Ordinal Conditional Functions for Nearly Counterfactual Revision
OOrdinal Conditional Functions for Nearly Counterfactual Revision
Aaron Hunter
BCITBurnaby, BC, Canadaaaron [email protected]
Abstract
We are interested in belief revision involving conditionalstatements where the antecedent is almost certainly false. Torepresent such problems, we use Ordinal Conditional Func-tions that may take infinite values. We model belief changein this context through simple arithmetical operations that al-low us to capture the intuition that certain antecedents cannot be validated by any number of observations. We frameour approach as a form of finite belief improvement, and wepropose a model of conditional belief revision in which onlythe “right” hypothetical levels of implausibility are revised.
Introduction
The theory of belief change is concerned with the way agentsincorporate new information. Typically, the focus is onnew information that is given as a propositional formula.In this paper, we are concerned with situations where anagent needs to revise by a conditional where the antecedentis almost certainly false. More precisely, we consider an-tecedents that will not be believed given any finite amountof “regular” supporting evidence. We represent the degree ofbelief in such formulas using Ordinal Conditional Functionsthat may take infinite values, and we provide an approach toconditional revision based on basic ordinal arithmetic.This paper makes several contributions to existing workon belief change. First, we demonstrate that a simple al-gebra of belief change in the finite case extends naturally tothe infinite case, giving a form of belief improvement. Inthe process, we demonstrate that there are natural examplesin commonsense reasoning where multiple levels of infiniteimplausibility are actually useful. In particular, we introducea natural approach to revision by conditional statements withlittle in the way of new formal machinery.
Motivating Example
Consider the following claims:1. heavy : Your dog is overweight.2. f ly : Your dog can fly.3. hollow | f ly : If your dog can fly, then it has hollow bones. This paper contains results that have been published in(Hun15) and (Hun16).
The first two claims are simple declarative statements. Butnote that there is a clear difference in the amount of evi-dence needed to convince the agent to believe each claim.For (1), it presumably takes some finite number of reportsfrom a trusted source. For (2), it seems unlikely that any fi-nite number of reports would be convincing. This statementis almost certainly false, though it is possible to imagine asituation that would convince an agent to believe it.The third statement is a conditional with a highly unlikelyantecedent. Nevertheless, the perceived “impossibility” of(2) does not mean that (3) is free of content. Revision by (3)should change an agent’s beliefs in a counterfactual sense;they may need to change their beliefs about hollow bones insome hypothetical scenario. Moreover, if ever the notion offlying dogs becomes believable, then this report will take onsignificance at the level of factual beliefs. In this paper, werefer to claims such as (3) as nearly counterfactual . We willprovide a formal characterization of such claims, as well asa suitable approach to revision.
Preliminaries
Belief Revision
Belief revision is the belief change that occurs when newinformation is presented to an agent with some prior, possi-bly contradictory, set of beliefs. We assume an underlyingpropositional signature P . An interpretation over P is calleda state , while a logically closed set of formulas over P iscalled a belief set . A belief revision operator is a functionthat combines the initial belief set and a formula to producea new belief set.Formal approaches to belief revision typically require anagent to have some form of ordering or ranking that givesthe relative plausibility of possible states. For example, inthe well-known AGM approach, total pre-orders over statesare used to represent the perceived likelihood of each state(AGM85; KM92). Unfortunately, this approach does nothandle the problem of iterated belief revision . Related workhas addressed iterated revision by explicitly specifying howthe ordering changes, rather than just the belief set (DP97;BM06; JT07). a r X i v : . [ c s . A I] M a r rdinal Conditional Functions An ordinal conditional function (OCF) is a function thatmaps each state to an ordinal (Spo88; Wil94). In this ap-proach, strength of belief is captured by ordinal precedence.Hence, if r is an OCF and r ( s ) < r ( t ) , then s is a moreplausible state than t . There is an obvious advantage to thisapproach in that a ranking function is clearly more expres-sive than a total pre-oder.While the orginal definition allows the range of an OCFto be the class of all ordinals, in existing work it is commonto restrict the range to the natural numbers, possibly with anadditional symbol ∞ representing impossibility. In this pa-per, we will actually use a slightly larger range; so we needto briefly review ordinal arithmetic. For our purposes, it issufficient to note that ordinals are actually sets defined byan “order type.” The finite ordinals are the natural numbers.The order type of the natural number n is unique, becauseit is the only ordinal that has exactly n − preceding ordi-nals. The first infinite