Ceteris paribs logic in conterfactal reasoning Patrick Girard University of Ackland p.girard@ackland.ac.nz Marcs Anthony Triplett University of Ackland mtri285@acklandni.ac.nz The semantics for conterfactals de to David Leis has been challenged on the basis of nlikely, or impossible, events. Sch events may ske a given similarity order in favor of those possible orlds hich exhibit them. By pdating the relational strctre of a model according to a ceteris paribs clase one forces ot, in a natral manner, those possible orlds hich do not satisfy the reqirements of the clase. We develop a ceteris paribs logic for conterfactal reasoning capable of performing sch actions, and offer several alternative (relaxed) interpretations of ceteris paribs. We apply this frameork in a ay hich allos s to reason conterfactally ithot having or similarity order skeed by nlikely events. This contines the investigation of formal ceteris paribs reasoning, hich has previosly been applied to preferences [2], logics of game forms [11], and qestions in decision-making [25], among other areas [18]. 1 Introdction The principal task of this paper is to ork toards integrating ceteris paribs modalities into conditional logics so that some dissonant analyses of conterfactals may be reconciled. We also sggest that ceteris paribs clases may be nderstood dynamically, in the sense of dynamic epistemic logic [8], and e interpret or reslting ceteris paribs logic accordingly. Ceteris paribs clases implicitly qalify many conditional statements that formlate las of science and economics. A ceteris paribs clase adds to a statement a proviso reqiring that other variables or states of affairs not explicitly mentioned in the statement are kept constant, ths rling ot benign defeaters. For instance, Avogadro s la says that if the volme of some ideal gas increases then, everything else held eqal, the nmber of moles of that gas increases proportionally. Varying the temperatre or pressre cold provide sitations that violate the plain statement of the la, bt the ceteris paribs clase acconts for those. It specifically isolates the interaction beteen volme and nmber of moles by keeping everything else eqal. In the same spirit, the Nash eqilibrim in game theory is a soltion concept that picks strategy profiles in hich none of the agents cold nilaterally (i.e., keeping the actions of others constant, or eqal) deviate to their on advantage. We may nderstand a ceteris paribs clase as a lingistic device intended to shrink the scope of the sentence qalified by the clase. For instance, hen I make the tterance I prefer fish to beef, ceteris paribs I may mean something different from if I simply ttered I prefer fish to beef. By enforcing the ceteris paribs condition I rle ot some sitations hich affect my preference. For example if, henever I eat fish I m beaten ith a mallet, hile henever I eat beef I m left in peace, I might retract the second tterance and maintain the first. The ceteris paribs clase redces the nmber of states of affairs nder consideration. For modal logicians, rling ot states of affairs amonts to strengthening an accessibility relation, conseqently changing the relational strctre of a model. This bears similarity to R. Ramanjam (Ed.): TARK 2015 EPTCS 215, 2016, pp. 176 193, doi:10.4204/eptcs.215.13
P. Girard and M. A. Triplett 177 the epistemological forcing of Vincent Hendricks [12], hich seeks to rle ot irrelevant alternatives in a ay hich allos knoledge in spite of the possibility of error. Wesley Holliday [13] develops several interpretations of the epistemic operator K based on the relevant alternatives epistemology; namely, that in order for an agent to have knoledge of a proposition, that agent mst eliminate each relevant alternative. Holliday s semantics are based on the semantics for conterfactals de to David Leis [15]. One cold see relevant orlds as those hich keep things eqal. When reasoning sing Avogadro s la, the relevant possible orlds are those here the temperatre and pressre have not changed. Ths, in order for an agent to have knoledge, that agent mst eliminate the alternatives among the orlds hich keep things eqal. Previosly, ceteris paribs formalisms have been given for logics of preference [2] and logics of game forms [11]. Here e extend the analysis to conterfactal reasoning. The importance of conterfactals in game theory is ell knon (see, for instance, [19]). For example, Bassel Tarbsh [23] arges that the Sre-Thing Principle 1 oght to be nderstood as an inherently conterfactal notion. We ill motivate or discssion by thinking throgh Kit Fine s ell-knon minor-miracles argment [10], a ptative conterexample to Leis semantics. We ill arge that ceteris paribs logic, sitably adapted to conditionals, provides a natral response to this kind of argment. Moreover, e ill see that ceteris paribs logic reveals a sefl featre missing from the standard formalisation of conterfactals; namely, the explicit reqirement that certain propositions mst have their trth remain fixed dring the evalation of the conterfactal. This is implicitly thoght to hold, to some degree, hen one orks ith models hich have similarity orders or systems of spheres. The conditional logic of Graham Priest [17] makes jst that assmption, bt ith no syntactic assrance. Ceteris paribs logic provides, in addition to the nderlying similarity order over possible orlds, a syntactic apparats to reason ith sch ceteris paribs clases directly in the object langage. 2 Conterfactals Here e shall formalise conterfactals in the style of Leis. Let Prop be a set of propositional variables. We are concerned ith models of the form M=(W,,V) sch that the folloing obtain. 1. W is a non-empty set of possible orlds. 2. is a family { } W of similarity orders, i.e., relations on W W (ith W W) sch that: W, is reflexive, transitive and total, and v for all v W \{}. 3. V is a valation fnction assigning a sbset V(p) W to each propositional variable p Prop. Intitively, W is the set of orlds hich are entertainable from. Worlds hich are not entertainable from are deemed simply too dissimilar from to be considered. Say that is at least as similar to as v is hen v, and that it is strictly more similar hen v. If M satisfies each of the three reqirements e call M a conditional model. A relation is said to be ell-fonded if for every non-empty S W the set 1 An otcome o of an action A is a sre-thing if, ere any other action A to be chosen, o old remain an otcome. The Sre-Thing Principle [20] states that sre-things shold not affect an agent s preferences.
