CPSC 121: Mode els of Computation Un nit 4 Propositiona l Logic - - PDF document

SLIDE 1

CPSC 121: Mode

Un Propositiona

Based on slides by Patrice Be Based on slides by Patrice Be

els of Computation

nit 4 l Logic Proofs

lleville and Steve Wolfman lleville and Steve Wolfman

Pre-Class Learning Pre-Class Learning

By the start of this class yo By the start of this class yo

Use truth tables to establish inference. Given a rule of inference an that correspond to the rule's t t t i li d b a new statement implied by

Unit 4 - Propositional Proofs

Goals Goals

• u should be able to
• u should be able to

h or refute the validity of a rule of nd propositional logic statements s premises, apply the rule to infer th i i l t t t the original statements.

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SLIDE 2

Quiz 4 Feedback: Quiz 4 Feedback:

Overall: Overall: Issues: We will discuss the open-e

p

Unit 4 - Propositional Proofs

ended question soon. q

3

In-Class Learning G In-Class Learning G

By the end of this unit you By the end of this unit, you

Determine whether or not a and explain why it is valid o p y Explore the consequences o statements by application of l i ll i d t rules, especially in order to desired form. Devise and attempt multiple Devise and attempt multiple for proving a propositional lo

• r premises.

Unit 4 - Propositional Proofs

• als
• als

u should be able to u should be able to

propositional logic proof is valid, r invalid.

• f a set of propositional logic

f equivalence and inference t t t i t massage statements into a e different appropriate strategies e different, appropriate strategies

• gic statement follows from a list

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SLIDE 3

Where We Are in Th Where We Are in Th

Theory: Theory:

How can we convince ourse what it's supposed to do? pp

In general

We need to prove that it wo p

We have done a few proof

p

Now we will learn Now we will learn

How to decide if a proof is v How to write proofs in Engli

• e p oo s

g

Unit 4 - Propositional Proofs

he Big Stories he Big Stories

elves that an algorithm does rks.

fs last week.

valid in a formal setting. sh. s

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What is Proof? What is Proof?

A rigorous formal argumen A rigorous formal argumen

truth of a proposition, given premises. premises.

In other words:

A proof is used to convince A proof is used to convince truth of a conditional propos Every step must be well just

Writing a proof is a bit like

you do it step by step, and make sure that you underst previous steps.

Unit 4 - Propositional Proofs

nt that demonstrates the nt that demonstrates the n the truth of the proof’s

• ther people (or yourself) of the
• ther people (or yourself) of the

sition. tified.

writing a function:

and how each step relates to the

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SLIDE 4

Things we'd like to p Things we d like to p

We can build a combinatio We can build a combinatio

truth table.

We can build any digital lo We can build any digital lo

NOR gates.

The maximum number of s The maximum number of s

students is n(n-1)/2.

No general algorithm exist No general algorithm exist

fewer than n log2n compar

There are problems that no There are problems that no

Unit 4 - Propositional Proofs

prove prove

• nal circuit matching any
• nal circuit matching any

gic circuit using only 2-input gic circuit using only 2-input swaps we need to order n swaps we need to order n s to sort n values using s to sort n values using risons.

• algorithm can solve
• algorithm can solve.

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What is a Propositio What is a Propositio

A propositional logic proof A propositional logic proof

propositions, where each p

a premise a premise the result of applying a logic inference to one or more ea

and whose last proposition

These are good starting po

simpler than the more free p later

Only a limited number of ch

Unit 4 - Propositional Proofs

• nal Logic Proof
• nal Logic Proof

consists of a sequence of consists of a sequence of proposition is one of

cal equivalence or a rule of arlier propositions.

n is the conclusion.

• int, because they are

e-form proofs we will discuss p

• ices at each step.

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SLIDE 5

Meaning of Proof Meaning of Proof

Suppose you W Suppose you

proved this:

Premise 1 W

A

Premise-1 Premise-2

B

... Premise-n

B C

• ∴Conclusion

D E

Unit 4 - Propositional Proofs

What does it mean? What does it mean?

A. Premises 1 to n can be true B Premises 1 to n are true

• B. Premises 1 to n are true
• C. Conclusion can be true

D. Conclusion is true

• E. None of the above.

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Meaning of Proof Meaning of Proof

What does this

A.

What does this

argument mean?

B

Premise-1 P i 2

B.

