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Discrete StructuresRules of inference

Dr. Muhammad HumayounAssistant Professor

COMSATS Institute of Computer Science, Lahore.mhumayoun@ciitlahore.edu.pk

https://sites.google.com/a/ciitlahore.edu.pk/dstruct/

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Rules of InferenceValid Arguments in Propositional Logic

Assume you are given the following two statements: “if you are in this class, then you will get a grade” “you are in this class”Therefore, “You will get a grade”

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Modus Ponens(Latin for “the way that affirms by affirming”

• If it snows today, then we will go skiing• Hypothesis: It is snowing today• By modus ponens, the conclusion is:• We will go skiing

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• If I smoke, then I cough• I Smoke_______________________• I cough

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Modus Tollens(Latin for "the way that denies by denying")

● Assume you are given the following two statements: “you will not get a grade” “if you are in this class, you will get a grade”

● Let p = “you are in this class”● Let q = “you will get a grade”

● By Modus Tollens, you can conclude that you are not in this class

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Addition

• If you know that p is true, then p q will ALWAYS be true i.e. p → p q

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Addition

• If you know that p is true, then p q will ALWAYS be true i.e. p → p q

• p : “It is below freezing now”• q : “It is raining now”• “It is below freezing or raining now”• “If it is below freezing now then it is below

freezing or raining now”

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Simplification

• If p q is true, then p will ALWAYS be true i.e. p q → p

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Simplification

• If p q is true, then p will ALWAYS be true i.e. p q → p

• p: “It is below freezing”• q: “It is raining now”

• p q : It is below freezing and raining now.

• p q → p: It is below freezing and raining now implies that it is below freezing

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Hypothetical syllogism

• If it rains today, then we will not have a barbecue today.

• If we do not have a barbecue today, then we will have a barbecue tomorrow.

• Therefore, if it rains today, then we will have a barbecue tomorrow.

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Disjunctive syllogism

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Resolution

• Computer programs have been developed to automate the task of reasoning and proving theorems.

• Many of these programs make use resolution

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Rules of Inference to Build Arguments• 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 (Conclusion)

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Rules of Inference to Build Arguments• 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 (Conclusion)• p: It is sunny this afternoon• q: It is colder than yesterday• r: We will go swimming• s: We will take a canoe trip• t: We will be home by sunset

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Rules of Inference to Build Arguments• 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 (Conclusion) • p: It is sunny this afternoon• q: It is colder than yesterday• r: We will go swimming• s: We will take a canoe trip• t: We will be home by sunset

Definitions

● An Argument in propositional logic is a sequence of propositions that end with c conclusion.

● All except the final proposition are called premises.

● The final proposition is called conclusion.● An argument is valid if the truth of all

premises implies that the conclusion is true. i.e. is a tautology.

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Rules of Inference to Build Arguments

[ (¬𝒑∧𝒒 )∧ (𝒓→𝒑 )∧ (¬𝒓→𝒔 )∧(𝒔→𝒕)]→𝒕𝒑∧𝒒→𝒑

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Rules of Inference to Build Arguments

[ (¬𝒑∧𝒒 )∧ (𝒓→𝒑 )∧ (¬𝒓→𝒔 )∧(𝒔→𝒕)]→𝒕

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Rules of Inference to Build Arguments

[ (¬𝒑∧𝒒 )∧ (𝒓→𝒑 )∧ (¬𝒓→𝒔 )∧(𝒔→𝒕)]→𝒕

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Rules of Inference to Build Arguments

[ (¬𝒑∧𝒒 )∧ (𝒓→𝒑 )∧ (¬𝒓→𝒔 )∧(𝒔→𝒕)]→𝒕

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• If you send me an e-mail message, then I will finish writing the program

• If you do not send me an e-mail message, then I will go to sleep early

• If I go to sleep early, then I will wake up feeling refreshed

____________________________________• If I do not finish writing the program, then I

will wake up feeling refreshed

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• If you send me an e-mail message, then I will finish writing the program

• If you do not send me an e-mail message, then I will go to sleep early

• If I go to sleep early, then I will wake up feeling refreshed

____________________________________• If I do not finish writing the program, then I will

wake up feeling refreshed

• p = You send me an e-mail• q = I will finish writing program• r = I will go to sleep early• s = l will wake up feeling refreshed

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• Hypotheses: and imply the conclusion:

• de Morgan’s law

• resolution

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Fallacies

• Several common fallacies arise in incorrect arguments.

• The proposition is not a tautology, because it is false when p is false and q is true

• There are many incorrect arguments that treat this as a tautology

• This type of incorrect reasoning is called the fallacy of affirming the conclusion

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Example

• If you do every problem in this book, then you will learn discrete mathematics.

• You learned discrete mathematics.• Therefore, you did every problem in this book.

• p: You did every problem in this book• q: You learned discrete mathematics

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• If you do every problem in this book, then you will learn discrete mathematics.

• You learned discrete mathematics.______________________________________• Therefore, you did every problem in this book.

• p: You did every problem in this book• q: You learned discrete mathematics

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• If you do every problem in this book, then you will learn discrete mathematics.

• You learned discrete mathematics.______________________________________• Therefore, you did every problem in this book.

• p: You did every problem in this book• q: You learned discrete mathematics

• If and then

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• If you do every problem in this book, then you will learn discrete mathematics.

• You learned discrete mathematics.______________________________________• Therefore, you did every problem in this book.

• p: You did every problem in this book• q: You learned discrete mathematics

• If and then • Fallacy• It is possible for you to learn discrete mathematics in

some way other than by doing every problem in this book (Reading, Listening Lectures, doing some but not all problems).

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Rules of Inference for

Quantified Statements

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Example• Show that the premises:• "Everyone in this discrete mathematics class has

taken a course in computer science" and “Aslam is a student in this class"

• Imply the conclusion “Aslam has taken a course in computer science.“

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• D(x): x is in this discrete mathematics class• C(x): x has taken a course in computer science• Premises: ∀x(D(x) → C(x)) and D(Aslam)• Conclusion: C(Aslam)

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• D(x): x is in this discrete mathematics class• C(x): x has taken a course in computer science• Premises: ∀x(D(x) → C(x)) and D(Aslam)• Conclusion: C(Aslam)

• Steps Reason• ∀x(D(x) → C(x)) Premise• D(Aslam) → C(Aslam) Universal instantiation

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• D(x): x is in this discrete mathematics class• C(x): x has taken a course in computer science• Premises: ∀x(D(x) → C(x)) and D(Aslam)• Conclusion: C(Aslam)

• Steps Reason• ∀x(D(x) → C(x)) Premise• D(Aslam) → C(Aslam) Universal instantiation• D(Aslam) Premise

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• D(x): x is in this discrete mathematics class• C(x): x has taken a course in computer science• Premises: ∀x(D(x) → C(x)) and D(Aslam)• Conclusion: C(Aslam)

• Steps Reason• ∀x(D(x) → C(x)) Premise• D(Aslam) → C(Aslam) Universal instantiation• D(Aslam) Premise• C(Aslam) Modus ponens•

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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 first exam has not read the book.”

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• 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 first 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”• Premises: ???

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• 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 first 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”• Premises: x(C(x)∃ ∧￢ B(x)) and x(C(x) → ∀

P(x)).• The conclusion???

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• 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 first 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”• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → P(x) ∀

).• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )

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• Premises: x(C(x) ∃ ∧￢ B(x)) and x( C(x) → ∀P(x) ).

• The conclusion: x( P(x) ∃ ∧￢ B(x) )