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1、Merging stit and counterfactual logic1Introduction to stit logic2Temporal StructuresA temporal structure is a tuple hMom,i,where Mom,(elementsof which are called moments)and Mom Mom(the temporalordering),where m m0means that m0occurs after m.Eric Pacuit3Some properties of Itransitive:for all m,m0,m0

2、0,if m m0and m0 m00,then m m00.Iirreflexive:for all m,m0,m m0Ilinear-past:for all m,m0,m00if m0 m and m00 m,thenm0=m00or m0 m00or m00 m0Ilinear-future:for all m,m0,m00if m m0and m m00,thenm0=m00or m0 m00or m00 m0Idiscrete:for all m,m0,if m m0,then there is an m00such thatm m00 m0and there is no m000

3、such that(m m000andm000 m00)where m m0if,and only if,m m0or m=m0Eric Pacuit4Some properties of Itransitive:for all m,m0,m00,if m m0and m0 m00,then m m00.Iirreflexive:for all m,m0,m m0Ilinear-past:for all m,m0,m00if m0 m and m00 m,thenm0=m00or m0 m00or m00 m0Ilinear-future:for all m,m0,m00if m m0and

4、m m00,thenm0=m00or m0 m00or m00 m0Idiscrete:for all m,m0,if m m0,then there is an m00such thatm m00 m0and there is no m000such that(m m000andm000 m00)where m m0if,and only if,m m0or m=m0Eric Pacuit4Some properties of Itransitive:for all m,m0,m00,if m m0and m0 m00,then m m00.Iirreflexive:for all m,m0

5、,m m0Ilinear-past:for all m,m0,m00if m0 m and m00 m,thenm0=m00or m0 m00or m00 m0Ilinear-future:for all m,m0,m00if m m0and m m00,thenm0=m00or m0 m00or m00 m0Idiscrete:for all m,m0,if m m0,then there is an m00such thatm m00 m0and there is no m000such that(m m000andm000 m00)where m m0if,and only if,m m

6、0or m=m0Eric Pacuit4Some properties of Itransitive:for all m,m0,m00,if m m0and m0 m00,then m m00.Iirreflexive:for all m,m0,m m0Ilinear-past:for all m,m0,m00if m0 m and m00 m,thenm0=m00or m0 m00or m00 m0Ilinear-future:for all m,m0,m00if m m0and m m00,thenm0=m00or m0 m00or m00 m0Idiscrete:for all m,m0

7、,if m m0,then there is an m00such thatm m00 m0and there is no m000such that(m m000andm000 m00)where m m0if,and only if,m m0or m=m0Eric Pacuit4Some properties of Itransitive:for all m,m0,m00,if m m0and m0 m00,then m m00.Iirreflexive:for all m,m0,m m0Ilinear-past:for all m,m0,m00if m0 m and m00 m,then

8、m0=m00or m0 m00or m00 m0Ilinear-future:for all m,m0,m00if m m0and m m00,thenm0=m00or m0 m00or m00 m0Idiscrete:for all m,m0,if m m0,then there is an m00such thatm m00 m0and there is no m000such that(m m000andm000 m00)where m m0if,and only if,m m0or m=m0Eric Pacuit4HistoryA history h is a maximally li

9、nearly ordered set of moments.An index is a pair(m,h)where m h.We write m/h when(m,h)isan indexHm=h|m h is the set of histories going through m.Eric Pacuit5Historieshh1,ih1hh2,ih2hh3,ih3hh4,ih4hh5,ih5hh6,ih6Each history is a linearly ordered set of momentsEric Pacuit6Historieshh1,ih1hh2,ih2hh3,ih3hh

10、4,ih4hh5,ih5hh6,ih6Each history is a linearly ordered set of momentsEric Pacuit6Historieshh1,ih1mhh2,ih2mhh3,ih3mhh4,ih4hh5,ih5hh6,ih6A moment m partitions the set of historiesEric Pacuit6Historieshh1,ih1mhh2,ih2mhh3,ih3mhh4,ih4mhh5,ih5hh6,ih6h and h0are equivalent given m when the initial segments

11、up to mare the sameEric Pacuit6Historieshh1,ih1hh2,ih2mhh3,ih3hh4,ih4hh5,ih5hh6,ih6h and h0are equivalent given m when the initial segments up to mare the sameEric Pacuit6Historic Necessityhh1,ih1hh2,ih2mhh3,ih3hh4,ih4hh5,ih5hh6,ih6The historic necessity modality(?)quantifies over the histories that