ordinal is ω , the set of all natural num-bers. Every countably infinite subset of the natural numbersis order-isomorphic to ω .It is easy to construct a countably infinite set that is notorder isomorphic to ω : just add another symbol ∞ at theend that is larger than every natural number. The ordinal thatdefines the order type of this set is written ω + 1 . Similarly,there exists a distinct ordinal ω + n for any natural number n . And if we add a complete copy of the natural numbers,then we have the ordinal ω + ω which is normally written as ω · . We can procede in this manner indefinitely to definea countably infinite sequence of ordinals. By taking powers,we can get even more order types; we will not delve furtherinto this topic.Ordinal addition can be understood in terms of the infor-mal discussion above. Given ordinals α and β , the ordinal α + β has the order type obtained by taking a set with ordertype α and then appending a set with order type β where allthe elements of β follow the elements of α in the underlyingordering. For finite ordinals, this coincides with the usualnotion of addition. For infinite valued ordinals it does not.Note for example that ω = ω ; adding a number that pre-cedes does not change the order type, because the resultingstructure is isomorphic to the natural numbers. On the otherhand ω + 1 (cid:54) = ω . So ordinal addition is not commutative.It is also worth noting that ordinal subtraction is, in general,not well defined. In particular, it is not possible to definesubtraction by ω . Belief Change as Ordinal Arithmetic
Although our goal is to address revision by conditionals, wefirst introduce a simple approach to belief change based onthe addition of ordinals. This will allow us to precisely de-fine the notion of a nearly counterfactual statement, which is It is beyond the scope of this paper to give a complete treat-ment of infinite ordinals, and ordinal arithmetic. In the discussionhere, we skip over fundamental set theory, and the fact that order-types are defined in terms of set-containment. We refer the readerto (Dev93) for an excellent introduction.
Figure 1: Visualizing ω important for the class of conditionals that we wish to con-sider. Restricted Domains
The following definition allows us to define conditionalfunctions over any set of ordinals.
Definition 1
Let S be a non-empty set of states and let Γ bea collection of ordinals. A Γ -CF ( Γ conditional function)over S is a function r : S → Γ such that r ( s ) = 0 for somestate s . Note that the definition of Γ -CFs does not actually specifythat Γ is a set , because we do not wish to specify the under-lying set theory in detail.Several special cases are immediate: • Spohn’s ordinal conditional functions are Ω -CFs, where Ω is the collection of all ordinals. • The class of ω -CFs coincides with the finite valued rank-ing functions common in the literature. • The class of ( ω + 1) -CFs is the set of ranking functionsthat can take finite values, as well as the single “impossi-ble” plausiblity value ∞ . This is essentially equivalent tothe possibilistic logic framework of (DP04), that uses the“necessity measure” of 0.In this paper, we are primarily interested in the class of ω -CFs. Note that ω can be specified as follows: ω = (cid:91) { ω · k + c | k, c ∈ ω } . Hence, every element of ω can be written as ω · k + c forsome k and c . We think of these conditional functions ashaving countably many infinite levels of implausibility. Apicture of ω is shown in Figure 1.If r is a Γ -CF, we write Bel ( r ) = { x | r ( x ) = 0 } . The degree of strength of a conditional function r is the least n such that n = r ( v ) for some v (cid:54)∈ Bel ( r ) . Hence, thedegree of strength is a measure of how difficult it would befor an agent to abandon the currently believed set of states. Finite Arithmetic on Conditional Functions
In the finite case, belief change can be captured through ad-dition on ranking functions. Some variant of the followingdefinition has appeared previously in published work by sev-eral authors; it is restated here and translated to our termi-nology. efinition 2
Let r and r be ω -CFs over S , and let m bethe minimum value of r + r . Then r ¯+ r is the functionon S defined as follows: r ¯+ r ( x ) = r ( x ) + r ( x ) − m. It is easy to check that this operation is associative, commu-tative, and that every element is invertible in the sense that,for each r there is an r (cid:48) such that r ¯+ r (cid:48) = 0 . Therefore, interms of algebra, we say that the class of ω -CFs is an abeliangroup under ¯+ .Note that Spohn’s conditionalization can be seen as a spe-cial case of this algebra on ranking functions. Let r be afinite plausibility function representing the initial beliefs ofan agent. Let φ be a formula, let d be a positive integer, andlet r be the ranking function defined as follows: r ( s ) = (cid:26) if s | = φd otherwiseThen r ¯+ r is equivalent to Spohn’s conditionalization of r by φ with strength d . Similarly, if r takes only two valuesand the degree of strength of r is strictly larger than thedegree of strength of r , then r ¯+ r is AGM revision.This approach does not extend to larger classes of ordi-nals. Proposition 1
Let β be an ordinal such that ω ∈ β . Then ¯+ is not well-defined over the class of β -CFs. The problem is that subtraction is not defined for all pairs of(infinite) ordinals.
