模态实在与模态非实在事件的逻辑.pdf
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1、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
2、 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 theirc
3、ompleteness.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 p
4、ossible 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 tha
5、t 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 eve
6、nt 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?W
7、e 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 andmo
8、dally 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 o
9、f 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 eve
10、nt,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,Mod
11、els,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
12、,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 possibi
13、lity 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 foll
14、owing 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
15、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
16、 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
17、 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
18、,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 tha
19、t 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
20、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()T
21、o 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 semiweakl
22、y 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(se
23、miweakly 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 symmetr
24、ic 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 s
25、ay 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
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