178 Ceteris paribs logic in conterfactal reasoning Min M (S)={v S W : there is no ith <v} (1) is non-empty. 2 We ill sppress the sperscript M if it is clear from the context hich model e re discssing. If a model M=(W,,V) has only ell-fonded similarity orders e say that M satisfies the limit assmption. For ease of exposition, e ill assme that or conditional models satisfy the limit assmption. Of corse, e may generalise the semantics for conterfactals in the sal ay [15], so that or reslts ork for models hich do not satisfy the limit assmption as ell. Definition 1 (Langage L ) The langage L of conterfactals is given by the folloing grammar ϕ ::= p ϕ ϕ ψ ϕ ψ. We define ϕ ψ := ( ϕ ψ), ϕ ψ := ϕ ψ, ϕ ψ := (ϕ ψ). Definition 2 (Semantics) Let M =(W,,V) be a ell-fonded conditional model. Then p M = V(p) ϕ M = W \ ϕ M ϕ ψ M = ϕ M ψ M ϕ ψ M = { W :Min ( ϕ M ) ψ M }. Let W. If ϕ e ritem, = ϕ, and if ϕ e ritem, = ϕ. 3 The Nixon argment There is a problem dating back to the 1970s [1, 4, 10] srronding the semantics for conterfactals proposed by Leis. We have fond that or ceteris paribs conterfactals (defined belo) provide a niqe perspective on the problem (a ptative conterexample). The argment goes as follos. Assme, dring the Cold War, that President Richard Nixon had access to a device hich lanches a nclear missile at the Soviets. All Nixon is reqired to do is press a btton on the device. Consider the conterfactal if Nixon had pshed the btton, there old have been a nclear holocast. Call it the Nixon coterfactal. It is not so difficlt to see that the Nixon conterfactal cold be tre, or cold be imagined to be tre. Indeed, one cold arge that the Nixon conterfactal oght to be tre in any sccessfl theory of conterfactals. Fine and Leis both agree (and so do e) that the conterfactal is tre ([10, p. 452], [16, p. 468]), bt Fine sed the Nixon conterfactal to arge that the Leis semantics yields the rong verdict. This is becase a orld ith a single miracle bt no holocast is closer to reality than one ith a holocast bt no miracle. [10, p. 452] In response, Leis arges that, provided the Nixon sitation is modelled sing a similarity relation hich respects a plasible system of priorities (see belo), the conterfactal ill emerge tre. We ill provide a different response sing ceteris paribs conterfactals, bt first let s see ho Fine and Leis model the sitation. 2 As sal, <v is defined as v and not v
P. Girard and M. A. Triplett 179 Consider to classes of possible orlds. One class,, consists of those orlds in hich Nixon pshes the btton, and the btton sccessflly lanches the missile. The second, v, consists of those orlds in hich Nixon pshes the btton, bt some small occrrence sch as a minor miracle prevents the btton s correct operation. Certainly those orlds here the btton does not lanch the missile bear more similarity to the present orld than those here it does. This is Fine s interpretation of Leis semantics. Any orld in has been devastated by nclear arfare, contless lives have been lost, there is nclear inter, etc., hereas orlds in v contine on as they old have done. To illstrate Fine s interpretation, let p, s, m, h be the propositions: p = Nixon pshes the btton, s = the missile sccessflly lanches, m = a miracle prevents the missile being lanched, h = a nclear holocast occrs, and consider the folloing model, the Fine model: F 1 2 n p,s,h v k v p,m An arro from x to y indicates relative similarity to, so 1 is more similar to than 2 is. Arros are transitive, and the snake arro beteen v indicates that v i j for every i, j. For each i, F, i = p s h; and for each v i v, F,v i = p m. World is intended to represent the real orld: Nixon did not psh any catastrophic anti-soviet bttons, 3 no nclear missile as sccessflly lanched at the Soviets, no miracle prevented any sch missile, and no nclear holocast occrred. World is more similar to than any orld in is, since in any -orld Nixon pshes the btton and begins a nclear holocast. By (1), is therefore the minimal p-orld. At the proposition h is false, and so F, = p h. Therefore, Fine concldes, the Nixon conterfactal is false in Leis semantics. In response, Leis arges that the proper similarity relation to model the Nixon conterfactal shold respect the folloing system of priorities: 1. It is of the first importance to avoid big, idespread, diverse violations of la. 2. It is of the second importance to maximize the spatio-temporal region throghot hich perfect match of particlar fact prevails. 3. It is of the third importance to avoid even small, localized, simple violations of la. 4. It is of little or no importance to secre approximate similarity of particlar fact, even in matters that concern s greatly. ([16, p. 472]) 3 Althogh there is no ay for s to kno this, for the sake of the argment e assme that it is so.
