模态实在与模态非实在事件的逻辑.pdf

Studies in Logic,Vol.16,No.3(2023):108118PII:16743202(2023)03010811Logics for Modally Real andModally Nonreal EventsXian ZhaoTianqun PanAbstract.An event is modally real in one world if it occurs either in the world or in one ofits possible worlds accordingly,an event is modally nonreal in one world if it does not occurin the world or in any one of its possible worlds.We call a place where all modally nonrealevents of a world occur or exist as a modally black hole.This paper presents logical systems formodally real events and modally nonreal events,proves their soundness,and establishes theircompleteness.1IntroductionModal realists,extreme or moderate,admit the reality of numerous worlds.Forexample,D.Lewissaid,“Possibleworldsarewhattheyare,andnotsomeotherthing.If asked what sort of thing they are,I cannot give the kind of reply my questionerprobably expects:that is,a proposal to reduce possible worlds to something else.Ican only ask him to admit that he knows what sort of thing our actual world is,andthen explain that possible worlds are more things of that sort,differing not in kindbut only in what goes on at them.”(5,p.85)Because any possible world constitutesthings,admitting that possible worlds are just as real as our world means admittingthat things in any possible world are just as real as things in our world.Hence,a thingor an event is regarded as modal reality if it exists or occurs either in our world or inone of the possible worlds of our world,and a thing or an event is regarded as modalnonreality if it does not exist or occur either in our world or in any of the possibleworlds of our world.Thus,we have two notions:modal reality and modal nonreality.Because a modally nonreal thing does not exist inthe world or any of its possibleworlds,where does it inhabit?We suppose there is such a place where all modallynonreal events of the world inhabit,and we call the place a modally black hole.Whatwe focus on here is not questions related to the modally black hole,such as whetherthe modally black hole exists,but the logical structures of modally real events andmodally nonreal events.Received 20220927Revision Received 20221219Xian ZhaoDepartment of Philosophy,Hebei UTianqun PanDepartment of Philosophy,Nanjing UXian Zhao,Tianqun Pan/Logics for Modally Real and Modally Nonreal Events1092Proof Systems for Modally Real and Modally Nonreal EventsThe definitions of modal reality and modal nonreality are as follows:an eventis modally real in our world if it occurs either in our world or in one of its possibleworlds,and an event is modally nonreal in our world if it does not occur either in ourworld or in any of its possible worlds.We use p to represent an event,R for a modalreality operator,and B for a modal nonreality operator.Rp and Bp represent that“pis modally real”and“p is modally nonreal”respectively.The formal language L isdefined as follows:=p|()|R|BThe language in L is interpreted by the standard possible world semantics.Definition 1(Frames,Models,and Satisfaction).A Kripke frame F=W,R isa tuple where W is a set of possible worlds and R W W is an accessibilityrelation.A Kripke model M=(F,)is a tuple where F is a Kripke frame and:P 2wis an interpretation for a set of propositional variables P.A formula is true in model M in the world w ifM,w|=p iff w (p),M,w iff it is not the case that M,w ,M,w|=iff M,w|=and M,w ,M,w R iff M,w ,or for some wwith Rww,M,w,andM,w B iff M,w ,and for any wwith Rww,M,w.Semantically,the relations between the modal reality operator R or the modalnonreality operator B and the necessity operator or the possibility operator are asfollows:Rp (p 3p)and Bp (p 2p)The relation between R and B is as follows:Rp BpBecause the modally real operator R