Download How logic works

Logic
Overview
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Scholastic logic
Propositional logic
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Predicate calculus
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Syntax and semantics
Inference rules
Syntax and semantics
Quantifiers
Higher-order logics and modal operators
Model theory
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The incremental, almost continuous nature of meaning
Knowledge representation as an evolutionary process
Features of Logic KR Languages
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Vocabulary
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Syntax
Semantics
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Logical symbols
Constants
Variables
Punctuation
Theory of reference
Theory of truth
Rules of inference
Automated Reasoning
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Premise: Given an appropriate KR language,
new knowledge can be inferred from existing
truths
Syllogisms
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Aristotle set out the concept of a threesentence structure to represent knowledge
All trailer trucks are 18 wheelers
Some Peterbilt is a trailer truck
Therefore, some Peterbilt is an 18 Wheeler
Sentence Types
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A: Universal affirmative – All a is b
I: Particular affirmative – Some a is b
E: Universal negative – No a is b
O: Particular negative – Some a is not b
Valid Syllogisms
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bArbArA
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A: All broad-leafed plants are deciduous
A: All vines are broadleaf plants
A: Therefore, all vines are deciduous
Valid Syllogisms (cont.)
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cElArEnt
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E: Nothing absent-minded is an elephant
A: All professors are absent-minded
E: Therefore, no professor is an elephant
Valid Syllogisms (cont.)
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dArII
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A: All trailer trucks are 18 wheelers
I: Some Peterbilt is a trailer truck
I: Therefore, some Peterbilt is an 18 Wheeler
Valid Syllogisms (cont.)
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fErIO
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E: No Corvette is a truck
I: Some Chevrolet is a Corvette
O: Therefore, some Chevrolet is not a truck
Boolean Algebra
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George Boole
System of representing and operating on
truth values
pxq
p+q
-p
AND
OR
NOT
First Order Logic (FOL)
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Extensions of Boolean Algebra and Aristotle’s
syllogisms
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Universal and existential quantifiers
Changes in syntax
Why FOL?
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Expressive power
Extensive Semantics
Best-defined model and proof theories
Syntax of Propositional Logic
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Symbols
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Logical constants True, False
Proposition symbols: P, Q, …
Logical connectives:
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not (aka “? ”)
and (aka “? ”)
or (aka “? ”)
=> (aka “? ”)
iff (aka “? ”)
Sentences are made from these symbols
Sentences
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True, False are sentences
Any proposition symbol is a sentence
Any sentences S1,… Sn combined with a
connective is a sentence
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(not S) is a sentence
(and S1 S2) is a sentence
(or S1 S2 S3 … Sn) is a sentence
(=> Sa Sc) is a sentence
(iff Sa Sc) is a sentence
Semantics of propositional logic
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Formal semantics = function from sentences to true,
false.
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True means that the sentence is true of the world
False means that the sentence isn’t true of the world
Truth tables define semantics of connectives
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See Sowa for details
Example
P
false
false
true
true
Q
false
true
false
true
(or P Q)
false
true
true
true
Rules of Inference: Examples
1.
Modus Ponens (aka implication-elimination)
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And-elimination
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Given P, Q, infer (and P Q)
Unit resolution
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Given (and P Q), infer P; or infer Q
And-introduction
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Given (=> P Q), P, infer Q
Given (or P Q), (not P), infer Q
Resolution
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Given (or P Q), (or (not Q) R), infer (or P R)
Using inference rules
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“On summer days in Memphis, it is either
scorching hot or overcast and muggy
outside. Today is a summer day in Memphis
and it is not scorching hot outside.”
Formalizing a situation
“On summer days in Memphis, it is either scorching hot or
overcast and muggy outside. Today is a summer day in
Memphis and it is not scorching hot outside.”
 Summer = “It is a summer day in Memphis”
 Hot = “It is scorching hot outside”
 Overcast = “It is overcast outside”
 Muggy = “It is muggy outside”
(=> Summer (or Hot (and Overcast Muggy)))
(and Summer (not Hot))
Using the Inference Rules
1.
2.
3.
4.
5.
6.
7.
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(=> Summer (or Hot
(and Overcast Muggy))) Given
(and Summer (not Hot )) Given
Summer [AE 2]
(or Hot (and Overcast Muggy)) [MP 1,3]
(not Hot) [AE 2]
(and Overcast Muggy) [UR 4,5]
Overcast [AE 6]
Muggy [AE 6]
Properties of inference rules
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They are sound if, when the premises are
true, the consequence is true
Many different sets of inference rules have
been created with the same expressive
power (i.e., complete)
Sometimes restricted sets of inference rules
used for tractability
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Example: Horn clauses
(=> (and A1 … An) C)
Predicate Calculus
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Propositions can have structure:
(<predicate> . <terms>)
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Terms are either
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Constants denoting objects in some universe (e.g.,
Block32)
Expressions denoting objects via a function (e.g., (color
Block32))
Predicates describe attributes and relationships
Examples:
(Red Block32)
(On Block32 Table)
(Above Block24 Block32)
Propositions express facts
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Concrete facts via appropriate vocabulary of
relationships
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Describing general facts of the world requires
variables
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Example: The things sitting on top of a table
We’ll use ?<symbol> to denote variables.
