Logic

1
Logic
2
Overview

• Scholastic logic
• Propositional logic
• Syntax and semantics
• Inference rules
• Predicate calculus
• Syntax and semantics
• Quantifiers
• Higher-order logics and modal operators
• Model theory
• The incremental, almost continuous nature of
  meaning
• Knowledge representation as an evolutionary
  process

3
Features of Logic KR Languages

• Vocabulary
• Logical symbols
• Constants
• Variables
• Punctuation
• Syntax
• Semantics
• Theory of reference
• Theory of truth
• Rules of inference

4
Automated Reasoning

• Premise Given an appropriate KR language, new
  knowledge can be inferred from existing truths

5
Syllogisms

• Aristotle set out the concept of a three-sentence
  structure to represent knowledge
• All trailer trucks are 18 wheelers
• Some Peterbilt is a trailer truck
• Therefore, some Peterbilt is an 18 Wheeler

6
Sentence Types

• 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

7
Valid Syllogisms

• bArbArA
• A All broad-leafed plants are deciduous
• A All vines are broadleaf plants
• A Therefore, all vines are deciduous

8
Valid Syllogisms (cont.)

• cElArEnt
• E Nothing absent-minded is an elephant
• A All professors are absent-minded
• E Therefore, no professor is an elephant

9
Valid Syllogisms (cont.)

• dArII
• A All trailer trucks are 18 wheelers
• I Some Peterbilt is a trailer truck
• I Therefore, some Peterbilt is an 18 Wheeler

10
Valid Syllogisms (cont.)

• fErIO
• E No Corvette is a truck
• I Some Chevrolet is a Corvette
• O Therefore, some Chevrolet is not a truck

11
Boolean Algebra

• George Boole
• System of representing and operating on truth
  values
• p x q AND
• p q OR
• -p NOT

12
First Order Logic (FOL)

• Extensions of Boolean Algebra and Aristotles
  syllogisms
• Universal and existential quantifiers
• Changes in syntax

13
Why FOL?

• Expressive power
• Extensive Semantics
• Best-defined model and proof theories

14
Syntax of Propositional Logic

• Symbols
• Logical constants True, False
• Proposition symbols P, Q,
• Logical connectives
• not (aka ? )
• and (aka ? )
• or (aka ? )
• (aka ? )
• iff (aka ? )
• Sentences are made from these symbols

15
Sentences

• True, False are sentences
• Any proposition symbol is a sentence
• Any sentences S1, Sn combined with a connective
  is a sentence
• (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

16
Semantics of propositional logic

• Formal semantics function from sentences to
  true, false.
• True means that the sentence is true of the world
• False means that the sentence isnt true of the
  world
• Truth tables define semantics of connectives
• See Sowa for details
• Example
• P Q (or P Q)
• false false false
• false true true
• true false true
• true true true

17
Rules of Inference Examples

• Modus Ponens (aka implication-elimination)
• Given ( P Q), P, infer Q
• And-elimination
• Given (and P Q), infer P or infer Q
• And-introduction
• Given P, Q, infer (and P Q)
• Unit resolution
• Given (or P Q), (not P), infer Q
• Resolution
• Given (or P Q), (or (not Q) R), infer (or P R)

18
Using inference rules

• 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.

19
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))

20
Using the Inference Rules

• ( 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

21
Properties of inference rules

• 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
• Example Horn clauses
• ( (and A1 An) C)

22
Predicate Calculus

• Propositions can have structure
• ( . )
• Terms are either
• 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)

23
Propositions express facts

• Concrete facts via appropriate vocabulary of
  relationships
• Example The things sitting on top of a table
• Describing general facts of the world requires
  variables
• Well use ? to denote variables.
• The meaning of variables are defined via
  quantifiers

24
Universal Quantification

• (forall )
• is true exactly when is true for every
  possible set of bindings for
• Examples
• (forall ?x ( (block ?x)
• (solid-object ?x)))
• (forall (?x ?y)
• ( (and (person ?x) (person ?y)
• (friend-of ?x ?y))
• (friend-of ?y ?x)))

25
Inference with universal quantifiers

• Elimination via substitution
• (forall ?x ( (block ?x)
• (solid-object ?x))),
• Let ?xBlock32
• ( (block Block32) (solid-object Block32))
• Introduction by proving statement for an
  arbitrary individual about which nothing else is
  known (a skolem)
• ( (even-number Foobar)
• (not (prime-number Foobar)))

26
Existential Quantification

• (exists )
• is true exactly when there is at least one set of
  bindings for that make true.
• Examples
• (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))))))

27
Using existential quantifiers

• Elimination by creating a skolem.
• (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)))

28
Tips for using quantifiers

• Be very careful of scoping when translating from
  natural language
• Wherever you go, there you are B. Banzi
• Heuristic Use inside forall
• (forall ?x (and (block ?x)
• (solid-object ?x)))
• Heuristic Use and inside exists
• (exists (?x ?y)
• ( (block ?x) (on ?x ?y)))
• Heuristic Use iff very sparingly
• (forall ?x (iff (block ?x)
• (solid-object ?x))

29
Higher-order logic

• 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)))))
• Omega-order logics have been used at times
• Even second order is undecidable, however

30
Aside Order in Structure-Mapping

• Not the same thing as the order of a logic
• Order of a logic indicates how expressive it can
  be
• In structure-mapping, the order of an expression
  is
• 0 for entities
• 1max(order(arguments)) for expressions
• Order of an expression indicates the degree of
  nesting

31
Types of Logic

• Variations of FOL
• Syntax
• Subsets
• Proof theory
• Model theory
• Ontology
• Metalanguage

32
Reification

• Formal trick
• For each predicate, create a constant that stands
  for it
• 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
• Reify facts, so one can say when they are relevant

33
Reification Example
Include state 1
Include state 1
34
Synthetic Item
The item node reifies the schema nodes
verification conditions
Wall to the left
35
Synthetic Item
Bump
Context spin-offs refine the verification
conditions
1) Applicability of host schema nodes
2) Successful execution of host schema nodes
3) Prediction by being in the result set of a
schema node
Wall to the left
36
Modal Operators

• Modal operators extend the expressive power of a
  logic by specifying context
• Metaphor quotation and backquote
• Can use reification of their semantics to drop
  the reasoning to first order, but can still be
  hairy
• Classic examples from logic (necessary P),
  (possible P)
• (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)))

37
Model Theory

• Meaning of a theory set of models that satisfy
  it.
• 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.
• E.g., above means above, sort of

38
Caution Meaning pertains to simplestmodel

• 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
• Example Classic blocks world axioms have ordered
  pairs of integers as a model
• ( ) ? 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

39
Misconceptions about meaning

• Predicates have definitions
• Most dont. 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
• 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.

40
Representations as Sculptures

• How does one make a statue of an elephant?
• Start with a marble block. Carve away everything
  that does not look like an elephant.
• How does one represent a concept?
• 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
• It isnt quick, but incremental additions lead to
  incremental progress
• All representations are by their nature imperfect