180 Ceteris paribs logic in conterfactal reasoning Based on this system of priorities orld 1 is more similar to than is becase perfect match of particlar fact conts for mch more than imperfect match, even if the imperfect match is good enogh to give s similarity in respects that matter very mch to s. [16, p. 470] That is, orlds in v in hich a small miracle prevents the missile being lanched may look qite similar to or orld, bt only approximately so. And in Leis system of priorities, perfect match oteighs approximate similarity. The Leis model, then, looks like this: L 1 2 n p,s,h v k v p,m In the Leis model, 1 is the orld most similar to, and in 1 the missile sccessflly lanches, there is a nclear holocast, and so the Nixon conterfactal is tre. Leis ths responds to Fine by defending a similarity order that favors 1 over. He is jstified by prioritising perfect over approximate match in a similarity relation according to the aforementioned system. The interpretation of the Nixon conterfactal e ill offer is in line ith Leis, thogh e do not rely on his system of priorities. We ill achieve a resoltion similar to his ithot having to defend a model different from Fine s. After all, as Leis says: I do not claim that this pre-eminence of perfect match is intitively obvios. I do not claim that it is a featre of the similarity relations most likely to gide or explicit jdgments. It is not; else the objection e are considering never old have been pt forard. [16, p. 470] Instead, e ill treat the Nixon conterfactal ith an explicit ceteris paribs clase, dispatching ith the nintitive pre-eminence of perfect match in constrcting the similarity relation. Or interpretation of the Nixon conterfactal is mch like in preference logic, here formal ceteris paribs reasoning as first applied [2, 9, 24]. Consider the folloing diagram, hich shos a preference of a raincoat to an mbrella, provided earing boots is kept constant: raincoat no boots raincoat boots mbrella no boots mbrella boots Arros point to more preferred alternatives, and are transitive. Evidently, having an mbrella and boots is preferred to having a raincoat and no boots. The variation of having boots skes the preference. If a ceteris paribs clase is enforced, garanteeing that in either case boots ill be orn or boots ill not be orn, then the correct preference is recovered. A similar sitation occrs in the logic of conterfactals. The variation of certain propositions can ske the similarity order. In Fine s argment, this is done by the variation of physical la, a miracle. If e ere to restrict the orlds considered dring the evalation of the conterfactal to those that agree ith on the proposition m, then in F the orld old no
P. Girard and M. A. Triplett 181 longer assme the role of minimal p-orld. Rather, 1 old. In orld 1 a nclear holocast does occr, hence the conterfactal becomes tre, as desired. This is or resoltion of the Nixon argment, hich e next formalise. 4 Ceteris paribs semantics We introdce into or langage a ne conditional operator hich generalises the sal one. In particlar, it accommodates explicit ceteris paribs clases. The athors in [2] ere the first to define object langages in this ay. They developed a modal logic of ceteris paribs preferences in the sense of von Wright [24]. For no e ill take the ordinary conditional operator and embed ithin it a finite set of formlas Γ nderstood as containing the other things to be kept eqal. 4 Definition 3 (Langage L CP ) Let Γ be a finite set of formlas. Then the langage L CP is given by the grammar 5 ϕ ::= p ϕ ϕ ψ [ϕ,γ]ψ. We nderstand the modality [ϕ,γ]ψ as the conterfactal ϕ ψ sbject to the reqirement that the trth of the formlas in Γ does not change. We define ϕ ψ := ( ϕ ψ), ϕ ψ := ϕ ψ, ϕ, Γ ψ := [ϕ, Γ] ψ. We call the conditional[ϕ, Γ]ψ a ceteris paribs conditional, or, if the antecedent is false, a ceteris paribs conterfactal. L CP is interpreted over standard conditional models, and ths reqires no additional semantic information. Some additional notation is reqired, hoever. Let M = (W,,V) be a conditional model and let,,v W. Let Γ L CP be finite. Define the relation Γ over W by Γ v if for all γ Γ, M, = γ iff M,v = γ. Then Γ is an eqivalence relation. 6 Set[] Γ ={ W : Γ }, the collection of -entertainable orlds hich agree ith on Γ. Define Γ := ([] Γ [] Γ ), the restriction of to the above orlds. Ths if,v [] Γ then either Γ v or v Γ. Definition 4 (Semantics) Let M =(W,,V) be a conditional model. Then [ϕ,γ]ψ M = { W :Min Γ ( ϕ M ) ψ M }. The semantics for the reglar connectives are the same as those in Definition 2. Notice that e recover the ordinary conterfactal ϕ ψ ith[ϕ, /0]ψ. 4 The choice of Γ finite is largely technical. We ill mention some possibilities and difficlties regarding the case here the ceteris paribs set Γ may be infinite in or conclding remarks. 5 We redefine the langage more precisely as Definition 8 in the appendix. For simplicity e ork ith the one no stated. 6 Technically, the relation Γ shold be defined together ith the semantics in Definition 4 by mtal recrsion. Again, e favor the simpler presentation.