and the modally nonreal operator B are interdefinable(i.e.,Rp Bp),we use B as the primitive operator,and R can be definedby B.Definition 2.System B0comprises the following axioms and transformation rules:Ax0 all tautologies of propositional logic.Ax1 B0B Ax2 B0B()B BAx3 B0B B B()MP B0,B0 B0RE B0 B BRC B0 B0B110Studies in Logic,Vol.16,No.3(2023)Note that B0is the propositional calculus plus the axioms Ax1,Ax2,and Ax3and the transformation rules RE and RC.Theorem 1.B0is sound w.r.t.arbitrary frames.Proof.We only demonstrate that Ax1,Ax2,Ax3,RE and RC are valid with respectto arbitrary frames.Suppose that M is a model that is based on an arbitrary frame and w is a worldin M.For Ax1,suppose that M,w B .Consequently,M,w (B).According to Ax0,M,w B.Hence,(a)M,w B and(b)M,w .From(a),according to the definition of B in Definition 1,M,w ,whichcontradicts(b).For Ax2,suppose that M,w B()B B.Therefore,M,w B()BB.Hence,(a)M,w B()Band(b)M,w B.Hence,from(a),M,w ()and M,w ,and for any world wwith Rww,M,w ()and M,w.Then,M,w ,and for any world wwithRww,M,w.Hence,we have M,w B,which contradicts(b).ForAx3,supposethatM,w BB B().Then,M,w BBB().Hence,M,w BB and M,w B().Consequently,fromM,w B B,M,w and M,w ,and for any world wwith Rww,M,w and M,w.Hence,M,w (),and for any world wwithRww,M,w().Thus,we have M,w B(),which contradictsM,w B().For RE,suppose that .Consequently,M,w ,and for anywsuch that Rww,M,w .(a)Assume that M,w B.Accordingto the definition of B,M,w ,and for any wsuch that Rww,M,w.Hence,according to M,w and M,w ,we have M,w andM,w.Therefore,according to the definition of B,M,w|=B is obtained.(b)Assume that M,w B.The same reason as that in(a)ensures that M,w B.Thus,by(a)and(b),we have M,w B B.For RC,suppose that .Then,M,w ,and for any wsuch that Rww,M,w.Hence,by Definition 1,M,w ,and for any wsuch that Rww,M,w.Thus,by the definition of B,M,w B.To obtain a new and useful derived rule,suppose that B0 .Consequently,by RC,B0B().Because()is equivalent to ,we,by RE and MP,obtain B0B().Applying Ax2 to B0B()and usingMP,we obtain B0 B B.Thus,we follow the derived rule:RC1 B0 ,B0B B.Theorem 2.The following formulae are provable in system B0:Xian Zhao,Tianqun Pan/Logics for Modally Real and Modally Nonreal Events1111.B0B()B()2.B0B B3.B0B()B B4.B0B B()To extend B0,a weak relation Rwover possible worlds must be defined by therelation R and the identical relation R0.Rwwwis defined as Rwwor R0ww.Formally,Rwwwdef.Rww R0ww.Definition 3(Weak Frames).1.A frame W,R is weakly transitive if for any w,w,w W,if RwwwandRwww,then Rwww.2.A frame W,R is semiweakly Euclidean if for any w,w,w W,if Rwwwand Rww,then Rwww.3.A frame W,R is weakly Euclidean if for any w,w,w W,if RwwwandRWww,then Rwww.4.A frame W,R is weakly dead if for any w,w W,if Rwww,then R0ww.Four notes:(a)Aframethatistransitive(semiweaklyEuclideanandweaklyEuclidean)mustbe weakly transitive(semiweakly Euclidean and weakly Euclidean),and not viceversa.(b)A semiweakly Euclidean frame must be weakly Euclidean,and not viceversa.(c)In a weakly Euclidean frame for any w,w W,if Rww,we,by R0ww,have Rww.It means that a weakly Euclidean frame must be symmetric,and not viceversa.(d)A weakly symmetric frame is identical to a symmetric frame.We do not present weakly reflexive frames in Definition 3.In fact,if we definea weakly reflexive frame in which Rwww holds for any w W,such a frame isarbitrary,and vice versa.This means that a frame is weakly reflexive if and only ifit is arbitrary.We can say that B0is sound with respect to weakly reflexive frames.This indicates that our language L is weaker and a model based on reflexive framesis indistinguishable.Theorem 3.1.The formula B BB is valid w.r.t.weakly transitive frames.2.BB is valid w.r.t.symmetric frames.3.B BB is valid