Download Rules of inference

Fall 2008/2009
The Foundations: Logic
and Proofs
Rules of inference
I. Arwa Linjawi & I. Asma’a Ashenkity
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Fall 2008/2009
Introduction
 Proofs in mathematics are valid arguments that
establish the truth of mathematical statement
 An argument in Propositional logic is a
sequence of propositions.
 An argument form in Propositional logic is a
sequence of compound propositions involving
propositional variables.
 All but the final proposition in the argument are
called premises and the final proposition is
called the conclusion.
I. Arwa Linjawi & I. Asma’a Ashenkity
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Introduction
 Example:
“If you have a current password,
then you can log onto the network”
p
q
“You have a current password”
premises
premises
Therefore,
“You can log onto the network”
I. Arwa Linjawi & I. Asma’a Ashenkity
conclusion
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Valid argument in Propositional
Logic
pq
p
______
q
 An argument is valid if the truth of all its premises
implies that the conclusion is true.
 The hypotheses are written in a rows, followed by
horizontal bar, followed by the  and the conclusion.
  means “therefore”
I. Arwa Linjawi & I. Asma’a Ashenkity
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Fall 2008/2009
Valid argument in Propositional
Logic
If we have an argument with premises p1, p2, p3, …,pn
and conclusion q
This argument is valid when
(p1  p2  p3  …………  pn)  q is a tautology
(p ( p  q))  q
p q p  q p(p  q)
pq
p
F F
T
F
T
_________
F T
T
F
T
q
T F
F
F
T
T
T
T
T
T
The tautology (p ( p  q))  q is the basis of the rule of
inference called Modus Ponens (or law of detachmentmode that affirms)
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I. Arwa Linjawi & I. Asma’a Ashenkity
Fall 2008/2009
Rules of Inference for propositional
logic
 We can use a truth table to show that an argument form
is valid. By showing that whenever the premises are true,
the conclusion must also be true.
 If an argument form involves 10 different propositional
variables, to use truth table, 210=1024 rows are needed.
This is a tedious (long and boring) approach.

Instead of using a truth table to show that an argument
form is valid, we cant use Rules of inference..
I. Arwa Linjawi & I. Asma’a Ashenkity
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Rules of Inference for propositional
logic
Modus Ponens
Modus Ponens tell us that if a conditional statement and
its hypothesis are both true, then the conclusion must also
be true.
 Example:
Conditional statement
“If you have a current password, then you can log onto the network”
Hypothesis
“You have a current password”
if conditional statement and hypothesis are true
Then the conclusion is true
“You can log onto the network”
I. Arwa Linjawi & I. Asma’a Ashenkity
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Rules of Inference for
propositional logic
 Examples:
“if 2  3 , then  2 
2
2
3
 
2
2
“
Determine whether the argument is valid, and determine whether its conclusion
must be true because of the validity of the argument.
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Let p be the proposition 2  2 and q be the proposition
 2
2
3
 
2
2
The premises of the argument are p  q , p and its conclusion is q.
This argument is valid (it is constructed by using Modus ponens), a valid
argument form.
3
The premise 2  2 is false, therefore we can not conclude that the conclusion
is true.
Conclusion
 2
2
3
 