182 Ceteris paribs logic in conterfactal reasoning Consider again the Fine model F. As before e have F, = p h, bt no F, =[p,{m}]h. (2) We ths think abot the Nixon conterfactal by ay of ceteris paribs reasoning. Alloing the trth of arbitrary formlas to vary dring the evalation of a conterfactal can distort the given similarity order, thereby attribting falsity to a sentence hich may be intitively tre. By forcing certain formlas to keep their trth stats fixed one can rle ot these cases, hich has jst been demonstrated ith (2). This ceteris paribs qalification is done in preference logic, and indeed in more general scientific and economic practice. 7 The Nixon conterfactal is simply a sitation involving a defeater, or an irrelevant alternative, hich oght to be forced ot. 5 Ceteris paribs as a dynamic action The modality [ϕ, Γ]ψ behaves like a dynamic operator, in the sense of dynamic epistemic logic. For modality-free formlas ϕ and ψ, evalating [ϕ, Γ]ψ at W amonts to transforming into M=(W,{ } W,V) [Γ]M=(W,{ Γ } W,V) and evalating ϕ ψ at [Γ]M,. This dynamic action is possible since e are altering the relational strctre of M ith only a finite amont of information from Γ. Note that the set W on hich is defined on may change after the pdate. By pdating the model M ith a ceteris paribs clase Γ, orlds hich disagree on Γ are relegated to the class W \W of infinitely dissimilar (indeed, irrelevant) orlds. Figre 1 shos ho the Fine model changes after being pdated by a ceteris paribs clase forcing agreement on m. This forces ot the v-orlds from consideration dring the evalation of the conterfactal; in some sense syntactically correcting the provided similarity order. Of corse, if each orld already agreed ith on {m} the ceteris paribs clase old have no effect. The modality-free condition on ϕ and ψ cannot be removed. In particlar, one cannot iterate the dynamic ceteris paribs action and retain agreement ith the static ceteris paribs conterfactal operator. To see this, consider the example in Figre 2. Taking Γ={s} and = /0, one has M, =[p,γ][q, ]r, bt [Γ]M, = p [q, ]r. 6 Uniformly selecting ceteris paribs clases Having created a formalism hich accommodates explicit ceteris paribs clases, one might desire a method for niformly selecting the ceteris paribs set Γ. For von Wright [24], ceteris paribs means fixing every propositional variable hich does not occr in the niverse of discorse of the ceteris paribs expression nder consideration. More precisely, let UD(ϕ) be the set of all propositional variables occrring in the formla ϕ, defined indctively as follos. 7 See Schrz [21] on comparative ceteris paribs las.
P. Girard and M. A. Triplett 183 Before F 1 2 n p,s,h v k v p,m After [{m}]f 1 2 n p,s,h v k v p,m Figre 1: The Fine model before and after is pgraded to {m}. UD(p) = {p} UD( ϕ) = UD(ϕ) UD(ϕ ψ) = UD(ϕ) UD(ψ) UD([ϕ, Γ]ψ) = UD(ϕ) UD(Γ) UD(ψ) UD({γ 1,...,γ n }) = UD(γ 1 ) UD(γ n ). Then the ceteris paribs conterfactal if ϕ ere the case then, ceteris paribs, ψ old be the case amonts to the expression [ϕ,prop\(ud(ϕ) UD(ψ))]ψ. (3) No all propositional variables not occrring in the niverse of discorse of the conterfactal antecedent or conseqent are fixed. Updating the Fine model ith respect to von Wright s ceteris paribs set yields the folloing model:
184 Ceteris paribs logic in conterfactal reasoning M [ ][Γ]M s p,s q,s q,r s p,s q,s q,r s p,s q,s q,r s p,s q,s q,r Figre 2: The horizontal panels labelled and define the similarity orders and respectively. [{s,m}]f 1 2 n p,s,h v k v p,m We have F, =[p,{m,s}]h, bt vacosly! It appears that the relation Γ is too strong to interact ith von Wright s definition. We are reqiring that everything else is kept eqal. This is qestionable metaphysics, to say the least. Leis made a similar observation in [15], abot the conterfactal if kangaroos had no tails, they old topple over : We might think it best to confine or attention to orlds here kangaroos have no tails and everything else is as it actally is; bt there are no sch orlds. Are e to sppose that kangaroos have no tails bt that their tracks in the sand are as they actally are? Then e shall have to sppose that these tracks are prodced in a ay qite different from the actal ay. [...] Are e to sppose that kangaroos have no tails bt that their genetic makep is as it actally is? Then e shall have to sppose that genes control groth in a ay qite different from the actal ay (or else that there is something, nlike anything there actally is, that removes the tails). And so it goes; respects of similarity and difference trade off. If e try too hard for exact similarity to the actal orld in one respect, e ill get excessive differences in some other respect. ([15, p. 9])
