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How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?

How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?. David Makinson (joint work with Jim Hawthorne). I. Uncertain Reasoning. Consequence Relations. Many ways of studying uncertain reasoning

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How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?

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  1. How Different are Quantitative and QualitativeConsequence Relations for Uncertain Reasoning? David Makinson (joint work with Jim Hawthorne)

  2. I Uncertain Reasoning

  3. Consequence Relations • Many ways of studying uncertain reasoning • One way: consequence relations (operations) and their properties • Two approaches to their definition: • Quantitative (using probability) • Qualitative (various methods) • Tend to be studied by different communities

  4. Behaviour Widely felt: quantitatively defined consequence relations rather less well-behaved than qualitative counterparts • Butexactlyhow much do they differ, and in what respects? • Are there any respects in which the quantititive ones are more regular?

  5. Tricks and Traps On quantitative side • Can simulate qualitative constructions On qualitative side • Behaviour varies considerably according to mode of generation

  6. Policy • Don’t try to twist one kind of approach to imitate the other • Take most straightforward version of each • Compare their behaviour as they are

  7. II Qualitative Side

  8. Recall Main Qualitative Account • Name: preferential consequence relations • Due to: Kraus, Lehmann, Magidor • Status: Industry standard • Our presentation: With single formulae (rather than sets of them) on the left

  9. Preferential models Structure S = (S, , |) where: • S is an arbitrary set (elements called states) •  is a transitive, irreflexive relation over S (called a preference relation) • | is a satisfaction relation between states and classical formulae (well-behaved on classical connectives )

  10. Preferential Consequence - Definition Given a preferential modelS = (S, , |), define consequence relation |~S by rule: a |~Sx iff x is satisfied by every state s that is minimal among those satisfying a state : in S satisfied : under | minimal : wrt <

  11. S = {s1, s2} s1 s2 s2 :p,q,r s1 : p,q, r p |~ r, but pq |~/ r Monotony fails Some other classical rules fail What remains? Example

  12. KLM Family P of Rules a |~ a reflexivity When a |~ x and x |y then a |~ y RW: right weakening When a |~ x and a||b then b |~ x LCE: left classical equivalence When a |~ xy then ax |~ y VCM: very cautious monotony When a |~ x and b |~ x, then ab |~ x OR: disjunction in the premises When a |~ x and a |~ y, then a |~ xy AND: conjunction in conclusion

  13. All Horn rules for |~(with side-conditions) Whenever a1 |~ x1, …., an |~ xn (premises with |~) and b1 |- y1, …., bm |- ym (side conditions with |-) then c |~ z (conclusion) (No negative premises, no alternate conclusions; finitely many premises unless signalled)

  14. KLM Representation Theorem A consequence relation |~ between classical propositional formulae is a preferential consequence relation (i.e. is generated by some stoppered preferential model) iff it satisfies the Horn rules listed in system P

  15. III Quantitative Side

  16. Ingredients and Definition • Fix a probability function p • Finitely additive, Kolgomorov postulates • Conditionalization as usual: pa(x) = p(ax)/p(a) • Fix a threshold t in interval [0,1] • Define a consequence relation |~p,t , briefly |~, by the rule: a |~p,tx iff either pa(x) t or p(a)  0

  17. Successes and Failures Succeed (zero and one premise rules of P) a |~ a Reflexivity When a |~ x and x |y then a |~ y RW: right weakening When a |~ x and a||b then b |~ x LCE: left classical equivalence When a |~ xy then ax |~ y VCM: very cautious monotony Fail (two-premise rules of P) When a |~ x and b |~ x, then ab |~ x OR: disjunction in premises When a |~ x and a |~ y, then a |~ xy AND: conjunction in conclusion

  18. IV Closer Comparison

  19. Two Directions Preferentially sound / Probabilistically sound • OR, AND • Look more closely later Probabilistically sound  Preferentially sound ? • Nobody seems to have examined • Presumed positive

