Logics for Modally Real and Modally Nonreal Events

Xian Zhao Tianqun Pan

Abstract. An event is modally real in one world if it occurs either in the world or in one of its possible worlds;accordingly,an event is modally nonreal in one world if it does not occur in the world or in any one of its possible worlds.We call a place where all modally nonreal events of a world occur or exist as a modally black hole.This paper presents logical systems for modally real events and modally nonreal events,proves their soundness,and establishes their completeness.

1 Introduction

Modal realists,extreme or moderate,admit the reality of numerous worlds.For example,D.Lewis said,“Possible worlds are what they are,and not some other thing.If asked what sort of thing they are,I cannot give the kind of reply my questioner probably expects:that is,a proposal to reduce possible worlds to something else.I can only ask him to admit that he knows what sort of thing our actual world is,and then explain that possible worlds are more things ofthatsort,differing not in kind but only in what goes on at them.”([5],p.85)Because any possible world constitutes things,admitting that possible worlds are just as real as our world means admitting that things in any possible world are just as real as things in our world.Hence,a thing or an event is regarded as modal reality if it exists or occurs either in our world or in one of the possible worlds of our world,and a thing or an event is regarded as modal nonreality if it does not exist or occur either in our world or in any of the possible worlds of our world.Thus,we have two notions:modal reality and modal nonreality.

Because a modally nonreal thing does not exist in the world or any of its possible worlds,where does it inhabit? We suppose there is such a place where all modally nonreal events of the world inhabit,and we call the place amodally black hole.What we focus on here is not questions related to the modally black hole,such as whether the modally black hole exists,but the logical structures of modally real events and modally nonreal events.

2 Proof Systems for Modally Real and Modally Nonreal Events

The definitions of modal reality and modal nonreality are as follows:an event is modally real in our world if it occurs either in our world or in one of its possible worlds,and an event is modally nonreal in our world if it does not occur either in our world or in any of its possible worlds.We usepto represent an event,Rfor a modal reality operator,andBfor a modal nonreality operator.RpandBprepresent that“pis modally real”and“pis modally nonreal”respectively.The formal languageLis defined as follows:

The language inLis interpreted by the standard possible world semantics.

Definition 1(Frames,Models,and Satisfaction).A Kripke frameF=〈W,R〉is a tuple whereWis a set of possible worlds andR ⊆W ×Wis an accessibility relation.A Kripke modelM=(F,π) is a tuple whereFis a Kripke frame andπ：P →2wis an interpretation for a set of propositional variablesP.A formulaφis true in modelMin the worldwif

Semantically,the relations between the modal reality operatorRor the modal nonreality operatorBand the necessity operator or the possibility operator are as follows:

The relation betweenRandBis as follows:

Because the modally real operatorRand the modally nonreal operatorBare interde finable(i.e.,Rp ↔¬Bp),we useBas the primitive operator,andRcan be defined byB.

Definition 2.SystemB0comprises the following axioms and transformation rules:

Ax0 all tautologies of propositional logic.

Note thatB0is the propositional calculus plus the axioms Ax1,Ax2,and Ax3 and 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 respect to arbitrary frames.

Suppose thatMis a model that is based on an arbitrary frame andwis a world inM.

For Ax1,suppose thatM,w⊭Bφ →¬φ.Consequently,M,w⊨¬(Bφ →¬φ).According to Ax0,M,w⊨Bφ ∧φ.Hence,(a)M,w⊨Bφand(b)M,w⊨φ.From (a),according to the definition ofBφin Definition 1,M,w⊨¬φ,which contradicts(b).

For Ax2,suppose thatM,w⊭B(φ ∧ψ)→¬B¬φ ∨Bψ.Therefore,M,w⊨B(φ∧ψ)∧B¬φ∧¬Bψ.Hence,(a)M,w⊨B(φ∧ψ)∧B¬φand(b)M,w⊨¬Bψ.Hence,from(a),M,w⊨¬(φ ∧ψ)andM,w⊨φ,and for any worldw′withRww′,M,w′⊨¬(φ ∧ψ)andM,w′⊨φ.Then,M,w⊨¬ψ,and for any worldw′withRww′,M,w′⊨¬ψ.Hence,we haveM,w⊨Bψ,which contradicts(b).