Premise-2 ...

C.

Premise-n

• D.

∴Conclusion

E.

Unit 4 - Propositional Proofs

Premise-1 ˄ … ˄ Premise-n ˄ Conclusion P i 1 ˅ ˅ P i ˅ Premise-1 ˅ … ˅ Premise-n ˅ Conclusion Premise-1 ˄ … ˄ Premise-n → Conclusion Conclusion Premise-1 ˄ … ˄ Premise-n ↔ Conclusion None of the above.

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SLIDE 6

Why do we want val Why do we want val

~p ~p______

∴ ~(p v q) a.

This is valid by generaliza

b.

This is valid because any also true. Thi i lid b h

c.

This is valid by some othe

d.

This is invalid because w t b t ( ) i f l true but ~(p v q) is false.

e.

None of these.

id rules? id rules?

ation (p ⇒ p v q). ytime ~p is true, ~(p v q) is l er rule. hen p = F and q = T, ~p is

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Basic Rules of Infere Basic Rules of Infere

Modus Ponens:

p → q p q p q

Generalization:

p p

Generalization:

p p p ˅ q q → p

Conjunction:

p q p ˄ q

Transitivity:

p → q q → r p → r p

Contradiction:

p → F ~p

Unit 4 - Propositional Proofs

ence ence

Modus Tollens:

p → q p q ~q ~p

Specialization:

p ˄ q p ˄ q

Specialization:

p ˄ q p ˄ q p q

Elimination:

p ˅ q p ˅ q ~p ~q q p

Proof by cases:

p ˅ q p → r q → r q r

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SLIDE 7

Onnagata Problem fr Onnagata Problem fr

Critique the following argu Critique the following argu

by Julian Baggini on logica

Premise 1: If women are to Premise 1: If women are to women then men must be to men, and vice versa. Premise 2: And yet, if the o too close to femininity to po too close to masculinity to p too close to masculinity to p Conclusion: Therefore, the women are not too close to

Note: onnagata are male a

characters in kabuki theatr

Unit 4 - Propositional Proofs

rom Online Quiz #4 rom Online Quiz #4

ment drawn from an article ment, drawn from an article al fallacies.

• close to femininity to portray
• close to femininity to portray
• o close to masculinity to play
• nnagata are correct, women are

rtray women and yet men are not play men play men. e onnagata are incorrect, and femininity to portray women.

actors portraying female re.

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Onnagata Problem Onnagata Problem

Which definitions should we Which definitions should we

a) w = women, m = men, f =

• nnagata, c = correct
• nnagata, c

correct b) w = women are too close close to masculinity, pw = men portray men, o = onna c) w = women are too close m = men are too close to m m = men are too close to m

• nnagata are correct

d) None of these, but anothe ) , e) None of these, and this pr with propositional logic.

Unit 4 - Propositional Proofs

e use? e use?

femininity, m = masculinity, o = to femininity, m = men are too women portray women, pm = agata are correct to femininity to portray women, masculinity to portray men o = masculinity to portray men, o = er set of definitions works well. roblem cannot be modeled well

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SLIDE 8

Onnagata Problem Onnagata Problem

Which of these is not an ac

the statements?

• A. w ↔ m
• B. (w → m) ∧ (m → w)
• C. o → (w ∧ ~m)
• D. ~o ∧ ~w
• E. All of these are accurate tr

S th t i

So, the argument is:

Unit 4 - Propositional Proofs

ccurate translation of one of

anslations.

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Onnagata Problem Onnagata Problem

Do the two premises con Do the two premises con

p1 ˄ p2 ≡ F)?

A Yes A. Yes B. No

• C. Not enough information

g

Is the argument valid? What can we prove? What can we prove?

We can prove that the On We can not prove that wo p femininity to portray wome

• What other scenario is

Unit 4 - Propositional Proofs

tradict each other (that is is tradict each other (that is, is

to tell

A: Yes B: No C:?

nagata are wrong. men are not too close to en. consistent with the premises?

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SLIDE 9

Onnagata Problem Onnagata Problem

Do the two premises con Do the two premises con

p1 ˄ p2 ≡ F)?

A Yes A. Yes B. No

• C. Not enough information

g

Is the argument valid?

g

A: Yes B: No C: ?