P. Girard and M. A. Triplett 185 In fact, for the logic of ceteris paribs conterfactals to fnction in a meaningfl fashion, every formla occrring in Γ mst be independent from the conterfactal antecedent. In the Fine model, e insist that the trth vales of s and m are kept fixed. These propositions, hoever, are nomologically related to p, so e can t change the trth vale of p ithot affecting the trth vales of s and m. This is hy the conterfactal [p,{m,s}]h is vacosly tre, bt then so is the conterfactal [p,{m,s}] h. To accommodate a niform method for selecting ceteris paribs clases, more flexibility is reqired. What oght to be kept eqal hen e can t keep everything else eqal? In the next section e ill consider to strategies for relaxing the interpretation of ceteris paribs to address this qestion. 7 Relaxing the ceteris paribs clase 7.1 Naïve conting We ill no introdce another interpretation for the modality [ϕ,γ]ψ. Let s rite [ϕ,γ]ψ M CP for the set [ϕ,γ]ψ M from Definition 4, and let = CP act as the ordinary satisfaction relation for Boolean formlas, bt ith M, = CP [ϕ,γ]ψ iff [ϕ,γ]ψ M CP. Whereas in Definition 4 e reqired strict agreement on the set Γ, in order to develop a logic for ceteris paribs conterfactals ith a eaker semantics e ill instead relax the reqirement to maximal agreement. The best e can do is preserve the set Γ as mch as possible for any given model. Let Γ L CP be finite, and let M=(W,,V) be a conditional model. Define A M Γ Define the relation Γ on W by Γ v iff : W W 2Γ by A M Γ (,v)={γ Γ :M, = γ iff M,v = γ}. (4) either A M Γ (,) > AM Γ (v,), or AM Γ (,) = AM Γ (v,) and v. The relation Γ can be seen as a transformed, reordering the similarity order so that orlds closer to preserve at least as mch of Γ as orlds frther aay, and if any to orlds agree on Γ to the same qantity, then the nearer orld is more similar to ith respect to. Definition 5 (Semantics) Let M = (W,,V) be a conditional model satisfying the limit assmption. Let Γ L CP be finite. Then [ϕ,γ]ψ M NC = { W :Min Γ ( ϕ M ) ψ M }. We rite M, = NC [ϕ,γ]ψ iff [ϕ,γ]ψ M NC. Fact 1 LetM=(W,,V) be a conditional model. Let W, and let X {CP,NC}. Then the folloing are tre, here ±α is shorthand hich niformly stands for either α or α: 1. M, = ϕ ψ iff M, = X [ϕ, /0]ψ
186 Ceteris paribs logic in conterfactal reasoning 2. M, = X (±α ϕ,γ (±α ψ)) ϕ,γ {α} ψ 3. M, = CP ϕ,γ ψ M, = NC ϕ,γ ψ 4. M, = NC [ϕ,γ]ψ M, = CP [ϕ,γ]ψ The original ceteris paribs preference logic [2] cold be axiomatised sing standard axioms together ith Fact 1.2 and its converse. A crcial difference ith NC semantics is that the converse of Fact 1.2 does not hold. The existence of a ϕ ψ-orld hich maximally agrees on Γ {α} does not ensre that α actally holds at that orld. In fact, it is not garanteed that any formla from Γ {α} is obtained. 7.2 Maximal spersets An approach to conterfactals familiar to the AI commnity [5 7, 14] makes se of a selection fnction hich chooses the closest orld according to maximal sets of propositional variables. More specifically, each orld satisfies some set P Prop of propositional variables, and a orld is a orld closest to if there is no v ith P P v P. Taking this as a kind of ceteris paribs formalism e obtain the folloing variant of or ceteris paribs conterfactals. First let s define the relation Γ on W by Γ v iff either A M Γ (v,) AM Γ (,), or AM Γ (v,)= AM Γ (,) and v. Definition 6 (Semantics) Let M = (W,,V) be a conditional model satisfying the limit assmption. Let Γ L CP be finite. Then [ϕ,γ]ψ M MS = { W :Min Γ ( ϕ M ) ψ M }. We ritem, = MS [ϕ,γ]ψ iff [ϕ,γ]ψ M MS. No Γ is maximally preserved in the sense that orlds hich preserve the same propositions as another, and frthermore preserve additional propositions from Γ, are deemed to approximate Γ more closely; hile orlds,v ith neither A M Γ (,) AM Γ (v,) nor A M Γ (v,) AM Γ (,) are considered incomparable. Fact 2 (Extends Fact 1) Let M =(W,,V) be a conditional model. Let W. Then the folloing are tre. 1. M, = ϕ ψ iff M, = MS [ϕ, /0]ψ 2. M, = MS (±α ϕ,γ (±α ψ)) ϕ,γ {α} ψ 3. M, = CP ϕ,γ ψ M, = MS ϕ,γ ψ 4. M, = MS [ϕ,γ]ψ M, = CP [ϕ,γ]ψ 8 Dynamics and the Nixon conterfactal Given a ceteris paribs interpretation X {CP,NC,MS}, let s rite [Γ] X M for the model M pdated ith a ceteris paribs clase Γ according to interpretation X. Specifically, e have the folloing definition.