  20. Yes and No Question Probabilistically sound  Preferentially sound ? Answer Yes and No – depends on what kind of rule

  21. Specifics Question • Prob. sound  Pref. sound ? Answer Yes and No – depends on what kind of rule Specifics • Finite-premise Horn rules: Yes • Alternative-conclusion rules: No • Countable-premise Horn rules: No

  22. Finite-Premise Horn rules Should have been shown c.1990…Hawthorne & Makinson 2007 If the rule is probabilistically sound (i.e. holds for every consequence relation generated by a prob.function, threshold) then it is preferentially sound (i.e. holds for every consequence relation generated by a stoppered pref. model)

  23. Alternate-Conclusion Rules Negation rationality (weaker than disjunctive rationality and rational monotony) When a |~ x, then ab |~ xorab |~ x Well-known: • Probabilistically sound • Not preferentially sound - fails in some stoppered preferential models

  24. Countable-Premise Horn Rules Archimedian rule (Hawthorne & Makinson 2007) Whenever a |~ ai(premises: i ) ai |~ xi(premises: i ) xi pairwise inconsistent(side conditions) thena |~  • Probabilistically sound Archimedean property of reals: t 0  n: n.t  1 • But not preferentially sound

  25.   :r, qi (i )  n :r, q1,.., qn,qn+1 2 : r, q1, q2,q3, ….  1 : r, q1,q2, … Put ar ai q1…qi xiq1…qiqi (1) a |/~  (2) a |~ ai for all i  (3) ai |~ xi for all i  (4) xi pairwise inconsistent Fails in this Preferential Model

  26. Corollary • No representation theorem for probabilistic consequence relations in terms of finite-premise Horn rules • Contrast with KLM representation theorem for preferential consequence relations

  27. Other Direction Pref. sound but not prob. sound: two-premise Horn rules: OR: When a |~ x and b |~ x, then ab |~ x AND: When a |~ x and a |~ y, then a |~ xy • Are there weakened versions that are prob. sound? • Can we get completeness over finite-premise Horn rules? • Representation no!, completeness maybe • Wedge between representation and completeness • Completeness relative to class of expressions

  28. Weakened Versions of OR, AND XOR:When a |~ x,b |~ x anda |b then ab |~ x • Requires that the premises be exclusive • Well-known WAND:When a |~ x,ay |~ , then a |~ xy • Requires a stronger premise • Hawthorne 1996

  29. Proposed Axiomatization for Probabilistic Consequence Hawthorne’s family O(1996): • The zero and one-premise rules of P • Plus XOR, WAND Open question: Is this complete for finite-premise Horn rules (possibly with side-conditions) ? Conjecture: Yes

  30. Partial Completeness Results The following are equivalent for finite-premise Horn rules with pairwise inconsistent premise-antecedents (1) Prob. sound (2a) Pref. sound (all stoppered pref.models) (2b) Sound in all linear pref. models at most 2 states (3) Satisfies ‘truth-table test’ of Adams (4a) Derivable from B{XOR} (when n 1, from B) (4b) Derivable from family O (4c) Derivable from family P for n 1: van Benthem 1984, Bochman 2001 Adams 1996 (claimed)

  31. V No-Man’s Land between O and P

  32. More about WAND: When a |~ x, ay |~ , then a |~ xy Second condition equivalent in O to each of: • ay |~ y • ay |~ z for all z • ab |~ y for all b (a |~ y ‘holds monotonically’) • (ay)b |~ y for all b

  33. What Does ay |~  mean ? • Quantitatively: Either t = 0 or p(ay) = 0 • Qualitatively: Preferential model has no (minimal) ay states • Intuitively: a givesindefeasiblesupport to y (certain but not logically certain)

  34. Between O and P Modulo rules in O: OR CM CT AND CT: when a |~ x and ax |~ y then a |~ y CM: when a |~ x and a |~ y then ax |~ y Modulo O: PAND  {CM, OR}  {CM, CT} (Positive parts Adams 1998, Bochman 2001; CM / AND tricky)