Example
Consider the motivating example. We can definethe following ( ω + 1) − CF s : r ( s ) = (cid:26) if s | = { f ly } ω otherwise r ( s ) = (cid:26) ω if s | = { f ly } otherwise.Normalized addition of r and r requires us to calculate ω − ω . But this subtraction is not defined, so the calculationcan not be completed.This problem could be avoided by removing the normal-ization, but the result would no longer be an OCF. If we wantto work with ranking functions that are closed under someform of addition, then we must either modify the definition,or we must relax the constraint that the pre-image of 0 isnon-empty. We opt for the former. Finite Zeroing
We define an algebra over ω -CFs based on finite zeroing .The following relation will be useful in proving results. Inthe definition, and in some future results, it is useful to con-sider functions over ordinals that do not necessarily take thevalue 0 for any argument. We use the general term Γ rankingto refer to an arbitrary function from S to Γ . Konieczny refers to this kind of OCF as a free OCF. (Kon09)
Definition 3
For Γ rankings r and r , we write r ∼ r just in case the following condition holds for every pair ofstates s, t r ( s ) < r ( t ) ⇐⇒ r ( s ) < r ( t ) . Clearly, ∼ is an equivalence relation.The intuition behind finite zeroing is that each conditionalfunction can be categorized by its minimum value, in a man-ner that is useful for revision. Given any ω ranking r , let min ( r ) denote the minimum value r ( s ) . Note that a min-imum is guaranteed by the fact that the ordinals are well-ordered. Definition 4
Let r be an ω ranking with min( r ) = ω · k + c .Then k is the degree of r and c is the finite shift , written deg ( r ) and f in ( r ) respectively. We can use the degree and the finite shift to define the fol-lowing operation.
Definition 5
Let r be an ω ranking with deg ( r ) = k and f in ( r ) = c . Define ¯ r as follows. Let s be a state with r ( s ) = ω · m + p .1. If m > k , then ¯ r ( s ) = ω · ( m − k ) + c .2. If m = k , then ¯ r ( s ) = ( p − c ) . We call ¯ r the finite zeroing of r . Intuitively, elements at the“lowest level” are normalized to zero and elements at higherlevels are shifted down by the degree of r . The followingresult is easy to prove. Proposition 2 If r is an ω ranking, then ¯ r is a ω -CF and r ∼ ¯ r . Hence, the finite zeroing of any ranking is an equivalent ω -CF. We can now extend the definition of ∗ to ω -CFs. Definition 6
Let r , r be ω -CFs. Then r ∗ r = r + r . Using this definition, ∗ is consistent with ¯+ for ω -CFs.Hence, ∗ can capture standard belief revision operators (e.g.,AGM, DP) by restricting to finite values and setting the de-gree of strength of each function appropriately. This is thenatural extension of revision, therefore, to the case that al-lows infinite plausibility values. Example
The motivating example over { heavy, f ly } canbe captured by the following function: r ( s ) = (cid:40) ω if s | = f ly if s | = heavy ∧ ¬ f ly otherwiseWe let ∗ n to denote a finite iteration of the ∗ operator. Sup-pose that, for each V ∈ { heavy, f ly } , r V is an OCF suchthat r V ( s ) = 2 if and only if s (cid:54)| = V . The following areimmediate: • r ∗ n r heavy ( s ) = 0 iff n ≥ . • r ∗ n r fly ( s ) (cid:54) = 0 for any n .ence, it takes 5 reports to convince the owner that theirdog is overweight. No finite number of reports will convincethem that the dog can fly.In the ω case, the algebra obtained is not identical to thefinite case. Proposition 3
The class of ω -CFs is a non-abelian groupunder ∗ . (i.e. it is closed, associative, and every element hasan inverse, but it is not commutative). The fact that ∗ is not commutative has interesting conse-quences, as illustrated in the following example. Example
Assume again that the vocabulary contains thepredicates { heavy, f ly } . Define r ( s ) = (cid:26) ω if s | = f ly otherwise r ( s ) = (cid:40) if s | = ¬ heavy ∧ f ly if s | = heavy ∧ f ly otherwise.Hence, r says that an agent believes dogs can not fly; more-over the agent essentially believes that a flying dog is an im-possibility. On the other hand, r says that an agent believesthat light dogs can fly - although the the strength of belief inthis claim is only finite. Moreover, r gives an ordering overless plausible states as well. Note that both r and r can beeither an initial belief state or an observation. The followingcalculations are immediate. r ∗ r ( s ) = (cid:40) ω if s | = ¬ heavy ∧ f lyω + 1 if s | = heavy ∧ f ly otherwise. r ∗ r ( s ) = (cid:26) ω if s = { f ly } otherwise.What is the significance of this example? It shows thatconditional beliefs from an observation can be maintainedat higher plausibility levels. In both cases, the underlyingagent will not believe dogs can fly following revision. Butthe first revision allows the ordering of states to be refinedsomewhat at the conditional level. The second revision, onthe other hand, washes away the finite level distinctions inthe original belief set. This is similar to AGM revision in thesense that recent information seems to carry some particularweight. However, the infinite jumps in plausibility outweighthe preference for recency. Nearly Counterfactual Reasoning
Motivation
In this section, we demonstrate how infinite-valued ordinalconditional functions can be useful for reasoning about con-ditional statements.
Example
We return to the flying-dog example. Supposethat we initially believe ¬ f ly and ¬ hollow ; in other words, we believe that dogs do not fly and that dogs do not have hol-low bones. Now suppose we are told that flying dogs havehollow bones. Informaly, we want to revise by the condi-tional statement ( hollow | f ly ) .Note that ( hollow | f ly ) actually does not give any newinformation about dogs. This revision should not changethe relative ordering of any worlds with a finite strength ofbelief. However, it does result in a change of belief. If one islater convinced of the existence of flying dogs, then the factabout hollow bones should be incorporated.We refer to the reasoning in the preceding example as nearly-counterfactual revision. It is essentially a form ofcounterfactual reasoning, in which hypothetical worlds areconsidered in isolation. At the same time, however, we keepa form of conditional memory at higher ordinal levels. Thisis not only useful for perspective altering revelations, but weargue it can also be useful for analogical reasoning.One important feature that is typically taken as a require-ment for conditional reasoning is the Ramsey Test. In thecontext of revision by conditional statements, Kern-Isbernerformulates the Ramsey Test as follows: when revising by aconditional, one would like to ensure that revision by ( ψ | φ ) followed by a revision by φ should guarantee belief in ψ (Ker99). We suggest that this formulation needs to be re-fined in order to be used in the case where infinite ranks arepossible.In the case of the flying dog, one is quite likely to acceptthe conditional ( hollow | f ly ) based on a single report withfinite strength. However, a single report of f ly with finitestrength will not be believed. If the antecedent of the con-ditional is “very hard” to believe, then we should not expectthe Ramsey Test to hold without some additional conditionon the strength of the subsequent report. The problem, in asense, is that the notion of believing a conditional is quitedifferent than the notion of believing a fact. In order to be-lieve ( hollow | f ly ) , we simply need to keep some kind ofrecord of this fact for the unlikely case where we discoverthat flying dogs happen to exist. On the other hand, in orderto believe f ly , we really need to make a significant changein our current world view. Levels of Implausibility
Approaches to counterfactual reasoning are typically in-spired to some degree by Lewis, who indicates that the truthof a counterfactual sentence is determined by its truth in al-ternative worlds (Lew73). We can represent this idea with ω -CFs. At each limit ordinal ω · k , we essentially havean entirely new plausibility ordering. As k increases, eachsuch ordering represents an increasingly implausible world.However, a sufficiently strong observation can force our be-liefs to jump to any of these unlikely worlds. As such, theseare not truly counterfactual worlds, because we admit thepossibility that they may eventually be believed.The important property that we can capture with ω -CFsis the following: there are some formulas that may be true,yet we can not be convinced to believe them based on anyfinite number of pieces of “weak evidence.” This allows uso give the following formal definition of the term nearlycounterfactual . Definition 7
Let r be an OCF. A formula φ is nearly coun-terfactual with respect to r just in case there is no ω -CF r (cid:48) such that Bel ( r ∗ r (cid:48) ) | = φ . The following is an immediate consequence of this defini-tion.