P. Girard and M. A. Triplett 187 Definition 7 LetM=(W,,V) be a conditional model, and let Γ L CP be a finite set of formlas. We define the pdated models [Γ] X M, for X {CP,NC,MS}, by [Γ] CP M := (W, Γ,V); [Γ] NC M := (W, Γ,V); [Γ] MS M := (W, Γ,V). This provides s ith three dynamic ceteris paribs pdates. Let s see ho they treat the Nixon conterfactal. We have already itnessed the CP pdate ith ceteris paribs sets {m} and {m, s}, and conclded that both make the conterfactal tre (vacos trth ith{m,s}). NC andms pdates agree on the trth of the Nixon conterfactal ith the CP pdate on {m}, bt disagree on {m,s}. Updating the Fine model ith von Wright s ceteris paribs clase {m, s} according to the NC interpretation yields F again. Ths F, = NC [p,{m,s}]h. Updating Fine s model ith {m,s} according to the MS interpretation gives the folloing model: [{m,s}] MS F 1 2 n p,s,h v k v p,m In [{m,s}] MS F the Nixon conterfactal is not tre, and neither is p h. We smmarise the trth of the Nixon conterfactals p h and p h in the varios pdated Fine models in the folloing table. Interpretation Conterfactal Clase CP NC MS p h {m} tre tre tre {m, s} tre false false p h {m} false false false {m, s} tre tre false The ros labelled ith p h and p h indicate the trth vale of those conterfactals in the pdated models [Γ] X F, here Γ is given by the cell in the Clase colmn and X is given by the Interpretation colmn. Formally, the table illstrates ho different trth vales for the Nixon conterfactal may be obtained by combining the varios interpretations of ceteris paribs (CP, NC, MS) ith the different ceteris paribs sets (the selected set {m} or von Wright s set {m,s}). Bt this doesn t mean that all combinations are legitimate formalisations of Fine s argment. Fine s story is abot small miracles that can interfere ith Nixon s ploy, not abot hether the missile old sccessflly lanch shold Nixon press the btton. That the proposition s mst be able to vary is crcial to the story, so one sholdn t attempt to keep it eqal, on a par ith m. We adhere to or favored formalisation of the Nixon argment in hich the
188 Ceteris paribs logic in conterfactal reasoning proposition m is the only one that needs to be kept eqal. We have given principled reasons for this choice, and or selection makes the conterfactal tre all interpretations agree on that. The point of the table is a formal one, namely that the trth-vales of conterfactals vary ith different ceteris paribs pdates according to their interpretation. 9 Theorems In the appendix (Corollary 1) e prove that the logic Λ L CP C of ceteris paribs conterfactals over the class of conditional frames C is complete for CP/NC/MS semantics. The proof orks by translating formlas of L CP into formlas of a comparative possibility langage, in the style of Leis, and axiomatising the eqivalent logic. This permits a clearer redction of ceteris paribs modalities to basic comparative possibility operators, albeit ith a translation exponential in the size of Γ. 10 Conclding remarks This paper has introdced a ceteris paribs logic for conterfactal reasoning by adapting the formalism in [2]. We have introdced some variants on ceteris paribs logic in light of philosophical difficlties arising in the application of conditionals. We apply or frameork to the Nixon conterfactal, and ith this bring a ne perspective to the problem. We have sggested and explored the dynamic perspective of or varios syntactic interpretations of ceteris paribs, hich has reslted in a richer nderstanding of so-called comparative ceteris paribs reasoning in formal settings. We have provided completeness theorems hich demonstrate that the ceteris paribs logics so obtained ltimately redce to the nderlying conterfactal logic; in or case Leis VC. With or frameork e defend Leisian semantics by appealing to examples from preference logic, here ceteris paribs reasoning is more idely discssed. Finally, e otline some limitations of or frameork and directions for ftre research. Iterated ceteris paribs actions. We sa in Section 5 that iterated ceteris paribs conterfactals deviate in trth-vale from the corresponding pdate-then-conterfactal seqence. Sch difficlties ith iterated conterfactals are not so ncommon. We leave the task of nderstanding the fll interaction beteen the to for frther investigation. Cardinality restrictions on Γ. In general, ceteris paribs reasoning reqires keeping eqal as mch information as possible, and sometimes nknon information (for example, nanticipated defeaters of las). Keeping everything else eqal may indeed mean keeping eqal an indefinite, and possibly infinite, set of things. Exploring ceteris paribs logic ithot cardinality restrictions to Γ is ths more than a mere technical exercise. Bt it is not so straightforard to extend the present frameork to accommodate the presence of infinite Γ. The translations presented in the appendix only carry over to the infinite case for infinitary langages, hich is not mch of a soltion. For the strict ceteris paribs semantics, e instead sggest folloing the δ-flexibility approach of [22]. For the relaxed ceteris paribs semantics, there are conceptal difficlties hich arise ith the comparison of infinite sets: hen shold e say of to infinite sets that one keeps more things eqal than the other? Clearly naïve conting ill not sffice. Minimising distance ith respect to Γ is more promising, bt has its on problems. We leave this challenging technical enterprise for ftre research.