12、are equivalent given a moment mEric Pacuit6Historic NecessityM,m/h|=?A iff M,m/h0|=A for all h0 HmThe logic of historic necessity is S5:Proppropositional tautologiesK?(A B)(?A?B)T?A A4?A?A5?A?ANecfrom A infer?AMPfrom A and A B infer BS5 is sound and strongly complete with respect to relationalstruct

13、ures in which the relation is an equivalence relationEric Pacuit6Historic NecessityM,m/h|=?A iff M,m/h0|=A for all h0 HmThe logic of historic necessity is S5:Proppropositional tautologiesK?(A B)(?A?B)T?A A4?A?A5?A?ANecfrom A infer?AMPfrom A and A B infer BS5 is sound and strongly complete with respe

14、ct to relationalstructures in which the relation is an equivalence relationEric Pacuit6Choiceshh1,ih1hh2,ih2mhh3,ih3hh4,ih4hh5,ih5hh6,ih6h and h0are equivalent given m when the initial segments up to mare the sameEric Pacuit7Choiceshh1,ih1hh2,ih2mhh3,ih3Choices at m is a partition that refines the h

15、istoric necessity partitionEric Pacuit7Choiceshh1,ih1hh2,ih2mhh3,ih3Choices at m is a partition that refines the historic necessity partitionEric Pacuit7Choiceshh1,ih1hh2,ih2mhh3,ih3The modality stit:quantifies over histories in a choice cell adsfasd f asdf asdf fafdsaEric Pacuit7stitM,m/h|=stit:A i

16、ff M,m/h0|=A for all h0 Choicem(h)The logic of stit is S5:Proppropositional tautologiesK stit:(A B)(stit:A stit:B)T stit:A A4 stit:A stit:stit:A5 stit:A stit:stit:ANecfrom A infer stit:AMPfrom A and A B infer BEric Pacuit7Deliberative stitM,m/h|=stit:A iff M,m/h0|=A for all h0 Choicem(h)M,m/h|=dstit

18、stit:A (dstit:A?A)Eric Pacuit7Historic Necessity and stitSince the choice partition refines the historic necessity partition,thefollowing is valid?A stit:A(For all relations R,R0,if R R0,then R0A RA is valid)Eric Pacuit8Historic Necessity and stitSince the choice partition refines the historic neces

19、sity partition,thefollowing is valid?A stit:A(For all relations R,R0,if R R0,then R0A RA is valid)Eric Pacuit8Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agentsEric Pacuit9Many AgentsAre there logical connections between the stit modality fo

20、r differentagents?Independence of agentsEric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agentsEric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agentsEric Pacuit9Many A

21、gentsAre there logical connections between the stit modality for differentagents?Independence of agentsEric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agentsABABABABAB stit:A stit:B(stit:A stit:B)Eric Pacuit9Many AgentsAre there logi

22、cal connections between the stit modality for differentagents?Independence of agentsABABABABAB stit:A stit:B(stit:A stit:B)Eric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agentsABABABABAB stit:A stit:B(stit:A stit:B)Eric Pacuit9Many

23、AgentsAre there logical connections between the stit modality for differentagents?Independence of agentsABABABABAB stit:A stit:B(stit:A stit:B)Eric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agents:V stit:A (VAgt stit:A)AAA stit:A st

24、it:B (stit:A stit:B)Eric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of agents:V stit:A (VAgt stit:A)AAABBB stit:A stit:BEric Pacuit9Many AgentsAre there logical connections between the stit modality for differentagents?Independence of a

25、gents:V stit:A (VAgt stit:A)AAABBB stit:A stit:B (stit:A stit:B)Eric Pacuit9Sound and Complete AxiomatizationIS5 for?IS5 for stit:I?A stit:AI(VAgt stit:A)(VAgt stit:A)IModus Ponens and Necessitation for?M.Xu.Axioms for deliberative STIT.Journal of Philosophical Logic,Volume 27(5),pp.505-552,1998.M.X

26、u.On the Basic Logic of STIT with a Single Agent.Journal of Symbolic Logic,60(2),pp.459-483,1995.Eric Pacuit10Alternative AxiomatizationP.Balbiani,A.Herzig and N.Troquard.Alternative axiomatics and complexity ofdeliberative STIT theories.Journal of Philosophical Logic,37:4,387-406,2008.H.Wansing.Tab

