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Inference Rules

CS160/CS122

Rosen: 1.5

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Whatislogic?

Logic is a truth-preserving system of inference

Inference: the process ofderiving (inferring) new

statements from old

statements

System: a set ofmechanistic

transformations, based

on syntax alone

Truth-preserving:If the initial

statements are

true, the inferred

statements will

be true

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Proposi0onalLogic

Anargumentisasequenceofproposi0ons:Premises(Axioms)arethefirstnproposi0ons

Conclusionisthen+1th(final)proposi0on.

Anargumentisvalidifisatautology,giventhatsarethepremises

(axioms)andistheconclusion

p1" p

2" ..." p

n( )# q

piq

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Hello

World

Proof By Truth Table

n If the conclusion is true in the truth tablewhenever the premises are true, it isprovedn Warning: when the premises are false, the

conclusion my be true orfalse

n Problem: given n propositions, the truthtable has 2

n

rowsn Proof by truth table quickly becomes

infeasible

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RulesofInference

n Aruleofinferenceisapre-provedrela0on:any0metheleHhandside(LS)istrue,the

righthandside(RS)isalsotrue.

n Therefore,ifwecanmatchapremisetotheLS(bysubs0tu0ngproposi0ons),wecan

assertthe(subs0tuted)RS

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ExampleRuleofInference

ModusPonensp" p# q( )( )# q

p

"

p# q

q

0 0 1 0 1

0 1 1 0 11 0 0 0 1

1 1 1 1 1

p

q

p" q

p" p# q( )

p" p# q( )( )# q

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Example

Given(p)Ifitisraining,then(q)thegrassiswet.(p)Itisraining.

Therefore,bymodusponens,(q)Thegrassiswet.

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Applyingrulesofinference

n Examplerule:A,ABBn ReadasAandAB,thereforeBn Thisrulehasaname:modusponens

n IfyouhavepremisesC,CDn Subs0tuteCforA,DforBnApplymodusponens

n ConcludeD

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RulesofInference

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LogicalEquivalences

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ASimpleProofn Given:X,XY,YZ,(Z)Wn Prove:W

Step Reason

1. Given (Premise)

2. Given (Premise)

3. Hypothetical Syllogism (1) & (2)

4. Given (Premise)

5. Modus Ponens (3) & (4)

6. Given (Premise)

7. Disjunctive Syllogism (5) & (6)

x! y

y! z

x! z

x

z

z!w

w

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ASimpleProofFromWords

InordertotakeCS161,ImustfirsttakeCS16andeitherM155orM16.IhavenottakenM155butIhavetakenCS161.ProvethatIhavetakenM16.

Firststep:assignproposi0onsn A:takeCS161n B:takeCS16n C:TakeM155n D:TakeM16

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Nowsetuptheproof

n Axioms:n AB(CD)n An C

n Conclusion:n D

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NowdotheProof

Step Reason

1. Given (Premise)

2. Given (Premise)

3. Modus Ponens (1) & (2)

4. Simplification

5. Given (Premise)

6. Disjunctive Syllogism (4) & (5)

A! B"(C#D)

A

B!(C"D)

C!D

D

C

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Example

Given: Conclude:p"q

r# p

r# s

s#

t

t

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ProofofExample

Step Reason

1. Premise

2. Simplification using (1)

3. Premise

4. Modus Tollens (2) & (3)

5. Premise

6. Modus Ponens (4) & (5)

7. Premise

8. Modus Ponens (6) & (7)

r" p

p"q

p

r

t

r" s

s

s" t

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AnotherExample

Given: Conclude:p" q

p" r

r" s

q" s

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ProofofAnotherExample

Step Reason

1. Premise

2. Implication law (1)

3. Premise

4. Hypothetical syllogism of(2) & (3)

5. Premise

6. Hypothetical syllogism of

(4) & (5)

p" r

p" q

q"p

q" r

r" s

q" s

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Proof by Rules of Inference (Example)

n By 2nd DeMorgansn By 1st DeMorgansn By double negationn By 2nd distributiven By definition ofn By commutative lawn By definition of

Prove: (p(pq)) (pq)

(p(pq)) p (pq)

p ((p)q)p(pq)

(pp) (pq) F (pq) (pq) F

(pq)