P. Girard and M. A. Triplett 189 11 Acknoledgments We ish to thank the participants at the Astralasian Association of Logic and the Analysis, Randomness and Applications meetings held in Ne Zealand in 2014. We also ish to thank Sam Baron, Andre Withy, and the anonymos referees for valable comments. References [1] Jonathan Bennett (1974): Conterfactals and possible orlds. The Canadian Jornal of Philosophy 4(2), pp. 381 402, doi:10.1080/00455091.1974.10716947. [2] Johan van Benthem, Patrick Girard & Olivier Roy (2009): Everything Else Being Eqal: A Modal Logic for Ceteris Paribs Preferences. Jornal of Philosophical Logic 38(1), pp. 83 125, doi:10.1007/ s10992-008-9085-3. [3] Patrick Blackbrn, Maarten de Rijke & Yde Venema (2001): Modal Logic. Cambridge University Press, Ne York, NY, USA, doi:10.1017/cbo9781107050884. [4] G. Lee Boie (1979): The similarity approach to conterfactals: Some problems. Noûs, pp. 477 498, doi:10.2307/2215340. [5] Lis Fariñas del Cerro & Andreas Herzig (1994): Interference logic = conditional logic + frame axiom. International Jornal of Intelligent Systems 9(1), pp. 119 130, doi:10.1002/int.4550090107. [6] Lis Fariñas del Cerro & Andreas Herzig (1996): Belief change and dependence. In: Proceedings of the 6th conference on Theoretical aspects of rationality and knoledge, Morgan Kafmann Pblishers Inc., pp. 147 161. [7] Mkesh Dalal (1988): Investigations into a theory of knoledge base revision: preliminary report. In: Proceedings of the Seventh National Conference on Artificial Intelligence, 2, pp. 475 479. [8] Hans van Ditmarsch, Wiebe van der Hoek & Barteld Pieter Kooi (2007): Dynamic epistemic logic. 337, Springer, doi:10.1007/978-1-4020-5839-4. [9] Jon Doyle & Michael P. Wellman (1994): Representing preferences as ceteris paribs comparatives. Ann Arbor 1001, pp. 48109 2110. [10] Kit Fine (1975): Revie of Leis conterfactals. Mind 84, pp. 451 458, doi:10.1093/mind/lxxxiv.1. 451. [11] Davide Grossi, Emiliano Lorini & Francois Scharzentrber (2013): Ceteris paribs strctre in logics of game forms. Proceedings of the 14th conference on theoretical aspects of rationality and knoledge. ACM. [12] Vincent F. Hendricks (2006): Mainstream and formal epistemology. Cambridge University Press. [13] Wesley H. Holliday (2014): Epistemic closre and epistemic logic I: Relevant alternatives and sbjnctivism. Jornal of Philosophical Logic, pp. 1 62, doi:10.1007/s10992-013-9306-2. [14] Hirofmi Katsno & Alberto O. Mendelzon (1991): Propositional knoledge base revision and minimal change. Artificial Intelligence 52(3), pp. 263 294, doi:10.1016/0004-3702(91)90069-v. [15] David Leis (1973): Conterfactals. Harvard University Press. [16] David Leis (1979): Conterfactal dependence and time s arro. Noûs, pp. 455 476, doi:10.1093/ 0195036468.003.0002. [17] Graham Priest (2008): An introdction to non-classical logic: From if to is. Cambridge University Press, doi:10.1017/cbo9780511801174. [18] Carlo Proietti & Gabriel Sand (2010): Fitch s paradox and ceteris paribs modalities. Synthese 173(1), pp. 75 87, doi:10.1007/s11229-009-9677-7.
190 Ceteris paribs logic in conterfactal reasoning [19] Dov Samet (1996): Hypothetical knoledge and games ith perfect information. Games and economic behavior 17(2), pp. 230 251, doi:10.1006/game.1996.0104. [20] Leonard J. Savage (1972): The fondations of statistics. Corier Corporation. [21] Gerhard Schrz (2002): Ceteris Paribs Las: Classification and Deconstrction. Erkenntnis (1975-) 57(3), pp. pp. 351 372, doi:10.1023/a:1021582327947. [22] Jeremy Seligman & Patrick Girard (2011): Flexibility in Ceteris Paribs Reasoning. The Astralasian Jornal of Logic 10(0). Available at http://ojs.victoria.ac.nz/ajl/article/vie/1826. [23] Bassel Tarbsh (2013): Agreeing on decisions: an analysis ith conterfactals. Proceedings of the 14th conference on theoretical aspects of rationality and knoledge. ACM. [24] Georg H. von Wright (1963): The Logic of Preference. Edinbrgh University Press. [25] Zojn Xiong & Jeremy Seligman (2011): Open and closed qestions in decision-making. Electronic Notes in Theoretical Compter Science 278, pp. 261 274, doi:10.1016/j.entcs.2011.10.020. A Appendix We first recast Definition 3 in a more formally precise manner. Definition 8 For each ordinal α let L α be given by ϕ ::= p ϕ ϕ ψ [ϕ,γ]ψ here Γ L β is finite and β < α. L CP is then defined to be α L α. This ensres the sets Γ are ell-defined. One can define a langage L of comparative possibility in a similar style, thogh e ill only give the folloing grammar We frther set ϕ ::= p ϕ ϕ ψ ϕ ψ ϕ Γ ψ ϕ Γ ψ ϕ Γ ψ. ϕ ψ := (ψ ϕ); ϕ Γ ψ := (ψ Γ ϕ); ϕ Γ ψ := (ψ Γ ϕ); ϕ Γ ψ := (ψ Γ ϕ); ϕ := ϕ ; ϕ := ϕ. Definition 9 (Semantics) Let M, be a conditional model. Then p M = V(p); M = /0; ϕ M = W \ ϕ M ; ϕ ψ M = ϕ M ψ M ; ϕ ψ M = { W : W v W sch that if ψ M then v ϕ M and v }; ϕ Γ ψ M = { W : W v W sch that if ψ M then v ϕ M and v Γ }; ϕ Γ ψ M = { W : [] Γ v [] Γ sch that if ψ M then v ϕ M and v Γ }; ϕ Γ ψ M = { W : W v W sch that if ψ M then v ϕ M and v Γ }. Lemma 1 The modal operator [ϕ, Γ]ψ nder NC semantics is definable in L.
