Argument and Inference - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics

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Argument and inference

• An argument is valid if and only if it is

impossible for all the premises to be true and

the conclusion to be false

• Rules of inference : use them to deduce

(construct) new statements from statements

that we already have

• Basic tools for establishing the truth of

statements

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Valid arguments

• is tautology

• When ((p→q)˄p) is true, both p→q and p are

ture, and thus q must be also be true

• This form of argument is true because when

the premises are true, the conclusion must be

true

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(( p → q )∧ p ) → q

Example

• p: “You have access to the network”

• q: “You can change your grade”

• p→q: “If you have access to the network, then

you can change your grade”

“If you have access to the network, then you

can change your grade” (p→q)

“You have access to the network” (p)

so “You can change your grade” (q)

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Rules of inference for propositional

logic

• Can always use truth table to show an

argument form is valid

• For an argument form with 10 propositional

variables, the truth table requires 2

rows

• The tautology is the rule of

inference called modus ponens ( mode that

affirms ), or the law of detachment

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(( p → q )∧ p ) → q

q p q p ∴ →

Example

• If both statements “If it snows today, then we

will go skiing” and “It is snowing today” are

true.

• By modus ponens, it follows the conclusion

“We will go skinning” is true

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10

Example

• “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” Can we conclude “We will be home by sunset”? 11 8 ) modusponensusing(6)and (7) 7 ) hypothesis 6 ) modusponensusing(4) 5 ) hypothesis 4 ) modustollensusing(2)and(3) 3 ) hypothesis 2 ) simplicationusing(1) 1 ) hypothesis t s t s r s r r p p p q → ¬ → ¬ → ¬ ¬ p ∧ q ¬ ∧ r → p ¬ r → s s → t t

Resolution

• Based on the tautology

• Resolvent:

• Let q=r, we have

• Let r=F, we have

• Important in logic programming, AI, etc.

13 (( p ∨ q )∧(¬ p ∨ r ))→( q ∨ r ) ( p ∨ q )∧(¬ p ∨ q )→ q ( p ∨ q )∧¬ p → q q ∨ r

Example

• “Jasmine is skiing or it is

not snowing”

• “It is snowing or Bart is

playing hockey”

imply

• “Jasmine is skiing or

Bart is playing hockey”

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q r

p r

q p

Fallacies

• Inaccurate arguments

• is not a tautology as it is false

when p is false and q is true

• 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

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(( p → q )∧ q )→ p

( p → q )∧ q

Example

• is it correct to conclude ┐q?

• Fallacy: the incorrect argument is of the form

as ┐p does not imply ┐q

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( p → q )∧¬ p

( p → q )≡¬ q →¬ p

Example

• “Everyone in this discrete mathematics has

taken a course in computer science” and

“Marla is a student in this class” imply “Marla

has taken a course in computer science”

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4. ( ) modusponensfrom(2)and (3)
5. ( ) premise
6. ( ) ( ) universalinstantiationfrom(1)
7. ( ( ) ( )) premise c Marla d Marla d Marla c Marla x d x c x →

Example

• “A student in this class has not read the book”, and “Everyone

in this class passed the first exam” imply “Someone who

passed the first exam has not read the book”

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9. ( ( ) ( )) existentialgeneralizationform (8)
10. ( ) ( ) conjunctionof (6)and(7)
11. ( ) simplicationfrom(2)
12. ( ) modusponensfrom(3)and(5)
13. ( ) ( ) universalinstantiationfrom(4)
14. ( ( ) ( ) premise
15. ( ) simpliciationfrom(2)
16. ( ) ( ) existentialinstantiationfrom(1)
17. ( ( ) ( )) premise x p x b x p a b a b a p a c a p a x c x p x c a c a b a x c x b x ∃ ∧ ¬