Proposition 4 If φ is nearly counterfactual with respect to r , then there is no finite sequence r , . . . , r n of ω -CFs suchthat Bel ( r ∗ r ∗ · · · ∗ r n ) | = φ . We introduce some useful notation.
Definition 8
Let φ be a formula. An OCF r is a φ -strengthening iff Bel ( r ) = { s | s | = φ } . So, a φ -strengthening is just a ranking function where theminimal states are exactly the models of φ . For any formula φ , let ( φ, n ) be the φ -strengthening of φ where models of φ have plausibility and every other state has plausibility n . Definition 9
Let r be an ω -CF. For any limit ordinal ω · k ,let r k be the following partial function: r k ( s ) = (cid:26) r ( s ) , if r ( s ) = ω · k + c for some c undefined otherwise Hence, r k is just the restriction of r to those states with plau-sibility values at level k . We say that φ is believed at level k if { s | s ∈ min( r k ) } | = φ . Let poss ( φ ) denote the set ofnatural numbers k such that s | = φ for some s in the domainof r k .We can now introduce a form of strengthening with nearlycounterfactual conditionals. In the definition, given an ω -CF r , we let deg ( s ) denote the value k such that r ( s ) = ω · k + c . Definition 10
Let r be an ω -CF and let ψ, φ be formulaswhere φ is nearly counterfactual with respect to r . Let n ∈ ω . r ∗ ( n, ψ | φ )( s ) = (cid:26) r ( s ) , if deg ( s ) (cid:54)∈ poss ( φ ) r ∗ ( ψ, n )( s ) otherwise We call this function the n -stengthening of ψ conditioned on φ . This function finds all levels of r where φ is possible, andthen strengthens ψ at only those levels. Example
Let r again be the plausibility function r ( s ) = (cid:40) ω if s | = f ly if s | = heavy ∧ ¬ f ly otherwiseIt is easy to verify that f ly is nearly counterfactual with re-spect to r . Now suppose that we extend the vocabulary toinclude the predicate symbol hollow . Define a new function r (cid:48) as follows: r (cid:48) ( s ) = (cid:26) r ( s ) , if s (cid:54)| = hollowr ( s ) + 1 , if s | = hollow This just says that we initally believe our dog does not havehollow bones; however, it is not particularly implausible. Itfollows that: • r (cid:48) ( s ) = ω if s | = f ly ∧ ¬ hollow . • r (cid:48) ( s ) = ω + 1 if s | = f ly ∧ hollow .From these results, it follows that: • r (cid:48) ∗ (2 , hollow | f ly )( s ) = r (cid:48) ( s ) , if s (cid:54)| = f ly . • r (cid:48) ∗ (2 , hollow | f ly )( s ) = ω , if s | = f ly ∧ hollow . • r (cid:48) ∗ (2 , hollow | f ly )( s ) = ω + 1 , s | = f ly ∧ ¬ hollow .So, roughly speaking, after strengthening by ( hollow | f ly ) ,we now believe that hollow bones are more plausible in allhypothetical situations where we believe flying dogs are pos-sible.Note that plausibility of a state is only changed at levelswhere φ is considered possible. Since the definition is onlyapplied to nearly counterfactual conditions, this means thatonly hypothetical states are affected by the strengthening.It remains to move from conditional strengthening to con-ditional revision. Recall that, for any ω -CF with min( r ) = ω · k + c , we write f in ( r ) = c . Definition 11
Let r be an ω -CF and let ψ, φ be formulaswhere φ is nearly counterfactual with respect to r . r ∗ ( ψ | φ )( s ) = (cid:26) r ( s ) , if deg ( s ) (cid:54)∈ poss ( φ ) r ∗ ( ψ, f in ( r k ))( s ) if r ( s ) = ω · k + c Hence, for revision, we strengthen belief in ψ by the leastvalue that will ensure ψ is believed at level k .Under this definition, we satisfy a modified form of theRamsey Test. Proposition 5
Let r be an ω -CF and let s be a state with r ( s ) = ω · k + c . If r (cid:48) is an ω -CF with degree of strengthlarger than k and Bel ( r (cid:48) ) | = φ , then Bel (( r ∗ ( ψ | φ )) ∗ r (cid:48) ) | = ψ. Hence, if we revise by ( ψ | φ ) followed by an OCF with “suf-ficiently strong” belief in φ , then ψ will be believed. Relation to Existing Work
Infinite Plausibility Values
There has been related work on the use of infinite valuedordinals in OCFs. In particular, Konieczny defines the no-tion of a level of belief explicitly in terms of limit ordi-nals(Kon09). In this work, different “levels” are used to rep-resent beliefs that are independent in a precise sense. Thelowest level is used for representing an agents actual beliefsabout the world, whereas higher levels are used to representintegrity constraints. Our approach here is different in thatwe explicitly use the ordering on limit ordinals to representinfinite leaps in plausibility. This work is also distinguishedby the fact that we use ordinal arithmetic on a small class ofordinals to define a simple algebra of belief change.