P. Girard and M. A. Triplett 191 PROOF: We sho that M, = NC [ϕ,γ]ψ iff M, = ϕ (ϕ ψ) Γ (ϕ ψ). : Assme M, = ϕ. Then there is a orld x W sch that M,x = ϕ. So, by assmption, Min Γ ( ϕ M ) /0 and Min Γ ( ϕ M ) ψ M. Hence, there exists y W sch that M,y = ϕ ψ and for every orld z W, if z Γ y then z ϕ ψ M. This is exactlym, =(ϕ ψ) Γ (ϕ ψ). : By contrapositive. Assme M, = [ϕ,γ]ψ. Then, by the semantic definition, there is an x Min Γ ( ϕ M ) sch that x ψ M. So M, = ϕ, and for every x W, there exists y W (namely v) sch that if x ϕ ψ M, then y Γ x and x ϕ ψ M. Hence, M, =(ϕ ψ) Γ (ϕ ψ), so M, =(ϕ ψ) Γ (ϕ ψ), and e are done. Lemma 2 The modal operator [ϕ, Γ]ψ nder CP semantics is definable in L. PROOF: Replace Γ ith Γ in the above proof to sho that the folloing eqivalence M, = CP [ϕ,γ]ψ iff M, = ϕ (ϕ ψ) Γ (ϕ ψ). holds. Lemma 3 The modal operator [ϕ, Γ]ψ nder MS semantics is definable in L. PROOF: Replace Γ ith Γ in the above proof to sho that the folloing eqivalence holds M, = MS [ϕ,γ]ψ iff M, = ϕ (ϕ ψ) Γ (ϕ ψ). Denote by L the L -fragment given by ϕ ::= p ϕ ϕ ψ ϕ ψ. Given a set Γ L or Γ L, let Γ be the set of all possible conjnctions of formlas and negated formlas from Γ; that is, the set of all ψ sch that ψ = ±γ, here +γ = γ and γ = γ. So if Γ={p, q} then γ Γ Γ ={p q, p q, p q, p q}. We ill often identity a conjnction ϕ 1 ϕ n ith the set {ϕ 1,...,ϕ n }. Lemma 4 The modal operator Γ of L is definable in L. PROOF: We sho that ϕ Γ ψ γ Γ [γ (ϕ γ) (ψ γ)]. (5) : Withot loss of generality rite M, = γ. Let W and sppose M, = ψ γ. By hypothesis there exists v [] Γ sch that M,v = ϕ and v Γ. No v Γ, so M,v = γ, and v as reqired. : Write M, = γ. Then M, =(ϕ γ) (ψ γ). Let [] Γ and sppose that M, = ψ. Then M, = ψ γ, so there exists v W ithm,v = ϕ γ and v. Then v Γ, and so v Γ.
192 Ceteris paribs logic in conterfactal reasoning Lemma 5 The modal operator Γ of L is expressible in L. PROOF: We sho that ϕ Γ ψ (γ [ γ Γ λ γ λ λ γ ]) (ϕ λ ) (ϕ λ) (ψ λ) (6) : Sppose M, = ϕ Γ ψ ithm, = γ, for some γ Γ. Take λ γ sch that M, = (ϕ λ ). (7) λ λ γ Take v W arbitrary sch that M,v = ψ λ. By the hypothesis there is W sch that Γ v and M, = ϕ. No, Γ v implies that ( ) either A M Γ (v,) AM Γ (,), or AM Γ (v,)= AM Γ (,) and v. If A M Γ (v,) AM Γ (,), then λ AM Γ (v,) implies that λ AM Γ (,). Frthermore, M, = ϕ A M Γ (,). Take λ := A M Γ (,), then M, = ϕ λ, and hence M, = (ϕ λ ), contradicting (7). λ λ γ Ths by ( ), A M Γ (v,)=am Γ (,) and v. Finally, since λ A M Γ (v,) andm, = ϕ AM Γ (,), e have that M, = ϕ λ, as desired. : Assme the right-hand side of (6). There is a niqe γ Γ for hich M, = γ. Take W sch that M, = ψ, and consider A M Γ (,). Case 1 of Γ. There is an x W sch that A M Γ (,) AM Γ (x,) and M,x = ϕ. Then x Γ, by definition Case 2 There is no x W sch that A M Γ (,) AM Γ (x,) and M,x = ϕ. No, if there is y W and a set of formlas λ ith A M Γ (,) λ γ sch that M,y = ϕ λ, then A M Γ (,) λ A M Γ (y,) and M, y = ϕ, contradicting or assmption. Hence M, = (ϕ λ ), A M Γ (,) λ γ and by taking λ := A M Γ (,), or initial assmption implies that M, =(ϕ A M Γ (,)) (ψ A M Γ (,)). Since M, = ψ A M Γ (,), there is an x sch that M,x = ϕ A M Γ (,). Hence AM Γ (,) A M Γ (x,), and also AM Γ (x,) AM Γ (,) as the containment cannot be proper by the case assmption. So A M Γ (,)=AM Γ (x,), and since x, one has that x Γ. Hence, both cases imply that there exists an x Γ sch that M,x = ϕ, as desired.
P. Girard and M. A. Triplett 193 Lemma 6 The modal operator Γ of L is definable in L. PROOF: Replace the sbset condition λ λ γ in (6) ith the cardinality condition λ < λ γ and repeat the above process. Notice that, if Γ {ϕ,ψ} L, then the right hand sides of the eqivalences established above are in L. This allos s to apply the translation to a formla from the inside-ot, the reslting formla belonging to L. By a conditional frame e mean a pair F =(W, ), sch that (F,V) is a conditional model for any valation fnction V. Let C be the class of conditional frames. Using the notation from [3], e rite Λ L C for the set of L-formlas valid over C. Theorem 1 The logic Λ L C is complete. PROOF: We take as or axiomatisation the axioms for VC [15], pls the translations from Lemmas 4, 5, and 6. Corollary 1 The logic Λ L CP C is complete for CP/NC/MS-semantics.