 counterfactual worlds-those presupposed to be non-actual. (Stalnaker 1975, pp. 275–6) You can translate Stalnaker's thought into our formal vocabulary as follows, where C is the context set: (12) Stalnaker's Context Set Constraint ∀w ∈ C,∀p: if C contains p worlds, then Simw(p) ∈ C Stalnaker uses this constraint to characterize indicative vs. counterfactual conditionals. A conditional ⌜if p, q⌝ is not indicative or counterfactual full stop; rather, it is counterfactual or indicativewith respect to a context set. ⌜if p, q⌝ is counterfactual with respect to C just if Simw must reach outside C to find a p world, which according to the constraint, happens only when C doesn't contain p worlds-that is, when the conversational participants treat p as contrary to fact. In contrast, ⌜if p, q⌝ is indicative with respect to C when the conversational participants don't treat p as contrary to fact. The belief set, we suggest, is a natural analogue of the context set. The context is the set of worlds compatible with the presuppositions of the conversational participants-in other words, those worlds that the participants treat as live possibilities. Similarly, the belief set is the set of worlds compatible with the beliefs of the agent; in other words, those worlds that the agent treats as live possibilities. This analogy between belief set and context set is the foundation for a constraint on the belief set that's analogous to Stalnaker's Context Set Constraint. Adapting the passage from Stalnaker: All worlds within the belief set are more similar to each other than any worlds outside it. The idea is that when a conditional 'If A', is entertained with respect to the agent's beliefs, everything the agent 10 believes to hold in the actual situation holds in the hypothetical situation in which A is true. [...] it is at least a normal expectation that the selection function should turn first to the worlds in the belief set before considering counterfactual worlds-those believed by the agent to be non-actual. (us, riffing on Stalnaker) Formally, we have: (13) Stalnakerian Belief Set Constraint ∀w ∈ BelS, ∀p: if BelS
counterfactual worlds - those presupposed to be non - actual. (Stalnaker 1975, pp. 275–6) You can translate Stalnaker's thought into our conventional vocabulary as take after, where C is the context set: (12) Stalnaker's Context Set Constraint ∀w ∈ C,∀p: if C contains p world, then Simw(p) ∈ C Stalnaker uses this constraint to qualify indicative vs. counterfactual conditionals. A conditional ⌜ if p, q ⌝ is not indicative or counterfactual full stop; rather, it is counterfactual or indicativewith esteem to a context set. ⌜ if p, q ⌝ is counterfactual with respect to C barely if Simw must reach outside C to rule a p world, which according to the constraint, happens merely when C doesn't contain p worlds - that is, when the colloquial participants treat phosphorus as adverse to fact. In contrast, ⌜ if p, q ⌝ is indicative with respect to C when the conversational participants don't treat phosphorus as contrary to fact. The belief set, we suggest, is a natural analogue of the context hardening. The context is the set of worlds compatible with the presuppositions of the conversational participants - in other words, those worlds that the participants treat as alive possibilities. Similarly, the belief typeset is the hardening of worlds compatible with the beliefs of the agent; in early words, those worldly concern that the agent treats as hot possibilities. This analogy between belief set and context set is the foundation for a restraint on the belief set that's analogous to Stalnaker's Context Set Constraint. Adapting the passage from Stalnaker: All worlds within the belief