27、leaux for multi-agent deliberative-STIT logic.in Advances in ModalLogic,Volume 6,503-520,2006.Eric Pacuit11 stit:stit:A?AK1K2AAAAAAEric Pacuit12 stit:stit:A?AK1K2AAAAAAEric Pacuit12 stit:stit:A?AK1K2AAAAAAEric Pacuit12 stit:stit:A?AK1K2AAAAAAEric Pacuit12 stit:stit:A?AK1K2AAAAAAEric Pacuit12 stit:st

28、it:A?AK1K2AAAAAAEric Pacuit12 stit:stit:A?AK1K2AAAAAAEric Pacuit12Alternative AxiomatizationIS5 for?IS5 for stit:I?A stit:AIA h stit:VAgt,h stit:AiiIModus Ponens and Necessitation for?P.Balbiani,A.Herzig and N.Troquard.Alternative axiomatics and complexity ofdeliberative STIT theories.Journal of Phi

29、losophical Logic,37:4,387-406,2008.Eric Pacuit13Another AxiomatizationIS5 axioms for stit:I?A stit:stit:AIh stit:h stit:Aii h stit:VAh stit:Aii for all A AgtIModus Ponens and Necessitation for stit:P.Balbiani,A.Herzig and N.Troquard.Alternative axiomatics and complexity ofdeliberative STIT theories.

30、Journal of Philosophical Logic,37:4,387-406,2008.Eric Pacuit14“Flat”SemanticsM=hW,R,Vi,whereIW,IR is a function assigning to each Agt,R W WIV:At (W).M,w|=A iff for all v W,if wRv,then M,v|=.Eric Pacuit15Generalized PermutationR satisfies the general permutation property iff for all w,v W andfor all,

31、Agt,if(w,v)R Rthen there is a u W such that(w,u)Rand(u,v)Rfor all Agt wvxuEric Pacuit16Generalized PermutationR satisfies the general permutation property iff for all w,v W andfor all,Agt,if(w,v)R Rthen there is a u W such that(w,u)Rand(u,v)Rfor all Agt wvxuEric Pacuit16Generalized PermutationR sati

32、sfies the general permutation property iff for all w,v W andfor all,Agt,if(w,v)R Rthen there is a u W such that(w,u)Rand(u,v)Rfor all Agt wvxuEric Pacuit16“Flat”SemanticsIThe logic of 2-agent stit is the product of S5(denoted S5 S5,need to show that the Church-Rosser axiom hiA hiAis derivable)IThe s

33、atisfiability problem for stit with more than two agents isNEXPTIME-completeIThe satisfiability problem for stit with one agent is NP-completeIThe satisfiability problem for stit with more than two agents andgroup stit modalities is undecidable.A.Herzig and F.Schwarzentruber.Properties of logics of

34、individual and groupagency.in Advances in Modal Logic,pp.133-149,2008.F.Schwarzentruber.Complexity Results of STIT Fragments.Studia Logica,100(5),1001-1045,2012.Eric Pacuit17Other AxiomatizationsA.Herzig and E.Lorini.A Dynamic Logic of Agency I:STIT,Capabilities and Powers.Journal of Logic,Language

35、and Information 19(1),89-121,(2010.S.W olfl.Propositional Q-logic.Journal of Philosophical Logic,31,387-414,2002.R.Ciuni and A.Zanardo.Completeness of a Branching-Time Logic with PossibleChoices.Studia Logica,96(3),393-420,2010.Eric Pacuit18Conditional LogicD.Lewis.Counterfactuals.Harvard University

36、 Press,1973.W.Starr.Counterfactuals.Stanford Encyclopedia of Philosophyhttps:/plato.stanford.edu/entries/counterfactuals/.Eric Pacuit19Conditionals1.If its a square,then its rectangle.2.If A B,then A B=A.3.If you strike the match,it will light.4.If you had struck the match,it would have lit.Conditio

37、nals play a role in mathematical,practical and causalreasoning.Eric Pacuit20Material conditional:is true if either the antecedent()is false or the consequent()is true.Conditional:?Eric Pacuit21IIf I weighed more than 300 pounds,I would weigh more than 200pounds.IIf I weighed more than 300 pounds,I w

38、ould weigh less than 10pounds.Eric Pacuit22(MP),(MT),(DS)Eric Pacuit23(FA)(TC)(C)(Mon)()(Trans),Eric Pacuit24?Beijing is not in Maryland.?So,if Beijing is in Maryland,then themoon is made of cheese.?Eric was in Maryland this morning.?So,if Eric was in Beijing thismorning,then Eric was in Maryland th

39、is morning.Eric Pacuit25?Beijing is not in Maryland.?So,if Beijing is in Maryland,then themoon is made of cheese.?Eric was in Maryland this morning.?So,if Eric was in Beijing thismorning,then Eric was in Maryland this morning.Eric Pacuit25?()?If I put sugar in my coffee,then it will taste good.?So,i

