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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
NOT¬A!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

AB¬A ∧ ¬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 Morgan’s 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 deMorgan’s 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)