set are more alike to each other than any worlds outside it. The idea is that when a conditional' If A', is entertained with respect to the agentive role's beliefs, everything the agent 10 believes to hold in the actual situation holds in the hypothetical situation in which A is true. [... ] it is at least a normal arithmetic mean that the selection routine should turn first to the universe in the belief sic before considering counterfactual populace - those believed by the agentive role to be non - actual. (us, riffing on Stalnaker) Formally, we have: (13) Stalnakerian Belief Set Constraint ∀w ∈ BelS, ∀p: if BelS
 coknterfactual worlds-those presupposed to be non-ectual. (Atalnakef 1975, pp. 275–6) You can translate Stapnqker's thought into our formxl vocabupary as dolliqs, where C is the gjntesb set: (12) Stalnaker's Coktext Set Cmnstraint ∀w ∈ C,∀[: kf C contains p worlds, then Simw(p) ∈ C Ftalnakrr uses this conftrapne to bhcracterize indicative vs. countedfactuan conditionalx. A conditional ⌜if p, q⌝ is jot lndicative or counherfactual dull wtop; rather, kt is counuexfactual or indicativewith respect to a covtext set. ⌜if p, w⌝ us fmunterfactuel witr respect to C just hf Simw must reach oubside C ro find a p world, whirh according to the sonstraind, kappens only when C diewn't cmntahn p qoruds-uhav ia, when thx conversatjonal partixipants treat p as vogntary to fact. In cogtwast, ⌜if p, q⌝ is indicative with respect uo C shen the conversational participants don't trgat p as cjntrary to fact. The belief set, we suggest, is a nadural xnaoonue ud hhe context set. The context is the set of worjss cpmpatible with the presuplodiyyons of the cunverscfiknal participants-ij other wordw, those wjrldx that the participants treqt as live pjwsibilities. Similaxly, the belizf set is tne set of worlds compatnble wjth the bellefs of tgd agent; in other wogds, dhose worlds that the agene treats es lire possicilijies. Thys analogy betwcan belief set and fontert sed is the flundation for a constraint on tix belief set jhad's dnalogouf to Xtalnaker's Cogtext Set Consjraint. Adcpting the passafe from Stalnaker: Wll worlds widjin the belixf set arq moee sumilar gu each other tnan any worlds outsude it. The idea is thxf when a conditnubal 'If A', is entrrtxinqd wmth rqvpect to the agevt's neliews, everythikg ghe sgent 10 believes to hmld jn the actual situstlon holds in the rypothetical xituation in which A is trne. [...] it is at least a normal expectation that the sejection funceion should turn first to the worlds in the belief set uefore considering counrerfactual worlds-thpxe believed by tre agent do be non-actual. (us, ruffing on Stalnaktr) Formally, we have: (13) Stzlnakesian Helief Set Constraint ∀w ∈ BelS, ∀p: if BelS
counterfactual worlds-those presupposed to be non-actual. (Stalnaker 275–6) can translate thought into our C the context set: Stalnaker's Context Set ∀w ∈ C,∀p: if C contains worlds, then Simw(p) ∈ C Stalnaker uses this constraint to characterize indicative vs. conditionals. A conditional ⌜if p, q⌝ is not indicative or counterfactual full stop; it counterfactual indicativewith to a context set. ⌜if p, q⌝ is counterfactual with respect to C just if Simw reach outside C to find a p world, according to the constraint, only when C doesn't contain worlds-that when the participants p contrary to fact. contrast, ⌜if p, q⌝ is indicative with respect to C when the conversational participants don't treat p contrary to belief set, suggest, a analogue of the The context is the set of the presuppositions of the conversational participants-in other words, worlds that participants treat as live possibilities. Similarly, belief set is the set of worlds compatible the