For Ax3,suppose thatM,w⊭Bφ∧Bψ →B(φ∨ψ).Then,M,w⊨Bφ∧Bψ∧¬B(φ ∨ψ).Hence,M,w⊨Bφ ∧BψandM,w⊨¬B(φ ∨ψ).Consequently,fromM,w⊨Bφ ∧Bψ,M,w⊨¬φandM,w⊨¬ψ,and for any worldw′withRww′,M,w′⊨¬φandM,w′⊨¬ψ.Hence,M,w⊨¬(φ ∧ψ),and for any worldw′withRww′,M,w′⊨¬(φ ∧ψ).Thus,we haveM,w⊨B(φ ∨ψ),which contradictsM,w⊨¬B(φ ∨ψ).

For RE,suppose that ⊨φ ↔ψ.Consequently,M,w⊨φ ↔ψ,and for anyw′such thatRww′,M,w′⊨φ ↔ψ.(a) Assume thatM,w⊨Bφ.According to the definition ofB,M,w⊨¬φ,and for anyw′such thatRww′,M,w′⊨¬φ.Hence,according toM,w⊨φ ↔ψandM,w′⊨φ ↔ψ,we haveM,w⊨¬ψandM,w′⊨¬ψ.Therefore,according to the definition ofB,M,w |=Bψis obtained.(b)Assume thatM,w⊨Bψ.The same reason as that in(a)ensures thatM,w⊨Bφ.Thus,by(a)and(b),we haveM,w⊨Bφ ↔Bψ.

For RC,suppose that ⊨φ.Then,M,w⊨φ,and for anyw′such thatRww′,M,w′⊨φ.Hence,by Definition 1,M,w⊨¬¬φ,and for anyw′such thatRww′,M,w′⊨¬¬φ.Thus,by the definition ofBφ,M,w⊨B¬φ.□

To obtain a new and useful derived rule,suppose that⊢B0φ →ψ.Conse quently,by RC,⊢B0B¬(φ →ψ).Because¬(φ →ψ)is equivalent to¬ψ ∧φ,we,by RE and MP,obtain⊢B0B(¬ψ ∧φ).Applying Ax2 to⊢B0B(¬ψ ∧φ)and using MP,we obtain⊢B0¬Bψ ∨Bφ.Thus,we follow the derived rule:

Theorem 2.The following formulae are provable in system B0:

To extendB0,a weak relationRwover possible worlds must be defined by the relationRand the identical relationR0.Rwww′is defined asRww′orR0ww′.Formally,Rwww′ ≡def. Rww′∨R0ww′.

Definition 3(Weak Frames).

1.A frame〈W,R〉is weakly transitive if for anyw,w′,w′′ ∈W,ifRwww′andRww′w′′,thenRwww′′.

2.A frame〈W,R〉is semiweakly Euclidean if for anyw,w′,w′′ ∈W,ifRwww′andRww′′,thenRww′w′′.

3.A frame〈W,R〉is weakly Euclidean if for anyw,w′,w′′ ∈W,ifRwww′andRW ww′′,thenRww′w′′.

4.A frame〈W,R〉is weakly dead if for anyw,w′ ∈W,ifRwww′,thenR0ww′.

Four notes:

(a)A frame that is transitive(semiweakly Euclidean and weakly Euclidean)must be weakly transitive (semiweakly Euclidean and weakly Euclidean),and not vice versa.

(b) A semiweakly Euclidean frame must be weakly Euclidean,and not vice versa.

(c)In a weakly Euclidean frame for anyw,w′ ∈W,ifRww′,we,byR0ww,haveRw′w.It means that a weakly Euclidean frame must be symmetric,and not vice versa.

(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 define a weakly reflexive frame in whichRwwwholds for anyw ∈W,such a frame is arbitrary,and vice versa.This means that a frame is weakly reflexive if and only if it is arbitrary.We can say thatB0is sound with respect to weakly reflexive frames.This indicates that our languageLis weaker and a model based on reflexive frames is indistinguishable.

Theorem 3.

1.The formula Bφ →B¬Bφ is valid w.r.t.weakly transitive frames.

2.φ →BBφ is valid w.r.t.symmetric frames.

3.¬φ ∧¬Bφ →BBφ is valid w.r.t.semiweakly Euclidean frames.

4.¬Bφ →BBφ is valid w.r.t.weakly Euclidean frames.

Proof.For 1.LetMbe an arbitrary model that is based on a weakly transitive frame andwbe a world inM.Suppose thatM,w⊭Bφ →B¬Bφ.Consequently,(a)M,w⊨Bφand(b)M,w⊨¬B¬Bφ.From(b),together with(a),we have for somew∗withRww∗,M,w∗⊨¬Bφ.Hence,M,w∗⊨¬BφandM,w∗⊨¬φ.Therefore,for somew∗∗withRw∗w∗∗,M,w∗∗⊨φ.BecauseRis weakly transitive,Rww∗∗orR0ww∗∗.However,from(a),we haveM,w⊨¬φ,and for anyw′withRww′,M,w′⊨¬φ.This means that we haveM,w∗∗⊨¬φandM,w⊨¬φ.A contradiction arises.

