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Logic

Planning
Logic
Planning

Lesson Preview

Formal notation

Conjunctions, disjunctions, negations, implications

Truth tables

Rules of inference

Resolution theorem proving

Why do we need formal logic?

Soundness: Only valid conclusions can be proven.

Completeness: All valid conclusions can be proven.

If an animal has feathers, then it is a bird

If an animal lays eggs and it flies, then it is a bird
Vertebrate
Bird
Bluebird
Penguin
Eagle

Brick
Block
Block
A Foo
Brick
touches
touches
supports
supports
supports
supports
Mark the sufficient conditions:
If
A brick supports two bricks
A brick supports two blocks
Those two blocks touch
Those two blocks do not touch
Those two blocks support a block
Those two blocks support a brick
then the object is a foo

Predicate:
A function that maps object arguments to true or false values

Feathers(bluebird)

Predicate:
A function that maps object arguments to true or false values

If an animal has feathers, then it is a bird

Feathers(animal)

If Feathers(animal):
Then Bird(animal)

If an animal lays eggs and it flies, then it is a bird

Lays-eggs(animal)

Flies(animal)

If Lays-eggs(animal)
Flies(animal):
Then Bird(animal)

If an animal lays eggs or it flies, then it is a bird

Lays-eggs(animal)

Flies(animal)

If Lays-eggs(animal)
Flies(animal):
Then Bird(animal)

Flies(animal)

Bird(animal)

If Flies(animal) Bird(animal):
Then Bat(Animal)

If an animal flies and is not a bird, it is a bat.

If Lays-eggs(animal) Flies(animal):
Then Bird(animal)
Lays-eggs(animal) Flies(animal) Bird(animal)

OperatorSymbolAccepted Symbol
ANDA BA & B
A && B
ORA BA | B
A || B
NOTA!A
~A
IMPLIESA BA = B
A == B
A => B

If an animal lays eggs and does not have feathers, it is a reptile.
Lays-eggs(animal) Feathers(animal) Reptile(animal)

If an animal has feathers or has talons, it is a bird.
Feathers(animal) Talons(animal) Bird(animal)

If an animal lays eggs, has a beak, and flies, it is a duck.
Lays-eggs(animal) Beak(animal) Flies(animal) Duck(animal)

If an animal lays eggs, has a beak, and do not fly, it is a platypus.
Lays-eggs(animal) Beak(animal) Flies(animal) Platypus(animal)

ABA B
TrueTrueTrue
TrueFalseTrue
FalseTrueTrue
FalseFalseFalse

ABA B
TrueTrueTrue
TrueFalseTrue
FalseTrueFalse
FalseFalseTrue

ABA B
TrueTrueFalse
TrueFalseFalse
FalseTrueFalse
FalseFalseTrue

ABCA (B C)
TrueTrueTrueTrue
TrueTrueFalseTrue
TrueFalseTrueTrue
TrueFalseFalseTrue
FalseTrueTrueFalse
FalseTrueFalseTrue
FalseFalseTrueFalse
FalseFalseFalseFalse

ABA BB A
TrueTrueTrueTrue
TrueFalseFalseFalse
FalseTrueFalseFalse
FalseFalseFalseFalse

Commutative Property

Distributive Property
ABCA (B C)(A B) (A C)
TrueTrueTrueTrueTrue
TrueTrueFalseTrueTrue
TrueFalseTrueTrueTrue
TrueFalseFalseFalseFalse
FalseTrueTrueFalseFalse
FalseTrueFalseFalseFalse
FalseFalseTrueFalseFalse
FalseFalseFalseFalseFalse

Associative Property
ABCA (B C)(A B) C
TrueTrueTrueTrueTrue
TrueTrueFalseTrueTrue
TrueFalseTrueTrueTrue
TrueFalseFalseTrueTrue
FalseTrueTrueTrueTrue
FalseTrueFalseTrueTrue
FalseFalseTrueTrueTrue
FalseFalseFalseFalseFalse

de Morgans Law
AB(A B)A B
TrueTrueFalseFalse
TrueFalseTrueTrue
FalseTrueTrueTrue
FalseFalseTrueTrue

Truth of Implications
ABA B
TrueTrueTrue
TrueFalseFalse
FalseTrueTrue
FalseFalseTrue

Implication Elimination
Given:
a b

Rewrite as:
a b
Given:
Feathers Bird

Rewrite as:
Feathers Bird

Modus Ponens
Sentence 1: p q
Sentence 2: p
Sentence 3: q

Feathers Bird
Feathers
Bird
Modus Tollens
Sentence 1: p q
Sentence 2: q
Sentence 3: p

Feathers Bird
Bird
Feathers
Rules of Inference: Instantiate general rules to prove specific claims.

Prove: Harry is a bird
By Modus Ponens
S1: Feathers(animal) Bird(animal)
S2: Feathers(Harry)
S3: Bird(Harry)

Prove: Harry is a bird
S1: Feathers(animal) Bird(animal)
S2: Feathers(Harry)
S3: Bird(Harry)

Prove: Buzz does not have feathers
S1: Feathers(animal) Bird(animal)
S2: Bird(Buzz)
S3: Feathers(Buzz)
By Modus Tollens

Prove: Buzz does not have feathers
S1: Feathers(animal) Bird(animal)
S2: Bird(Buzz)
S3: Feathers(Buzz)

For one animal:

Lays-eggs(animal) Flies(animal) Bird(animal)

For all animals:

x[Lays-eggs(x) Flies(x) Bird(x)]

Universal Quantifier

For one animal:

Lays-eggs(animal) Flies(animal) Bird(animal)

For at least one animal:

y[Lays-eggs(y) Flies(y) Bird(y)]

Existential Quantifier

We know:
S1: can-move liftable

We find:
S2: can-move

How do we prove the box is not liftable?

We know:
S1: can-move liftable

By implication elimination:
S1: can-move liftable

We find:
S2: can-move

We assume:
S3: liftable

How do we prove the box is not liftable?

How do we prove the box is not liftable?
S1: can-move liftable
S2: can-move
S3: liftable

We know:
S1: can-move battery-full liftable

We find:
S2: can-move
S3: battery-full

How do we prove the box is not liftable?

We know:
S1: can-move battery-full liftable

By implication elimination:
S1: (can-move battery-full) liftable

We find:
S2: can-move
S3: battery-full

How do we prove the box is not liftable?

We know:
S1: can-move battery-full liftable

By implication elimination:
S1: (can-move battery-full) liftable

By deMorgans Law:
S1: can-move battery-full liftable

We find:
S2: can-move
S3: battery-full

How do we prove the box is not liftable?

How do we prove the box is not liftable?
S1: can-move liftable battery-full
S2: can-move
S3: battery-full

How do we prove the box is not liftable?
S1: can-move liftable battery-full
S2: can-move
S3: battery-full
S4: liftable

How do we prove this is a bird?
If an animal has wings and does not have fur, it is a bird.

Write in formal logic:
has-wings has-fur bird

(Use the predicates has-wings, has-fur, and bird)