Unit 4 - Propositional Proofs

tradict each other (that is is tradict each other (that is, is

to tell

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Onnagata Problem Onnagata Problem

What can we prove? What can we prove? Can we prove that the On

A Yes

• A. Yes
• B. No
• C. Not enough information
• C. Not enough information

Can we prove that wome

femininity to portray wom y p y

• A. Yes
• B. No
• C. Not enough information

What other scenario is co

Unit 4 - Propositional Proofs

nnagata are wrong. en are not too close to men?

• nsistent with the premises?

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SLIDE 10

Proof Strategies Proof Strategies

Look at the information you Look at the information you

Is there irrelevant informatio Is there critical information y Is there critical information y

Work backwards from the

Especially if you have made Especially if you have made a step or two.

Don't be afraid of inferring

g you are not quite sure whe get to the conclusion you w

Unit 4 - Propositional Proofs

u have u have

• n you can ignore?

you should focus on? you should focus on?

end

e some progress but are missing e some progress but are missing

new propositions, even if p p , ether or not they will help you want.

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Proof strategies (con Proof strategies (con

If you are not sure of the c

between between

trying to find an example tha using the place where your g p y the counterexample trying to prove it, using your it th f you write the proof.

Unit 4 - Propositional Proofs

ntinued) ntinued)

• nclusion, alternate

at shows the statement is false, proof failed to help you design p p y g r failed counterexample to help

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SLIDE 11

Example Example

To prove: Wha To prove:

~(q ∨ r)

Wha

A. ~

(u ∧ q) ↔ s ~s → ~p

B.

s

s → p___ ∴ ~p

s

• C. D

~

• D. A

E N

• E. N

Unit 4 - Propositional Proofs

at will the strategy be? at will the strategy be?

Derive ~u so you can derive ~s s Derive u ∧ q so you can get

s

Derive ~s by deriving first ~(u ∧ q) (u ∧ q) Any of the above will work None of the above will work None of the above will work

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Example (cont') Example (cont )

Proof:

• 1. ~(q ∨ r)

Premise 2 (u ∧ q) ↔ s Premise

• 2. (u ∧ q) ↔ s

Premise

• 3. ~s → ~p

Premise

• 4. ~q ∧ ~r

De Morgan’s (1)

• 5. ~q

Specialization (4)

• 6. ((u ∧ q) → s) ∧ Bicond (2)

(s → (u ∧ q)) (s → (u ∧ q))

• 7. s → (u ∧ q)

Specialization (6)

• 8. ????

????

• 9. ~(u ∧ q)

????

• 10. ~s

Modus tollens (7, 9)

• 11. ~p

Modus ponens (3,10) p p ( , )

Unit 4 - Propositional Proofs

~(q ∨ r) (u ∧ q) ↔ s

What is in step 8?

~s → ~p___ ∴ ~p

What is in step 8?

A.

u ∧ q

• B. ~u ∨ ~q
• C. s
• D. ~s
• E. None of the above

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SLIDE 12

Example (cont') Example (cont )

Proof:

• 1. ~(q ∨ r)

Premise 2 (u ∧ q) ↔ s Premise

• 2. (u ∧ q) ↔ s

Premise

• 3. ~s → ~p

Premise

• 4. ~q ∧ ~r

De Morgan’s (1)

• 5. ~q

Specialization (4)

• 6. ((u ∧ q) → s) ∧ Bicond (2)

(s → (u ∧ q)) (s → (u ∧ q))

• 7. s → (u ∧ q)

Specialization (6)

• 8. ????

????

• 9. ~(u ∧ q)

????

• 10. ~s

Modus tollens (7, 9)

• 11. ~p

Modus ponens (3,10) p p ( , )

Unit 4 - Propositional Proofs

~(q ∨ r) (u ∧ q) ↔ s

Which rule was used

~s → ~p___ ∴ ~p

Which rule was used

in step 8?

A.

modus ponens

B De Morgan's

• B. De Morgan s
• C. modus tollens
• D. generalization

E N f th b

• E. None of the above

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Example (cont') Example (cont )

Proof:

• 1. ~(q ∨ r)

Premise 2 (u ∧ q) ↔ s Premise

• 2. (u ∧ q) ↔ s

Premise

• 3. ~s → ~p

Premise

• 4. ~q ∧ ~r

De Morgan’s (1)

• 5. ~q

Specialization (4)

• 6. ((u ∧ q) → s) ∧ Bicond (2)

(s → (u ∧ q)) (s → (u ∧ q))

• 7. s → (u ∧ q)

Specialization (6)

• 8. ????