For 2.LetMbe an arbitrary model that is based on a symmetric frame andwbe a world inM,and suppose thatM,w⊨φ.Then,M,w⊭Bφ,andM,w′⊭Bφfor any worldw′inMthat“sees”w.BecauseMis symmetrical,the fact thatM,w′⊭BφandM,w⊭BφcausesBBφto be false inw.Hence,we haveM,w⊨BBφ.

For 3.LetMbe an arbitrary model that is based on a semiweakly Euclidean frame andwbe a world inM,and suppose thatM,w⊨¬φ ∧¬Bφ.Consequently,M,w⊨¬φ,andM,w⊨¬Bφfrom which eitherM,w⊨φor there must be a worldw∗inMsuch thatRww∗andM,w∗⊨φ.BecauseM,w⊨¬φ,we haveM,w∗⊨φ.To demonstrate thatM,w⊨BBφ,because we knowM,w⊨¬Bφ,we must demonstrate that for any worldw′inMsuch thatRww′,M,w′⊨¬Bφ.It is the case because for anyw′inMsuch thatRww′,we haveRw′w∗orR0w′w∗,andw∗is a world in whichφis true.

For 4.LetMbe an arbitrary model that is based on a weakly Euclidean frame andwbe a world inM,and suppose thatM,w⊨¬Bφ.There exist two cases:(a)M,w⊨φand(b)for somew∗withRww∗,M,w∗⊨φ.

Case(a).Because a Euclidean frame is symmetric,for anyw′withRww′,we have thatRw′w,andM,w′⊨¬Bφ.Thus,M,w⊨BBφ.

Case(b).BecauseMis weakly Euclidean,for anyw′such thatRwww′,Rww′w∗.ByM,w∗⊨φ,M,w′⊨¬Bφ.Therefore,we haveM,w⊨BBφ.□

Definition 4.We have the extensions ofB0:

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.

Note that becauseφ →BBφand¬φ ∧¬Bφ →BBφare provable inB4,B4is stronger than bothB2andB3.

There exists a trivial axiomφ →B¬φto the systems.Ifφ →B¬φis added toB0or other systems,the resulting system will collapse into the propositional calculus.This can be demonstrated as follows.Because of Ax1 in Definition 1,we haveφ ↔B¬φ,by which the operatorBin all formulae will be replaced.If a weakly dead end frame is defined as for anyw ∈W,ifRww′,thenR0ww′; the trivial axiomφ →B¬φis valid in weakly dead end frames.1Let M be an arbitrary model based on a weakly dead end frame and w be a world in M.Assume that M,w ⊨φ.Because w is only a possibly accessible world of w and M,w ⊨φ,we obtain M,w ⊨B¬φ.

ByRφ ↔φandφ →B¬φ,Rφ →φ.Philosophically,if our world is the only world,Rφ →φmeans that the modally real thing must be actual,and not vice versa.However,ifφis actual,its negation is modally nonreal.

3 Completeness

The modal logics we present in the previous section are nonstandard.Follow ing logicians who have dealt with other nonstandard modal logics such as logics of contingency 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 modal nonreality.

For the purpose of presenting a general result,we useSto stand forB0or one of its extensions.

A set Γ of well formed formulae is maximally consistent with respect to a systemS,if and only if for every formulaα,eitherα ∈Γ or¬α ∈Γ,and there is no finite collectionα1,α2,...,αn ∈Γ such that⊢S ¬(α1∧α2∧···∧αn).We simply call Γ a maximallySconsistent set.

To establish the completeness of the systems,we construct the successor of a maximallySconsistent set.

Definition 5.Let Γ be a maximallySconsistent set.The successor of Γ,D(Γ),is defined asD(Γ)={α|for everyα,B¬α ∈Γ}.

Lemma 1.For a maximally S consistent setΓof formulae,the successor D(Γ)is closed under conjunction.

Proof.Let Γ be a maximallySconsistent set andD(Γ)be the successor of Γ.

Suppose thatα ∈D(Γ) andβ ∈D(Γ).According to the construction ofD(Γ),B¬α ∈Γ andB¬β ∈Γ.By Ax3,B¬α ∧B¬β →B(¬α ∨¬β),and by the maximality of Γ,B(¬α ∨¬β)∈Γ.By RE,B(¬α ∨¬β)↔B¬(α ∧β).Hence,B¬(α ∧β)∈Γ.Thus,(α ∧β)∈D(Γ),which is required.□

Lemma 2.LetΓbe a maximally S consistent set of formulae.Assume that for some β,¬Bβ ∈Γ.Then,{β}∪D(Γ)is S consistent.