  35. Moral • AND serves as a watershed condition between family O (sound for probabilistic consequence) and family P(characteristic for qualitative consequence) • No other single well-known rule does the same

  36. VI Open Questions

  37. Mathematical • Is Hawthorne’s family O completefor prob. consequenceover finite-premise Horn rules ? Conjecture: positive • Can we give a representation theorem for prob.consequence in terms of O + NR + Archimedes + …? Conjecture: negative

  38. Philosophical • Pref. consequence, as a formal modelling of qualitative uncertain consequence, validates AND • So do most others, e.g. Reiter default consequence • But do we really want that? • Perhaps it should fail even for qualitative consequence relations • Example: paradox of the preface

  39. Paradox of the preface(Makinson 1965) An author of a book making a large number n of assertions may check and recheck them individually, and be confident ofeach that it is correct. But experience teaches that inevitably there will be errors somewhere among the n assertions, and the preface may acknowledge this. Yet these n+1 assertions are together inconsistent. • Inconsistent belief set, whether or not we accept AND • Inconsistent belief, if we accept AND

  40. VII References

  41. References James Hawthorne & David Makinson The quantitative/qualitative watershed for rules of uncertain inference Studia Logica Sept 2007 David Makinson Completeness Theorems, Representation Theorems: What’s the Difference? Hommage à Wlodek: Philosophical Papers decicated to Wlodek Rabinowicz, ed. Rønnow-Rasmussen et al., www.fil.lu.se/hommageawlodek

  42. VIII Appendices

  43. What is Stoppering? To validate VCM: When a |~ xy then ax |~ y, we need to impose stoppering (alias smoothness) condition: Whenever state s satisfies formula a, either: • s is minimal under  among the states satisfying a • or there is a state ss that is minimal under  among the states satisfying a Automatically true in finite preferential models. Also true in infinite models when no infinite descending chains

  44. Derivable from Family P Can derive SUP: supraclassicality: When a |x, then a |~ x CT: cumulative transitivity: When a |~ x and ax |~ y, then a |~ y Can’t derive Plain transitivity: When a |~ x and x |~ y, then a |~ y Monotony When a |~ x then ab |~ x

  45. VCM versus CM KLM (1990) use CM: cautious monotony: When a |~ x and a |~ y, then ax |~ y instead of VCM When a |~ xy then ax |~ y These are equivalent in P (using AND and RW) But not equivalent in absence of AND

  46. Kolmogorov Postulates Any function defined on the formulae of a language closed under the Boolean connectives, into the real numbers, such that: (K1) 0 p(x)  1 (K2) p(x) = 1 for some formula x (K3) p(x)p(y) whenever x |- y (K4) p(xy)= p(x) p(y) whenever x |- y

  47. Conditionalization • Let p be a finitely additive probability function on classical formulae in standard sense (Kolmogorov postulates) • Let a be a formula with p(a)  0 • Write pa alias p(•|a)for the probability function defined by the standard equation pa(x) = p(ax)/p(a) • pacalled the conditionalization of p on a

  48. What is System B? • Burgess 1981 • May be defined as the 1-premise rules in O and P plus 1-premise version of AND: VWAND: When a |~ x and a |y then a |~ xy • AND WAND  VWAND

  49. What is Adams’ Truth-Table Test ? There is some subset I {1,..,n} such that both by |iI(ai xi) and iI(aixi) |by • When n = 0 this reduces to: b |y • For n = 1, reduces to: either b |y or both ax |by and ax |by • Proof of 134ain Adams 1996 has serious gap

  50. Some Alternate-Conclusion Rules • Negation rationality when a |~ x then ab |~ xorab |~ x • Disjunctive rationality when ab |~ x then a |~ xorb |~ x • Rational monotony when a |~ x then ab |~ xora |~ b • Conditional Excluded Middle a |~ xora |~ x Of these, NRalone holds for probabilistic consequence

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