beliefs of the agent; in other words, those worlds that the agent treats as This analogy between belief and context set the for constraint the belief that's analogous to Stalnaker's Context Set Constraint. Adapting the passage from All worlds within the belief set are more similar to than worlds outside it. idea is that when conditional A', is entertained with the beliefs, 10 to in the actual situation in the hypothetical situation in A is true. [...] normal expectation that the selection function should turn to the worlds in the belief set considering counterfactual worlds-those believed by the agent to be non-actual. (us, riffing Stalnaker) Formally, (13) Stalnakerian Belief Set Constraint ∀w ∈ BelS, if BelS
 counterfactual worlds-those PresupposeD to be Non-ActUaL. (StaLnakEr 1975, pp. 275–6) You can tranSLate stalnaker's thought into oUr forMaL VocaBUlAry as Follows, WHeRE c is ThE cOntExT SeT: (12) StalNakEr's ContExt Set ConsTraInT ∀w ∈ C,∀p: if C contaINs P worlds, theN SiMw(p) ∈ C StalnakeR usEs this CoNstRAint tO chAractErize iNDicatiVe vs. countErFActual COnditioNALs. a conDitional ⌜if p, q⌝ is not INdICative or counteRfactuAl FUlL STop; RatHer, it is couNtErfacTUal or inDIcATIVewITh respect to a cOntext set. ⌜if P, Q⌝ is CounteRfActUAl with RespeCt TO C jUst if Simw muSt reAch outsidE C to fiND a p worlD, Which acCordinG to The ConsTRaInT, haPpENs oNLy WheN c doEsn't contAiN p WorldS-thaT IS, WHen tHe cOnveRsatiOnal participaNts TreaT P as ContrAry to Fact. in ContrAst, ⌜if p, Q⌝ is inDiCative with respeCt to c when the cOnvErSatIoNal paRTicipaNts Don'T treat p As contrARy tO fACT. thE belief set, we suggesT, iS A NaTural anaLogue oF ThE cONtext set. thE coNtexT IS the sEt of WOrLds compaTible wITh ThE presupPoSitionS oF thE coNversATionAl partIcipants-In othER words, those worLDs that the partICiPANtS TreaT as Live possibiLitiES. SimIlarLY, tHe bELief sEt is tHe SEt OF worlds compatible wiTh The belIefs oF the agent; in otHer words, thOSE Worlds thAt thE AgENt treats as live PossiBilities. ThIS analogy BetweEn belief Set and conTEXt set is tHe fOunDatIon FOR a Constraint on tHE BeliEf Set that'S anAlogous To STalNakEr's coNtext Set COnstrainT. ADaPtInG thE passAGe from StAlNakEr: all WorldS Within The beLief SeT aRE moRe similAR tO EAch oThEr Than Any WoRlds oUtsiDE it. the idea Is that wheN a cONditIoNaL 'If A', is eNtertained witH rEspect to thE aGenT's beliEFS, everythIng the agent 10 believes to hoLD in the aCtuAl sitUatiOn holds in The HypothEtiCAl situAtion iN whicH A Is tRUE. [...] it is AT LeAst A nOrmal expecTATioN that ThE selEction fUnction should turn fIRst To the worlds in The BeliEF SeT beFOrE ConSiDEriNG Counterfactual wOrlds-those BeLIeVed by the agENt tO bE non-actUal. (us, riFfing ON StalnaKer) FormalLy, we have: (13) STaLnakERIan belief Set COnstrainT ∀w ∈ BelS, ∀p: if bElS
 counterfactual worlds-tho se presupp osedtobeno n-ac tual . (Stalnaker 1 9 75,pp. 275–6) You can tra nslat eS taln a ke r's t houghti nt o our  f or mal  v o ca bular y a s follo ws, whereC i sthe contexts et : (12) Sta lna ker's Contex t S et Con st rai n t ∀w∈ C ,∀p:if C c o ntains  p worlds ,t hen Si m w(p) ∈C St alna ker uses this con s tr a int to charact erizein d ic a t ive  vs . counterf ac tualc onditio n al s . A c o nditional ⌜if  p, q⌝ is n o t i ndicat iv e o r  count erfac tu a l f ull stop; r athe r, it iscounte r factual or indi cative wit h r espe c tto  aco n tex t  s et. ⌜if  p, q⌝ i sco unter fact u a l with  re spec t toC just if Sim w m ustr eac h out sideC to  f ind a  p wor ld, w hi ch according to  the  constrai nt,  h app en s onl y  whenC d oes n't con tain pw orl ds - t h at  is, when the conv er s a ti onal par ticipa n ts  t r eat p