2
2
is false. 2< 9/4
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Inference Rule – general form
Pattern establishing that if we know that a set of
hypotheses all are true, then a certain related conclusion
statement is true.
Hypothesis 1
Hypothesis 2
…….
Hypothesis n
 Conclusion
Each logical inference rule corresponds to an implication
that is a tautology.
[(Hypothesis 1)  (Hypothesis 2)  …  (Hypothesis
n)]  conclusion
I. Arwa Linjawi & I. Asma’a Ashenkity
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Some Inference Rules
Rule of inference
pq
p
Tautology
Name
[ p  ( p  q )]  q
Modus ponens
q
q
pq
[q  ( p  q )]  p
Modus tollen
 p
pq
qr
[( p  q )  (q  r )]  ( p  r )
Hypothetic al syllogism
pr
pq
p
(( p  q )  p )  q
Disjunctiv e syllogism
p  ( p  q)
Addition
( p  q)  p
Simplifica tion
(( p )  (q ))  ( p  q )
Conjunctio n
[( p  q )  (p  r )]  ( p  r )
Resolution
q
p
pq
pq
p
p
q
pq
pq
p  r
q  r
I. Arwa Linjawi & I. Asma’a Ashenkity
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Some Inference Rules
Examples:
 “It is below freezing now (p). Therefore, it is either below freezing or raining
now (q).”
 “It is below freezing (p). It is raining now (q). Therefore, it is below freezing
and it is raining now.
 “if it rains today (p), then we will not have a barbecue today (q). if we do not
have a barbecue today, then we will have a barbecue tomorrow (r) .
Therefore, it is rains today, then we will have a barbecue tomorrow.
 “If it is snows today (p), then we will go skiing (q)”. “It is snowing today”.
Therefore (or imply that) “We will go skiing”
 “It is below freezing (p) and raining now (q). Therefore, it is below freezing
now.”
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I. Arwa Linjawi & I. Asma’a Ashenkity
Fall 2008/2009
Using Rules of Inference to Build
Arguments
 A formal proof of a conclusion C, given
premises p1, p2,…,pn consists of a sequence of
steps, each of which applies some inference
rule to premises or to previously-proven
statements (as hypotheses) to yield a new true
statement (the conclusion).
 A proof demonstrates that if the premises are
true, then the conclusion is true (i.e., valid
argument).
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Using Rules of Inference to Build
Arguments
Example:
1.
2.
3.
4.
5.
It is not sunny this afternoon and it is colder than yesterday.
We will go swimming only if it is sunny.
If we do not go swimming then we will take a canoe trip.
If we take a canoe trip then we will be home by sunset.
We will be home by sunset
p
q
r
s
t
It is sunny this afternoon
It is colder tha n yesterday
We go swimming
We will take a canoe trip
We will be home by sunset (the conclusion )
1.  p  q
2. r  p
3.  r  s
4. s  t
5.
t
propositions
I. Arwa Linjawi & I. Asma’a Ashenkity
hypotheses
Fall 2008/2009
Using Rules of Inference to Build
Arguments
Step
Step
Reason
Reason
Reason
1. 
pp 
 qq Hypothesis
Hypothesis
Hypothesis
2.  p
Simplifica
Simplifica tion
tion using
using (1)
(1)
3. r  p
4.  r
Hypothesis
Hypothesis
Modus
Modustollens
tollens using
using (2)
(2)and
and(3)
(3)
5. r  s Hypothesis
6. s
Modus ponens using (4) and (5)
7. s  t
8. t
Hypothesis
Modus ponens using (6) and (7)
I. Arwa Linjawi & I. Asma’a Ashenkity
Fall 2008/2009
Resolution

Computer programs have been developed to
automate the task of reasoning and proving
theorem. Many of these programs make use
of rule of inference as resolution
((p  q) ( p  r)) (q  r) and it is tautology
Even if we let q=r then
((p  q) ( p  r)) q
Or when r=F then
(p  q) ( p) q because q  FΞq
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Resolution Example
1. Anna is skiing or it is not snowing.
2. It is snowing or Bart is playing hockey.
3. Consequently Anna is skiing or Bart is playing hockey.
propositions
hypotheses
1. p   r
2. r  q
3. p  q
p  r
rq
p Anna is skiing
q Bart is playing hockey
r it is snowing
Resolution rule
pq
Consequently Anna is skiing or Bart is playing hockey
I. Arwa Linjawi & I. Asma’a Ashenkity
Fall 2008/2009
Fallacies
Fallacy of affirming the conclusion
Is this argument is valid?
“If you do every problem in this book, then you will learn discrete mathematics
(q).” “You learned discrete mathematics.” “Therefore, you did every problem in
this book (p).”





