. ¬(¬p∨¬q) Modus tollens
(5) ∧(6) →(7)
8. p ∧ q Demorgan law
(7) →(8)
9. p Simplification
(8) →(9)

7. Using the rule of inferences construct a valid argument using to prove
that.
“petter has two legs”
is a consequence off the premises,
“Every human being has two legs”
and “petter is a human being”
 Solution :-
Let Hx:- x is a human being.
Lx:- x has two legs.
Now premises and conclusion are translated to the following logical form
Step No of inferences Rule of inferences
2 H(p) premises
3 H(p)->L(p) Rule Us (1)->(3)
4 L(P) Modune pones (2)^(3)-
>(4)
1 ∀ x (H(x)->L(x)) Premises

8. Prove that C v D is derived as a conclusion from the given set of
premises.
A v B, A→C, B→D
Solution:-
AvB, A → C, B → D
Steps No of inferences Rule of inferences
1 A->C premises
2 AvB->CvB Addition(1)->(2)
3 B->D premises
4 CvB->CvD Addition(3)->(4)
5 AvB->CvB Hypothesis
slogans(2)^(4)->(5)
6 AvB premises
7 CvD Module pones(5)^(6)
->(7)

9. Using the rules of inferences construct a valid argument to prove that
“Some fierce creatures do not drink coffee.”
is a consequence (conclusion) of the premises
“All lions are fierce.”
“Some lions do not drink coffee.”
Let,
L(x): “x is a lion.”
F(x): “x is fierce.” and
C(x): “x drinks coffee.”
Now, the premises and conclusion are
translated into the following logical form.
Premises: ∀x(L(x)→ F(x))
∃x (L(x) ∧ ¬C(x))
Conclusion:
∃x (F(x) ∧ ¬C(x))
Step No. of inferences Rule of inferences
1. ∃x (L(x) ∧ ¬C(x)) Premise
2. L(x) ∧ ¬C(x) E.s. (1) →(2)
3. L(x) Simplification
(2) →(3)
4. ¬C(x) Simplification
(2) →(4)
5. ∀x (L(x)→ F(x)) Premise
6. L(x) → F(x) U.s. (5) →(6)
7. F(x) Modus ponens
(3) ∧(6) →(7)
8. F(x) ∧ ¬C(x) Conjunction
(4) ∧(7) →(8)
9 ∃x (F(x) ∧ ¬C(x)) E.g. (8) →(9)

10. Prove that CvD is derived as a conclusion from the given set of
premises.
AvB, A->C, B->D
Solution:-
AvB, A->C, B->D
Steps No of inferences Rule of inferences
1 A->C premises
2 AvB->CvB Addition(1)->(2)
3 B->D premises
4 CvB->CvD Addition(3)->(4)
5 AvB->CvB Hypothesis
slogans(2)^(4)->(5)
6 AvB premises
7 CvD Module pones(5)^(6)
->(7)

11. prove that:-
“Some one who passed the exam has not
attend the class”
Follows logically from the premises:
“a student in this class has not attened the class”
“everyone in this class pass the exam”
solution:- Let,
C(x):x student in class.
P(x):x is passed the exam.
R(x):x is not attend the class.
Now,
the premise and conclusion are converted into
into logic form.
Premise:
∃x(c(x) ∧ ~r(x))
∀x (c(x) →p(x)
Conclusion:
∃x(p(x) ∧ ~r(x))
Step No of inferences Rules of inferences
1. ∃x(c(x) ∧ ~r(x)) premise
2. ∀x (c(x) →p(x) premise
3. C(a) ∧~r(a)) Rule Eg (1)
→(3)when it is a
free variable
4. C(a) →p(a) Us(2) →(4)
5. C(a) Simplification:(3)
→(5)
6. p(a) Module ponens(4)
∧(5) →(6)
7. ~R(a) Simplification:(3)
→(7)
8. P(a) ∧ ~r(a)) Conjunction (6)
∧(7) →(8)
9. ∃P(x) ∧ ~r(x)) Rule Eg(8) →(9)

12. THANK YOU!!!!