Rules of Inference - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

Download Rules of Inference - Discrete Mathematics - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity!

CSE115/ENGR160 Discrete Mathematics 01/26/

1.6 Rules of Inference

• Proof : valid arguments that establish the truth of a mathematical statement
• Argument : a sequence of statements that end with a conclusion
• Valid : the conclusion or final statement of the argument must follow the truth of proceeding statements or premise of the argument

Valid arguments in propositional

logic

• Consider the following arguments involving propositions “If you have a correct password, then you can log onto the network” “You have a correct password” therefore, “You can log onto the network”

4

premises

conclusion

q

p

p q

∴

→

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

5

(( p → q )∧ p ) → q

Example

“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)

• Valid arguments
• But the conclusion is not true
• Argument form : a sequence of compound propositions involving propositional variables

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 10 rows
• The tautology is the rule of inference called modus ponens ( mode that affirms ), or the law of detachment

8

(( p → q )∧ p ) → q

q

p q

p

∴

→

Example

• The premises of the argument are p→q and p, and q is the conclusion
• This argument is valid by using modus ponens
• But one of the premises is false, consequently we cannot conclude the conclusion is true
• Furthermore, the conclusion is not true

10

Isitavalidargument?Isconclusion true?

4 )^9 2 Consequently,( 2 ) 2 (^3

2 ). Weknowthat 2 3 2 then( 2 ) (^3 2 If 2 3 2 2

2 2 = > =

Example

• “If you send me an email message, then I will finish my program”
• “If you do not send me an email 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” 13

6 ) hypotheticalsyllogismusing(4)and (5)

5 ) hypothesis

4 ) hypotheicalsyllogismusing(2)and(3)

3 ) hypothesis

2 ) contrapositiveof (1)

1 ) hypothesis

q s

r s

q r

p r

q p

p q

¬ →

→

¬ →

¬ →

¬ →¬

→

p → q

¬ p → r

r → s

¬ q → s

Resolution

• Based on the tautology
• Resolvent:
• Let q=r, we have
• Let r=F, we have
• Important in logic programming, AI, etc.

14

(( p ∨ q )∧(¬ p ∨ r ))→( q ∨ r )

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

q ∨ r

Example

• To construct proofs using resolution as the only rule of inference, the hypotheses and the conclusion must be expressed as clauses
• Clause : a disjunction of variables or negations of these variables

16

r s r s

p q r p r q r

p q r r s p s

→ ≡¬ ∨

∧ ∨ ≡ ∨ ∧ ∨

∧ ∨ → ∨ ( ) ( ) ( )

Show ( ) and imply

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

17

(( p → q )∧ q )→ p

( p → q )∧ q

Inference with quantified

statements

19

Instantiation: c is one particular member of the domain

Generalization: for an arbitrary member c

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”

20

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 →

∀ →