Belief Improvement
The success postulate ( K ∗ φ (cid:96) φ ) of the AGM framework isclearly incorrect in cases where evidence is additive. That isto say, there are situations where a single observation is notsufficient to convince an agent to believe a particular fact. mprovement operators (KP08) are belief change operatorsthat address this issue by introducing a new set of postulates.The most important postulate states that an improvement op-erator ◦ must have the property that: (I1) There exists n ∈ N such that B (Ψ ◦ n φ ) (cid:96) φ .Here Ψ is an epistemic state, and B ( · ) maps an epistemicstate to the minimal elements of the underlying ordering.Hence, an improvement operator has the property that anagent will be convinced to believe φ after a finite numberof improvements. The remaining postulates for a weak im-provement operator are essentially the DP postulates appliedto the operation ◦ n obtained from (I1) . We refer the readerto (KP08) for the complete list of postulates.We define an analog of (I1) as follows. If r φ denotes a φ -strengthening, we can express the condition as follows.(I ∗ ) There exists n ∈ N such that Bel ( r ∗ n r φ ) | = φ .The truth of this property depends on the degrees of strengthof the functions. Proposition 6 If r is an ω -CF and r φ is a φ -strengtheningwith finite strength, then I ∗ holds. For an epistemic state Ψ defined by ≺ Ψ , let r Ψ be the canon-ical representation of Ψ . Define ◦ n such that Ψ ◦ φ is ob-tained by taking the ordering induced by r Ψ ∗ r ( φ, n ) . Proposition 7
For any n ∈ N , the operator ◦ n is a weakimprovement operator. We call ◦ n a finite improvement operator , because the de-gree of strength is finite. This result is essentially a corollaryof Proposition 6, and it suggests that our ∗ operation basedon normalized addition is actually the natural extension of improvement to the setting of ω -CFs.The advantage of infinite plausibility values is that theygive us greater flexibility in modelling improvement. Proposition 8 If r is an ω -CF and r φ is a φ -strengtheningwith finite strength, then I ∗ does not hold. This result essentially states that ( I1 ) is not a sound propertyfor ∗ if we allow infinite plausibility values. This distinctioncan be seen in our running example. There is no finite num-ber of improvements that will force the agent to believe thatdogs can fly.It is actually difficult to express the analog of Proposition7 in the context of ω -CFs, because a total pre-order overstates can not capture the “infinite jumps” in plausibility en-coded by ω ordinal ranks. But it is possible to define a cor-respondence between sequences of orderings and ordinals in ω . Definition 12
Let r be an ω -CF where max( r ) = ω · d + b for some d, b . For i ≤ d , let r i denote the function definedas follows:1. domain ( r i ) = { s | r ( s ) = ω · i + c } .2. If r ( s ) = ω · i + c , then r i ( s ) = c . The following propositions are immediate. If s is in the n th level of Ψ , then r Ψ ( s ) = n Proposition 9
Each r i is a ω ranking, and there exists an ω -CF such that r (cid:48) i ∼ r i Proposition 10
For any ω -CF r over a vocabulary P with deg ( r ) = d , there is an extended vocabulary P and a se-quence r , . . . , r d of ω -CFs such that, for each i ≤ d , the r i is equivalent (i.e. ∼ ) to the restriction of r to ordinals ofdegree i . This result is proved by just extending the vocabulary appro-priately with propositional variables that make each infinitejump in the ordinal value definable. By breaking r into a setof ω -CFs, it follows that (I ∗ ) holds at level d when r φ hasdegree of strength ω · d . Therefore, belief change by nor-malized addition on ω -CFs can really just be seen as a fi-nite collection of improvements as each level. The importantpoint, however, is that no finite sequence of improvementsat level d will ever impact the actual beliefs at lower levels. Conditional Belief Revision