This argument is of the form p q and q then p.
This is an example of an incorrect argument using the Fallacy of affirming
the conclusion.
pq and q does not imply p.
The proposition [(pq) q) p] is not a tautology (its false if p is F and q
is T) check the truth table
You can learn discrete mathematics in some way other than doing every
problem in this book (by reading- listening to lectures-doing some (but not
all) the problems in this book)
I. Arwa Linjawi & I. Asma’a Ashenkity
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Fallacies
Fallacy of denying the hypothesis
Is this argument is valid?
“If you do every problem in this book (p), then you will learn discrete
mathematics (q)”. “You did not do every problem in this book. Therefore, You
did not learn discrete mathematics.”
this argument is of the form pq and p imply q .
This is an example of an incorrect argument using the Fallacy of denying the
hypothesis.
pq and  p does not imply  q.
The proposition [(pq)  p)  q] is not a tautology (its false if p is F and q
is T).
It is possible that you learned discrete even you did not do every problem in
this book
I. Arwa Linjawi & I. Asma’a Ashenkity
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Fall 2008/2009
Fallacies
 A fallacy is an inference rule or other proof
method that is not logically valid. It may yield a
false conclusion!
 Fallacy of affirming the conclusion:
“pq is true, and q is true, so p must be true.”
(No, because FT is true.)
 Fallacy of denying the hypothesis:
“pq is true, and p is false, so q must be false.”
(No, again because FT is true.)
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Inference Rules for Quantifiers
Universal instantiation
x P(x)
P(c)
(Substitute any object c)
We can conclude that P(c) is true, where c is a particular member of the
domain, if we know that x P(x) is true.
Example
“all students have internet account”
“Ahmed has internet account”
Where, Ahmed is a member of the domain of all students.
Universal generalization
P(c)
(for an arbitrary c)
 x P(x)
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Inference Rules for Quantifiers
Existential instantiation
x P(x)
 P(c) (Substitute for some element c )
We can conclude that there is an element c in the domain for which P(c) is
true, if we know that x P(x) is true.
Existential generalization
P(c) (for some element c )
 x P(x)
I. Arwa Linjawi & I. Asma’a Ashenkity
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Fall 2008/2009
Inference Rules for Quantifiers
Example:
Show that the premises
“Everyone in this discrete math class has taken a course in
computer science” and “Ali is a student in this class”
Imply the conclusion “Ali has taken a course in computer science”
D(x): “x is in discrete math class”
C(x): “x has taken a course in computer science”
x (D(x)  C(x))
Premise #1
D(Ali)
Premise #2
 C(Ali)
Step
Reason
1 x (D(x)  C(x))
Premise #1
2 D(Ali)  C(Ali)
Univ. instantiation
3 D(Ali)
Premise #2
4 C(Ali)
Modus ponens on 2,3
I. Arwa Linjawi & I. Asma’a Ashenkity
Fall 2008/2009
Inference Rules for Quantifiers
Example:
Show that the premises
“A student in this class has not read the book” and “Everyone in
this class passed the first exam “
Imply the conclusion “someone who passed the exam has not
read the book”
C(x) = “x is in this class”
B(x) = “ x has read the book”
P(x)= “ x passed the first exam “
I. Arwa Linjawi & I. Asma’a Ashenkity
Fall 2008/2009
Inference Rules for Quantifiers
x(C(x)  B(x))
x(C(x) P(x))
  x(P(x)  B(x))
Step
1.  x(C(x)  B(x))
2. C(a)  B(a)
3. C(a)
4. x(C(x) P(x))
5. C(a) P(a)
6. P(a)
7. B(a)
8. P(a)  B(a)
9.  x(P(x)  B(x))
Premise #1
Premise #2
Reason
premise 1
Existential instantiation from (1)
Simplification from 2
premise 2
Universal instantiation from (4)
Modus ponens from (3) and (5)
Simplification from 2
conjunction from 6 and 7
Existential generalization from 8
I. Arwa Linjawi & I. Asma’a Ashenkity