w.r.t.semiweakly Euclidean frames.4.B BB is valid w.r.t.weakly Euclidean frames.112Studies in Logic,Vol.16,No.3(2023)Proof.For 1.Let M be an arbitrary model that is based on a weakly transitiveframe and w be a world in M.Suppose that M,w B BB.Consequently,(a)M,w B and(b)M,w BB.From(b),together with(a),we have forsome wwith Rww,M,w B.Hence,M,w B and M,w.Therefore,for some wwith Rww,M,w.Because R is weakly transitive,Rwwor R0ww.However,from(a),we have M,w ,and for any wwithRww,M,w.This means that we have M,w and M,w .Acontradiction arises.For2.LetM beanarbitrarymodelthatisbasedonasymmetricframeandw beaworldinM,andsupposethatM,w .Then,M,w B,andM,w Bforanyworld win M that“sees”w.Because M is symmetrical,the fact that M,w Band M,w B causes BB to be false in w.Hence,we have M,w BB.For 3.Let M be an arbitrary model that is based on a semiweakly Euclideanframe and w be a world in M,and suppose that M,w B.Consequently,M,w ,and M,w B from which either M,w or there must be aworld win M such that Rwwand M,w.Because M,w ,we haveM,w.To demonstrate that M,w BB,because we know M,w B,wemust demonstrate that for any world win M such that Rww,M,w B.It isthe case because for any win M such that Rww,we have Rwwor R0ww,andwis a world in which is true.For 4.Let M be an arbitrary model that is based on a weakly Euclidean frameand w be a world in M,and suppose that M,w B.There exist two cases:(a)M,w and(b)for some wwith Rww,M,w.Case(a).Because a Euclidean frame is symmetric,for any wwith Rww,wehave that Rww,and M,w B.Thus,M,w BB.Case(b).BecauseM isweaklyEuclidean,foranywsuchthatRwww,Rwww.By M,w,M,w B.Therefore,we have M,w BB.Definition 4.We have the extensions of B0:1.B1=B0 B BB.2.B2=B0 BB.3.B3=B0 B BB.4.B4=B0 B BB.By Theorem 2 and 3,we obtain the following:Theorem 4(Soundness).1.B1is sound w.r.t.weakly transitive frames.2.B2is sound w.r.t.symmetric frames.3.B3is sound w.r.t.semiweakly Euclidean frames.4.B4is sound w.r.t.weakly Euclidean frames.Xian Zhao,Tianqun Pan/Logics for Modally Real and Modally Nonreal Events113Note that because BB and B BB are provable in B4,B4is stronger than both B2and B3.There exists a trivial axiom B to the systems.If B is added toB0orothersystems,theresultingsystemwillcollapseintothepropositionalcalculus.This can be demonstrated as follows.Because of Ax1 in Definition 1,we have B,by which the operator B in all formulae will be replaced.If a weakly deadend frame is defined as for any w W,if Rww,then R0ww the trivial axiom B is valid in weakly dead end frames.1By R and B,R .Philosophically,if our world is the onlyworld,R means that the modally real thing must be actual,and not vice versa.However,if is actual,its negation is modally nonreal.3CompletenessThe modal logics we present in the previous section are nonstandard.Following logicians who have dealt with other nonstandard modal logics such as logics ofcontingency and noncontingency(2,3,4,9)and the ones of essence and accident(6,7),we establish ad hoc canonical models for logics of modal reality and modalnonreality.For the purpose of presenting a general result,we use S to stand for B0or oneof its extensions.AsetofwellformedformulaeismaximallyconsistentwithrespecttoasystemS,if and only if for every formula,either or ,and there is no finitecollection 1,2,.,n such that S(1 2 n).We simply call a maximally Sconsistent set.To establish the completeness of the systems,we construct the successor of amaximally Sconsistent set.Definition 5.Let be a maximally Sconsistent set.The successor of,D(),isdefined as D()=|for every,B .Lemma 1.For a maximally Sconsistent set of formulae,the successor D()isclosed under conjunction.Proof.Let be a maximally Sconsistent set and D()be the successor of.Suppose that D()and D().According to the construction ofD(),B and B .By Ax3,B B B(),andby the maximality of,B().By RE,B()B().Hence,B().Thus,()D(),which is required.1LetM beanarbitrarymodelbasedonaweaklydeadendframeandw beaworldinM.AssumethatM,w .Becausew isonlyapossiblyaccessibleworldofw andM,w ,weobtainM,w B.114Studies in Logic,Vol.16,No.3(2023)Lemma 2.Let be a maximally Sconsistent set of formulae.Assume that for some,B .Then,D()is Sconsistent.Proof.Suppose that B but D()is not Sconsistent.Consequently,there must be 1,2,.,n D()such thatS(1 2 n).