40、f I put sugarand gasoline in my coffee,then it will taste good.If this match is struck,then it will light.?So,if this match is soakedovernight and struck,then it will light.Eric Pacuit26?,?If I quit my job,I wont be able to afford my apartment.But if I win 10million dollars,I will quit my job.?So,if

41、 I win 10 million dollars,Iwont be able to afford my apartment.Eric Pacuit27Similarity models:M=hW,R,V,wwWi,where hW,R,Vi is aKripke model and wis a centered total pre-order on worlds:Iwis transitiveIu wv or v wuIx ww implies x=wM,w|=(?)iff if M R(w),then there is av R(w)Msuch that there is no u suc

42、h that u wv andu M.Eric Pacuit28(K?)(?(1 2)(?1)(?2)(DefA)A (?)(Cond)(1 2?)(1?(2)(Suc)?(Ext)A(1 2)(1?)(2?)(Cen0)(?)(S)(1?2)(1?)(1 2?)(RN?)From infer?Eric Pacuit29I.Canavotto and EP.Counterfactuals in STIT with action types.manuscript,2020.Eric Pacuit30m1K1K2m2K3K4m3K5K6Ah1Ah2Ah3Ah4m2/h1|=i stit:AEric

43、 Pacuit31m1K1K2m2K3K4m3K5K6Ah1Ah2Ah3Ah4m2/h16|=?i stit:AEric Pacuit31m1K1K2m2K3K4m3K5K6Ah1Ah2Ah3Ah4m2/h1?|=Y i stit:A Eric Pacuit31m1K1K2m2K3K4m3K5K6Ah1Ah2Ah3Ah4m2/h16|=Yj stit:XAEric Pacuit31Example 1Imagine that David and Max have been playing a simple coin-tossingand betting game.Max flips the co

44、ins.Unknown to David,Max hastwo coins,one with heads on each side and one with tails on eachside.If David bets tails,Max flips the first coin;if he bets heads,Maxflips the second.Two minutes ago David bet that the coin would comeup heads on the next flip.Max flipped the coin and it came up tails.R.T

45、homason and A.Gupta.A Theory of Conditionals in the Context of BranchingTime.The Philosophical Review,Vol.89,No.1,pp.65-90,1980.Eric Pacuit32i1i2i3m1BHBTm2HTm3HTWm4Wm5Wm6Wm7h1h2h3h4Eric Pacuit33Imagine that David and Max have been playing a simple coin-tossingand betting game.Max flips the coins.Unk

46、nown to David,Max hastwo coins,one with heads on each side and one with tails on eachside.If David bets tails,Max flips the first coin;if he bets heads,Max flips the second.Two minutes ago David bet that the coinwould come up heads on the next flip.Max flipped the coin and itcame up tails.Eric Pacui

47、t33T=BTH=BHW=(H BH)(T BT)J.Halpern.Actual Causality.The MIT Press,2016.J.Halpern.From causal models to counterfactual structures.Review of SymbolicLogic 6(2),pp.305-322,2013.Eric Pacuit33i1i2i3m1BHBTm2HTm3HTWm4Wm5Wm6Wm7h1h2h3h4Eric Pacuit33DBT structure with instantshMom,m0,ti,where hMom,m0i is a di

48、screte tree-like temporalstructure with root m0.The instant relation t Mom Mom is anequivalence relation satisfying:T1.If m1,m2 succ(m),then m1tm2.T2.If m1 succ(m)and m1tm2,then there is m0s.t.m0tm andm2 succ(m0).T3.If m tm0and m1 succ(m0),then there is m2 succ(m)s.t.m2tm1.IFor each m Mom,succ(m)is

49、the(non-empty)set of immediatesuccessors and pred(m)the unique predecessor of m(if apredecessor exists).IIf h Hm,then let succh(m)be the unique successor of m on h.Eric Pacuit34DBT structure with instantshMom,m0,ti,where hMom,m0i is a discrete tree-like temporalstructure with root m0.The instant rel

50、ation t Mom Mom is anequivalence relation satisfying:T1.If m1,m2 succ(m),then m1tm2.T2.If m1 succ(m)and m1tm2,then there is m0s.t.m0tm andm2 succ(m0).T3.If m tm0and m1 succ(m0),then there is m2 succ(m)s.t.m2tm1.IFor each m Mom,succ(m)is the(non-empty)set of immediatesuccessors and pred(m)the unique

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