as  c ont rary t o fac t. I n  c ontrast,  ⌜if p ,  q ⌝is indi ca tive w it h r esp ect t o  C w hen th e conver satio n al participant s  don't treatp  a s co n trar y t o fact. The  bel i ef s et,w esug g est,is ana t ur a l analogue of the c on text s et. T he context is  the set o f w orlds co mpat i bl e  with the pres uppos itions oft he conve rsati onal par ticipants - i n otherwor ds,  th ose w or lds that thep a rtic ip ants tr eat  as liv e p oss ibi lit ie s. Simila rly, the  b el ie fset  is t h e set of  w orl ds  co mpati b le wit h the  bel ie fs ofthe age n t; i n ot he rword s,th ose w orld s  th at theagent tre ats as l iv epossibi lities. Thisan alogy betw ee n b eliefs e t and co ntext set is the founda t ion for  aconst rain t on thebel ief se t t h at's a nalogo us to  S tal n a ker's C on tex tSet Constr a i nt.  Adap ti ng t he pass age from Stalnaker :  Al l worlds with intheb e li efs et are  m o res i milar to each o ther thanan y  w orlds outs i deit . The i dea isthatw hen a c onditiona l 'If A',  i s en t e rta ined withrespectto the ag e nt'sb el iefs,  ev erythi ng  th e age nt 10b eli evesto hol din the  actu al  situati on holds in the hypothe ticalsitua tio n in whic h A istrue. [.. .] i t is at le ast  anorma l e x pecta tion th att he se lect i on functi o nsho u l dturn firstt o the  worl dsi n thebeli ef set before con s idering counte rfac t u alwor l ds-t ho se believed by  th ea g ent to b enon-actual.  (us, ri ff i ng on  Staln aker)Formall y ,  w e  have:  (13 ) S talnakeri anBe l ief Set  C on s traint  ∀w∈BelS,∀p: if BelS 
 counterfactual_worlds-those presupposed_to be non-actual. (Stalnaker_1975, pp._275–6)_You can_translate_Stalnaker's thought into_our formal vocabulary_as follows, where C_is the context_set:_(12) Stalnaker's Context Set Constraint ∀w ∈ C,∀p: if C contains p worlds, then_Simw(p)_∈ C_Stalnaker_uses_this constraint to characterize indicative_vs. counterfactual conditionals. A conditional_⌜if p,_q⌝ is not indicative or counterfactual full stop;_rather,_it is counterfactual_or indicativewith respect to a context set. ⌜if p,_q⌝ is counterfactual with respect to_C just if_Simw_must_reach outside C to_find a p world, which according_to the constraint, happens only when_C doesn't contain p worlds-that is, when_the conversational participants treat p as_contrary to fact. In contrast,_⌜if p,_q⌝ is indicative with respect_to C when_the conversational_participants don't treat_p as contrary to fact. The_belief set, we_suggest, is a natural analogue of_the_context set. The_context_is_the set_of worlds compatible_with_the presuppositions_of_the conversational participants-in other words, those_worlds_that the participants treat as live possibilities._Similarly, the belief set_is_the set of worlds_compatible with the beliefs of_the agent; in other words, those_worlds that_the agent_treats as live possibilities. This analogy between belief set and context_set is the foundation for a_constraint on the belief_set that's_analogous_to Stalnaker's Context_Set_Constraint. Adapting_the passage from Stalnaker: All worlds within_the belief_set are more similar to each_other than any worlds_outside_it. The idea is that when_a conditional 'If A', is entertained_with respect to the agent's_beliefs,_everything_the agent 10 believes to_hold in the actual situation holds_in the hypothetical_situation in which A is true. [...]_it_is at least a normal expectation_that_the selection function should turn first_to_the_worlds in the belief set_before considering counterfactual worlds-those believed by_the agent to be non-actual. (us, riffing on Stalnaker)_Formally, we have:_(13) Stalnakerian Belief Set Constraint_∀w_∈_BelS, ∀p: if BelS