(a)By Ax0,S (1 2 n).(b)By RC1,from(b),SB(1 2 n)B.(c)According to 1,2,n D()and Lemma 1,1 2 n D()(d)By the construction of D(),B(1 2.n).(e)B (f)(f)means B/,which contradicts the supposition of B .ToprovethecompletenessofB0anditsextensions,wemustconstructthespecialcanonical model for them.Definition 6(Canonical Model).The canonical model MC=WC,RC,C forthe logic S is defined as follows:1.WCis the set of all maximally Sconsistent sets of formulae2.RC WCXWCis defined by RC1or R01iff D()1 and3.p C()iff p .The canonical model for S is ad hoc.However,according to Definition 6,wehave D()1by RC1 we cannot specify RC1by D()1,which isdifferent from normal modal logics.Lemma 3(Truth Lemma).Let MC=(WC,RC,C)be the canonical model for S.For all formulae and all maximally Sconsistent sets s,MC,S .Proof.The theorem can be proven by induction on the structure of wff.Here,weprove the case of B only:MC,SB B .We assume that the theorem holds for and for all:MC,S .Suppose that B .By the definition of D(),D().According toDefinition 6,we have that if RC1or R01,1.This means that if RC1,1and .Hence,according to the assumption in the beginning,we haveXian Zhao,Tianqun Pan/Logics for Modally Real and Modally Nonreal Events115(a)if RC1,MC,1 and(b)MC,S.According to the definition ofthe truth value of B,(a)and(b)yield MC,|=B.Suppose that B/.Subsequently,because is maximal and Sconsistent,B .According to Lemma 2,D()is Sconsistent.Thus,there is some1such that(a)D()1and(b)1.According to the definition of RC,(a)yields RC1or R01,but according to the assumption,(b)yields MC,1.Hence,by the definition of the truth value of B,MC,SB.Theorem 5(Completeness).Given the system S,for any formula,we haveS S.Proof.Soundness(i.e.,S S)is illustrated in Theorem 1 and Theorem 4.For completeness,we suppose that S.Then,is maximally Sconsistent.Therefore,by the Lindenbaum theorem,there is some maximally Sconsistent set in WCsuch that .By Lemma 3,MC,S.Thus,S.Different systems are complete with respect to different frames.Thus,we havethe following:Theorem 6.1.B0is complete w.r.t.arbitrary frames.2.B1is complete w.r.t.weakly transitive frames.3.B2is complete w.r.t.symmetric frames.4.B3is complete w.r.t.semiweakly Euclidean frames.5.B4is complete w.r.t.weakly Euclidean frames.Note that the definitions of weakly transitive frames,semiweakly Euclideanframes,and weakly Euclidean frames are described in Definition 3.Proof.For 1.The axioms and transformation rules of B0require no informationabout its canonical model,which means that B0is complete with respect to arbitraryframes.For 2.We must demonstrate that the canonical model of B1is weakly transitive.Assume that Rwand Rw thus,D()and D().Let bean arbitrary element in D().Hence,B .According to axiom B BB,B B .Therefore,B D(),and because D(),itfollows that B .Again,we have D(),and because D(),itfollows that .Because is arbitrary,we have D(),and this meansRor R0.For 3.We must demonstrate that the canonical of B2is symmetric.Assumethat R.Hence,D().Let be an arbitrary element in D().Hence,B .Because D(),B/D().According to the definition of116Studies in Logic,Vol.16,No.3(2023)D(),it implies that BB/.Then,by axiom BB,we have /,which means .Because is arbitrary,we have D(),and this meansR or R0.Thus,R.For 4.Assume that Rwan