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Transcript of 1 How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?...
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How Different are Quantitative and Qualitative
Consequence Relations for Uncertain Reasoning?
David Makinson
(joint work with Jim Hawthorne)
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I
Uncertain Reasoning
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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
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Behaviour
Widely felt: quantitatively defined consequence relations rather less well-behaved than qualitative counterparts
• But exactly how much do they differ, and in what respects?
• Are there any respects in which the quantititive ones are more regular?
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Tricks and Traps
On quantitative side
Can simulate qualitative constructions
On qualitative side
Behaviour varies considerably according to mode of generation
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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
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II
Qualitative Side
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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
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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 )
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Preferential Consequence - Definition
Given a preferential model S = (S, , |), define consequence relation |~S by rule:
a |~S x iff x is satisfied by every state s that is
minimal among those satisfying a
state : in S
satisfied : under |
minimal : wrt <
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Example
S = {s1, s2}
s1 s2
s2 : p,q,r
s1 : p,q, r
p |~ r, but pq |~/ r
Monotony fails
Some other classical rules fail
What remains?
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KLM Family P of Rules
a |~ a reflexivity
When a |~ x and x | y then a |~ yRW: right weakening
When a |~ x and a || b then b |~ xLCE: left classical equivalence
When a |~ xy then ax |~ yVCM: very cautious monotony
When a |~ x and b |~ x, then ab |~ xOR: disjunction in the premises
When a |~ x and a |~ y, then a |~ xyAND: conjunction in conclusion
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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)
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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
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III
Quantitative Side
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Ingredients and Definition
• Fix a probability function p
– Finitely additive, Kolgomorov postulates
• Conditionalization as usual: pa(x) = p(ax)/p(a)
– Fix a threshold t in interval [0,1]
• Define a consequence relation |~p,t , briefly |~, by the rule:
a |~p,t x iff either pa(x) t or p(a) 0
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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 |~ xy then ax |~ y VCM: very cautious monotony
Fail (two-premise rules of P)
When a |~ x and b |~ x, then ab |~ x OR: disjunction in premises
When a |~ x and a |~ y, then a |~ xy AND: conjunction in conclusion
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IV
Closer Comparison
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Two Directions
Preferentially sound / Probabilistically sound– OR, AND– Look more closely later
Probabilistically sound Preferentially sound ?– Nobody seems to have examined– Presumed positive
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Yes and No
Question
Probabilistically sound Preferentially sound ?
Answer
Yes and No – depends on what kind of rule
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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
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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)
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Alternate-Conclusion Rules
Negation rationality (weaker than disjunctive rationality and rational monotony)
When a |~ x, then ab |~ x or ab |~ x
Well-known:
– Probabilistically sound
– Not preferentially sound - fails in some stoppered preferential models
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Countable-Premise Horn Rules
Archimedian rule (Hawthorne & Makinson 2007)
Whenevera |~ ai (premises: i )
ai |~ xi (premises: i ) xi pairwise inconsistent (side conditions)
then a |~ – Probabilistically sound
Archimedean property of reals: t 0 n: n.t 1
– But not preferentially sound
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Fails in this Preferential Model
: 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…qiqi
(1) a |/~
(2) a |~ ai for all i
(3) ai |~ xi for all i
(4) xi pairwise inconsistent
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Corollary
• No representation theorem for probabilistic consequence relations in terms of finite-premise Horn rules
• Contrast with KLM representation theorem for preferential consequence relations
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Other Direction
Pref. sound but not prob. sound: two-premise Horn rules:
OR: When a |~ x and b |~ x, then ab |~ x AND: When a |~ x and a |~ y, then a |~ xy
• 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
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Weakened Versions of OR, AND
XOR: When a |~ x, b |~ x and a | b then ab |~ x
– Requires that the premises be exclusive
– Well-known
WAND: When a |~ x, ay |~ , then a |~ xy
– Requires a stronger premise
– Hawthorne 1996
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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
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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 2001Adams 1996 (claimed)
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V
No-Man’s Land
between O and P
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More about WAND: When a |~ x, ay |~ , then a |~ xy
Second condition equivalent in O to each of:
• ay |~ y
• ay |~ z for all z
• ab |~ y for all b (a |~ y ‘holds monotonically’)
• (ay)b |~ y for all b
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What Does ay |~ mean ?
• Quantitatively: Either t = 0 or p(ay) = 0
• Qualitatively: Preferential model has no (minimal) ay states
• Intuitively: a gives indefeasible support to y (certain but not logically certain)
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Between O and P
Modulo rules in O:
OR CM
CT
AND
CT: when a |~ x and ax |~ y then a |~ y
CM: when a |~ x and a |~ y then ax |~ y
Modulo O: P AND {CM, OR} {CM, CT}
(Positive parts Adams 1998, Bochman 2001; CM / AND tricky)
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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
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VI
Open Questions
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Mathematical
• Is Hawthorne’s family O complete for prob. consequence over finite-premise Horn rules ?
Conjecture: positive
• Can we give a representation theorem for prob.consequence in terms of O + NR + Archimedes + …?
Conjecture: negative
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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
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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 of each 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
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VII
References
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References
James Hawthorne & David Makinson The quantitative/qualitative watershed for rules of uncertain inference Studia Logica Sept 2007
David MakinsonCompleteness 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
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VIII
Appendices
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What is Stoppering?
To validate VCM: When a |~ xy then ax |~ 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
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Derivable from Family P
Can derive
SUP: supraclassicality: When a | x, then a |~ x
CT: cumulative transitivity:When a |~ x and ax |~ y, then a |~ y
Can’t derive
Plain transitivity:When a |~ x and x |~ y, then a |~ y
MonotonyWhen a |~ x then ab |~ x
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VCM versus CM
KLM (1990) use CM: cautious monotony:
When a |~ x and a |~ y, then ax |~ y
instead of VCM
When a |~ xy then ax |~ y
These are equivalent in P (using AND and RW)
But not equivalent in absence of AND
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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(xy) = p(x) p(y) whenever x |- y
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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(ax)/p(a)
• pa called the conditionalization of p on a
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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 |~ xy
• AND WAND VWAND
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What is Adams’ Truth-Table Test ?
There is some subset I {1,..,n} such that both by | iI(ai xi) and iI(aixi) | by
– When n = 0 this reduces to: b | y
– For n = 1, reduces to: either b | y or both ax | by and ax | by
– Proof of 134a in Adams 1996 has serious gap
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Some Alternate-Conclusion Rules
• Negation rationalitywhen a |~ x then ab |~ x or ab |~ x
• Disjunctive rationalitywhen ab |~ x then a |~ x or b |~ x
• Rational monotonywhen a |~ x then ab |~ x or a |~ b
• Conditional Excluded Middlea |~ x or a |~ x
Of these, NR alone holds for probabilistic consequence