????

• 9. ~(u ∧ q)

????

• 10. ~s

Modus tollens (7, 9)

• 11. ~p

Modus ponens (3,10) p p ( , )

Unit 4 - Propositional Proofs

~(q ∨ r) (u ∧ q) ↔ s

Which rule was used

~s → ~p___ ∴ ~p

Which rule was used

in step 9?

A.

modus ponens

B De Morgan's

• B. De Morgan s
• C. modus tollens
• D. generalization

E N f th b

• E. None of the above

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SLIDE 13

Another Example Another Example

Prove the following argume Prove the following argume

p p → r p → r p → (q ˅ ~r) ~q ˅ ~s q ∴ s

Unit 4 - Propositional Proofs

ent: ent:

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Limitations of Truth T Limitations of Truth T

Why can we not just use tr Why can we not just use tr

propositional logic theorem

A

No reason; truth tables

• A. No reason; truth tables
• B. Truth tables scale poo

C Rules of inference and

• C. Rules of inference and

prove theorems that c tables tables.

• D. Truth tables require in

inference can be applie inference can be applie

Unit 4 - Propositional Proofs

Tables Tables

ruth tables to prove ruth tables to prove ms? s are enough s are enough.

• rly to large problems.

d equivalence rules can d equivalence rules can cannot be proven with truth nsight to use, while rules of ed mechanically ed mechanically.

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SLIDE 14

Limitations of Logica Limitations of Logica

Why not use logical equiva Why not use logical equiva

conclusions follow from the A No reason; logical equ

• A. No reason; logical equ
• B. Logical equivalences

problems problems.

• C. Rules of inference and

theorems that cannot b theorems that cannot b equivalences. D Logical equivalences

• D. Logical equivalences

rules of inference can

Unit 4 - Propositional Proofs

al Equivalences al Equivalences

alences to prove that the alences to prove that the e premises? uivalences are enough uivalences are enough. scale poorly to large d truth tables can prove be proven with logical be proven with logical require insight to use while require insight to use, while be applied mechanically.

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One More Remark One More Remark

Consider the following: Consider the following:

George is rich If George is rich then he will p If George is rich then he will p ∴ George will pay your tuition

Is this argument valid? Is this argument valid?

• A. Yes

B No

• B. No
• C. Not enough information

Sh

ld t iti

Should you pay your tuition

that George will pay it for y

Unit 4 - Propositional Proofs

ay your tuition ay your tuition .

n to tell h ld n, or should you assume you? Why?

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SLIDE 15

Exercises Exercises

Prove that the following argum

Prove that the following argum

p → q q → (r ^ s) q → (r s) ~r v (~t v u) p ^ t p ∴ u

Given the following premises,

p → q p v ~q v r (r ^ ~p) v s v ~p ~r

Unit 4 - Propositional Proofs

ment is valid: ment is valid: what can you prove?

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Further Exercises Further Exercises

Hercule Poirot has been as Hercule Poirot has been as

• ut who closed the lid of hi

cat inside. Poirot interroga cat inside. Poirot interroga Akilna and Eiluj. One and o in the piano. Plus, one alw

Eiluj: I did not put the cat in less than $60 to help him st Akilna: Eiluj did it. Urquhart

Who put the cat in the pian

Unit 4 - Propositional Proofs

sked by Lord Maabo to find sked by Lord Maabo to find is piano after dumping the tes two of the servants, tes two of the servants,

• nly one of them put the cat

ways lies and one never lies.

the piano. Urquhart gave me tudy. $ paid her $50 to help him study.

no?

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SLIDE 16

Reading for Next Le Reading for Next Le

Online quiz #5 is tentativel Online quiz #5 is tentativel

at 5:00pm

Assigned reading for the q Assigned reading for the q

Epp, 4th edition: 3.1, 3.3 Epp 3rd edition: 2 1 2 3 Epp, 3rd edition: 2.1, 2.3 Rosen, 6th edition: 1.3, R h di i 1 4 Rosen, 7th edition: 1.4,

Unit 4 - Propositional Proofs

ecture ecture

y due Monday October 8th y due Monday, October 8th uiz: uiz: 3 3 1.4 1.5

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