The meaning of variables are defined via
quantifiers
Universal Quantification
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(forall <vars> <sentence>)
is true exactly when <sentence> is true for every possible set
of bindings for <vars>
– Examples:
(forall ?x (=> (block ?x)
(solid-object ?x)))
(forall (?x ?y)
(=> (and (person ?x) (person ?y)
(friend-of ?x ?y))
(friend-of ?y ?x)))
Inference with universal quantifiers
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Elimination via substitution
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(forall ?x (=> (block ?x)
(solid-object ?x))),
Let ?x=Block32
(=> (block Block32) (solid-object Block32))
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Introduction by proving statement for an arbitrary
individual about which nothing else is known (a
skolem)
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(=> (even-number Foobar)
(not (prime-number Foobar)))
Existential Quantification
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(exists <varlist> <sentence>)
is true exactly when there is at least one set of bindings for <varlist>
that make <sentence> true.
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Examples
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(for-all ?x
(=> (number ?x)
(exists ?y (> ?y ?x))))
(for-all ?x
(=> (block ?x)
(or (on ?x table) (held ?x)
(exists ?y (and (block ?y)
(on ?x ?y))))))
Using existential quantifiers
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Elimination by creating a skolem.
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(exists ?y (and (block ?y)
(on Block32 ?y)))
create novel entity, G0012, and substitute:
(and (block G0012)
(on Block32 G0012))
Introduction by generalizing from a specific
example
(and (block Block18)(on Block32 Block18))
justifies the introduction of
(exists (?x ?y)
(and (block ?x) (on ?y ?x)))
Tips for using quantifiers
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Be very careful of scoping when translating from
natural language
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“Wherever you go, there you are” B. Banzi
Heuristic: Use => inside forall
(forall ?x (and (block ?x)
(solid-object ?x)))
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Heuristic: Use and inside exists
(exists (?x ?y)
(=> (block ?x) (on ?x ?y)))
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Heuristic: Use iff very sparingly
(forall ?x (iff (block ?x)
(solid-object ?x))
Higher-order logic
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First order logic ? variables range over objects
Second order logic ? variables can range over
predicates also
(for-all ?r
(iff (transitive ?r)
(forall (?x ?y ?x)
(=> (and (?r ?x ?y) (?r ?y ?z))
(?r ?x ?z)))))
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Omega-order logics have been used at times
Even second order is undecidable, however
Aside: Order in Structure-Mapping
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Not the same thing as the order of a logic
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In structure-mapping, the order of an
expression is
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Order of a logic indicates how expressive it can
be
0 for entities
1+max(order(arguments)) for expressions
Order of an expression indicates the degree
of nesting
Types of Logic
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Variations of FOL
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Syntax
Subsets
Proof theory
Model theory
Ontology
Metalanguage
Reification
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Formal trick:
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For each predicate, create a constant that stands for it
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In KIF, it is the tuple semantics of the predicate
Requires an axiom schema (think “macro”)
Use those constants in axioms
Since they are constants, the resulting theory is first-order
Useful for expressing control knowledge
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Reify facts, so one can say when they are relevant
Reification Example
See Glimmer
Feel Breeze
Include state = 1
Include state = 1
Move Forward
Schema Node
Synthetic Item
Bump
The
itemreliability
node reifies
General
of
the
this schema
schema node’s
node is
verification
conditions
low, but local
reliability
Host link
is high
Schema Node
Item Node
“Wall to the left”
Move Left
Synthetic Item
Bump
Bump
Item Node
“Wall to the left”
Schema Node
Schema Node
Context
spin-offs
Item
Node
1)
3) Applicability
Prediction
2) Successful
byofbeing
host
refine the verification
“Wall to the left”
inexecution
the
schema
resultnodes
of
set
host
of a
conditions
schema
schema
nodes
node
Move Left
At pos. 1, 3
Move Left
Modal Operators
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Modal operators extend the expressive power of a
logic by specifying context
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Classic examples from logic: (necessary P),
(possible P)
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Metaphor: quotation and backquote
Can use reification of their semantics to drop the reasoning
to first order, but can still be hairy
(iff (necessary P) (not (possible (not P))))
Classic examples from AI/Philosophy: (Knows A
P), (Believes A P)
(=> (knows John (visible morning-star))
(knows John (visible evening-star)))
Model Theory
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Meaning of a theory = set of models that satisfy it.
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Model = set of objects and relationships
If statement is true in KB, then the corresponding
relationship(s) hold between the corresponding objects in
the modeled world
The objects and relationships in a model can be formal
constructs, or pieces of the physical world, or whatever
Meaning of a predicate = set of things in the models
for that theory which correspond to it.
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E.g., above means “above”, sort of
Caution: Meaning pertains to simplest
model
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There is usually an intended model, i.e., what one is
representing.
A sparse set of axioms can be satisfied by
dramatically simpler worlds than those intended
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Example: Classic blocks world axioms have ordered pairs of
integers as a model
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(<position on table> <height>) ? block
(on A B) ? p(A) = p(B) & h(A) = h(B)+1
(above A B) ? p(A) = p(B) & h(A) > h(B)
Moral: Use dense, rich set of axioms and make sure
you are defining what you think you are defining
Misconceptions about meaning
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“Predicates have definitions”
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Most don’t. Their meaning is constrained by the sum total of
axioms that mention them.
“Logic is too discrete to capture the dynamic fluidity
of how our concepts change as we learn”
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If you think of the set of axioms that constrain the meaning
of a predicate as large, then adding (and removing)
elements of that set leads to changes in its models.
Sometimes small changes in the set of axioms can lead to
large changes in the set of models. This is the logical
version of a discontinuity.
Representations as Sculptures
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How does one make a statue of an elephant?
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How does one represent a concept?
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Start with a marble block. Carve away everything that does
not look like an elephant.
Start with a vocabulary of predicates and other axioms. Add
axioms involving the new predicate until it fits your intended
model well.
Knowledge representation is an evolutionary process
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It isn’t quick, but incremental additions lead to incremental
progress
All representations are by their nature imperfect