Bridges from Classical to Nonmonotonic Logic

1
Bridges from Classical to Nonmonotonic Logic

• David Makinson
• Kings College London

2
Purpose Message

• Take mystery out of nonmonotonic logic
• Not so unfamiliar
• Easily accessible given classical logic

• There are natural bridge systems
• Monotonic
• Supraclassical
• Stepping stones

3
Some Misunderstandings about NMLs
4
A Habit to Suspend

• Bridge logics supraclassical closure opns
• But how is this possible?
• Not closed under substitution
• Nor are the nonmonotonic ones

5
General Picture
6
First Bridge Using Additional Assumptions

• Pivotal-assumption consequence
• Fixed set of background assumptions
• Monotonic

• Default-assumption consequence
• Vary background set with current premises
• Nonmonotonic

7
Pivotal-Assumption Consequence

• Fix background set K of formulae
• Define  A -K x iff K?A - x
• Alias x ? CnK(A)
• Class pivotal-assumption consequence relations
  
• -K for some set K

8
Pivotal-Assumption Consequence (ctd)

• Properties
• Paraclassical
• Supraclassical (includes classical consequence)
• Closure operation (reflexivity idempotence
  monotony)
• Disjunction in premises (alias OR)
• Compact
• Representation
• Pivotal-assumption consequence iff above three
  properties

9
Default-Assumption Consequence

• Idea
• Allow background assumptions K to vary with
  current premises A
• Diminish K when inconsistent with A
• Work with maximal subsets of K that are
  consistent with A
• Define
• A K x iff K??A - x for every subset K? ? K
  maxiconsistent with A
• Alias x ? CK(A)
• Known as Poole consequence

10
Second Bridge Restricting the Valuation Set

• Pivotal-valuation consequence
• Fixed subset of the set of all Boolean valuations
• Monotonic

• Default-valuation consequence
• Vary valuation set with current premises
• Nonmonotonic

11
Pivotal-Valuation Consequence

• Idea exclude some of the valuations
• Fix subset W ? V
• Define  A -W x iff no v ? W v(A) 1 v(x)
  0
• Class pivotal-valuation consequence relations
  -W for some set W ? V

12
Pivotal-Valuation Consequence (ctd)

• Properties
• Paraclassical
• Disjunction in premises
• But not compact
• Fact
• pivotal assumption pivotal
  valuation?compact
• Representation
• Open (when infinite premise sets allowed)

13
Default-Valuation Consequence

• Idea
• allow set W? V to vary with current premises A
• put WA set of valuations in W minimal among
  those satisfying premise set A
• Require the conclusion to be true under all
  valuations in WA
• Define A W x iff no v ? WA v(A) 1 v(x)
  0
• Alias x ? CW(A)
• Known as preferential consequence (Shoham,
  KLM.)

14
Third Bridge Using Additional Rules

• Pivotal-rule consequence
• Fixed set of rules
• Monotonic

• Default-rule consequence
• Vary application of rules with current premises
• Nonmonotonic

15
Pivotal-Rule Consequence

• Rule any ordered pair (a,x) of formulae
• Fix set R of rules
• Define  A -R x iff x ? every superset of A
  closed under both Cn and R
• Class pivotal-rule consequence relations
  -R for some set R of rules

16
Pivotal-Rule Consequence (ctd)

• Properties
• Paraclassical
• Compact
• But not Disjunction in premises
• Facts
• pivotal assumption pivotal rule?OR
• pivotal rule?pivotal
  valuation
• Representation
• Pivotal-rule consequence iff above two properties

17
Pivotal-Rule Consequence (ctd)

• Equivalent definitions of CnR(A)
• ? X ? A X Cn(X) R(X)
• ?An n ? ?, where A1 A and An1
  Cn(An?R(An))
• ?An n ? ? with A1 A and An1 Cn(An?x)
• where (a,x) is first rule in ?R? such that a
  ? An but x ? An
• (in the case that there is no such rule An1
  Cn(An))

18
Default-Rule Consequence

• Fix an ordering ?R? of R
• Define C?R?(A)
• ?An n ? ? with A1 A and An1 Cn(An?x)
• where (a,x) is first rule in ?R? such that
• a ? An , x ? An , and x is consistent with An
• (if no such rule An1 Cn(An))

19
Default-Rule Consequence (ctd)

• Facts
• The sets C?R?(A) for an ordering ?R? of R are
  precisely the Reiter extensions of A using the
  normal default rules (a,x) alias (ax/x)
• The ordering makes a difference
• Standard inductive definition versus fixpoints
• Sceptical operation
• CR(A) ?C?R?(A) ?R? an ordering of R

20
Summary Table
21
Further reading

• Makinson, David 2003. Bridges between classical
  and nonmonotonic logic
• Logic Journal of the IGPL 11 (2003) 69-96.
• Free access http//www3.oup.co.uk/igpl/Volume_11
  /Issue_01/
• Makinson, David 1994. General Patterns in
  Nonmonotonic Reasoning
• pp 35-110 in Handbook of Logic in Artificial
  Intelligence and Logic Programming, vol. 3, ed.
  Gabbay, Hogger and Robinson. Oxford University
  Press.