Conditional belief revision was previously addressed byKern-Isberner, who proposes a set of rationality postulatesfor conditional revision (Ker99). A concrete approach toconditional revision is also proposed, through the following ω -CF : r ∗ ( ψ | φ )( s ) = (cid:40) r ( s ) − r ( ψ | φ ) , if s | = φ ∧ ψr ( s ) + α + 1 , if s | = φ ∧ ¬ ψr ( s ) , if s | = ¬ φ where α = − if r ( { φ, ψ } ) < r ( { φ } ) , and α = 0 other-wise. This operation satisfies all of the postulates for con-ditional revision, as well as the Ramsey Test. We remark,however, that this approach is not well-defined if we allowinfinite plausibility values because of the ordinal subtractionon the right hand side. We suggest that this is not just aformal artefact of the theory; conditionals that are ”almostcertainly” false actually must be treated slightly differently.In our approach, we essentially require the evidence for φ to be substantially stronger than the evidence for the con-ditional. We suggest that our beliefs following conditionalrevision should be changed in sort of an infinitesimally smallway. While our beliefs about the actual world do not change,our beliefs about some (nearly) impossible world do, in fact,change.Note that it is actually possible to reconcile our approachwith Kern-Isberner’s approach, by using the conditional re-vision above on each level r k of the initial OCF r . Atpresent, we are using a simple strengthening on each level,which actually flattens the plausibility structure after ordi-nal addition. A combined approach could respect the infi-nite jumps in plausibility, while satisfying the postulates forconditional revision at each level. We leave an investigationof this combined approach for future work. Discussion
Conclusion
In this paper, we have explored the use of infinite ordinalsfor reasoning about belief change and conditional reasoning.We have shown that allowing plausibility values to rangever ω results in a belief algebra that is only slightly morecomplicated, and we gain an expressive advantage. In par-ticular, we can represent situations where stubbornly heldbeliefs are resistant to evidence to the contrary. We havedemonstrated that this results in a slightly more expressiveclass of improvement operators where evidence increasesrelative belief, but no finite number of improvements willactually lead to a change in the belief state. Finally, weaddressed so-called “nearly counterfactual” revision, wherewe incorporate information that is conditional on a highlyunlikely statement. Future Work
This paper is a preliminary exploration into different appli-cations and formal properties of infinite valued ordinal con-ditional functions. It remains to move beyond ω -CFs, tocompletely characterize the relationship with improvementoperators, and to consider further practical applications.In the present framework, we have discussed nearly coun-terfactual reasoning as a tool for keeping a sort of “memory”about unlikely situations, in order to incorporate this infor-mation later if necessary. But there is also a natural kind ofreasoning that would allow us to use conditionals to reasonby analogy about the actual state of the world. Consider thefollowing well-known ambiguity from (Lew73), and origi-nally attributed to Quine:1. If Caesar was president, he would use nuclear weapons.2. If Caesar was president, he would use catapults.As a conditional, we could write both as ( W | C ) , where W stands for a weapon that would be used and C is the condi-tion “Caesar is president.” But (1) suggests that we condi-tion by imagining Caesar alive in the current world. So thisis a conditional statement interpreted in the current state ofthe world. On the other hand, (2) suggests that we considerwhat would happen in some past world where Caesar exists.Now suppose that we believe a certain politician is actu-ally very similar to Caesar. If we believe that Caesar woulduse nuclear weapons, then we may conclude that this “real”politician would also use nuclear weapons. Formally, wecould proceed as follows: if some hypothetical world is iso-morphic to the current state of the world when we restrictthe vocabulary (to not include Caesar), then we can use in-ferences about the hypothetical world to draw conclusionsabout the actual world. This is a form of ampliative rea-soning that we intend to explore through ω -CFs in futurework. References [AGM85] C.E. Alchourr´on, P. G¨ardenfors, and D. Makin-son. On the logic of theory change: Partial meet functionsfor contraction and revision.
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