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

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