Proof.Suppose that¬Bβ ∈Γ but{β}∪D(Γ)is notSconsistent.Consequently,there must beψ1,ψ2,...,ψn ∈D(Γ)such that

To prove the completeness ofB0and its extensions,we must construct the special canonical model for them.

Definition 6(Canonical Model).The canonical modelMC=〈WC,RC,πC〉for the logicSis defined as follows:

1.WCis the set of all maximallySconsistent sets of formulae;

2.RC ⊆WCXWCis defined byRCΓΓ1orR0ΓΓ1iffD(Γ)⊆Γ1;and

3.p ∈πC(Γ)iffp ∈Γ.

The canonical model forSisad hoc.However,according to Definition 6,we haveD(Γ)⊆Γ1byRCΓΓ1; we cannot specifyRCΓΓ1byD(Γ)⊆Γ1,which is different from normal modal logics.

Lemma 3(Truth Lemma).Let MC=(WC,RC,πC)be the canonical model for S.For all formulae φ and all maximally S consistent setsΓs,MC,Γ ⊨S φ ⇔φ ∈Γ.

Proof.The theorem can be proven by induction on the structure of wff.Here,we prove the case ofBφonly:

We assume that the theorem holds forφand for all Γ:MC,Γ ⊨S φ ⇔φ ∈Γ.Suppose thatBφ ∈Γ.By the definition ofD(Γ),¬φ ∈D(Γ).According to Definition 6,we have that ifRCΓΓ1orR0ΓΓ1,¬φ ∈Γ1.This means that ifRCΓΓ1,¬φ ∈Γ1and¬φ ∈Γ.Hence,according to the assumption in the beginning,we have(a)ifRCΓΓ1,MC,Γ1⊨¬φand(b)MC,Γ ⊨S ¬φ.According to the definition of the truth value ofBφ,(a)and(b)yieldMC,Γ|=Bφ.

Suppose thatBφ/∈Γ.Subsequently,because Γ is maximal andSconsistent,¬Bφ ∈Γ.According to Lemma 2,{φ}∪D(Γ)isSconsistent.Thus,there is some Γ1such that(a)D(Γ)⊆Γ1and(b)φ ∈Γ1.According to the definition ofRC,(a)yieldsRCΓΓ1orR0ΓΓ1,but according to the assumption,(b)yieldsMC,Γ1⊨φ.Hence,by the definition of the truth value ofBφ,MC,Γ ⊭S Bφ.□

Theorem 5(Completeness).Given the system S,for any formula φ,we have

Proof.Soundness(i.e.,⊢S φ ⇒⊨S φ)is illustrated in Theorem 1 and Theorem 4.

For completeness,we suppose that ⊬S φ.Then,¬φis maximallySconsistent.Therefore,by the Lindenbaum theorem,there is some maximallySconsistent set Γ inWCsuch that¬φ ∈Γ.By Lemma 3,MC,Γ ⊨S ¬φ.Thus,⊭S φ.□

Different systems are complete with respect to different frames.Thus,we have the 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 Euclidean frames,and weakly Euclidean frames are described in Definition 3.

Proof.For 1.The axioms and transformation rules ofB0require no information about its canonical model,which means thatB0is complete with respect to arbitrary frames.

For 2.We must demonstrate that the canonical model ofB1is weakly transitive.Assume thatRwΓΓ′andRwΓ′Γ′′; thus,D(Γ)⊆Γ′andD(Γ′)⊆Γ′′.Letαbe an arbitrary element inD(Γ).Hence,B¬α ∈Γ.According to axiomB¬α →B¬B¬α,B¬ B¬α ∈Γ.Therefore,B¬α ∈D(Γ),and becauseD(Γ)⊆Γ′,it follows thatB¬α ∈Γ′.Again,we haveα ∈D(Γ′),and becauseD(Γ′)⊆Γ′′,it follows thatα ∈Γ′′.Becauseαis arbitrary,we haveD(Γ)⊆Γ′′,and this meansRΓΓ′′orR0ΓΓ′′.

For 3.We must demonstrate that the canonical ofB2is symmetric.Assume thatRΓΓ′.Hence,D(Γ)⊆Γ′.Letαbe an arbitrary element inD(Γ′).Hence,B¬α ∈Γ′.BecauseD(Γ)⊆Γ′,¬B¬α/∈D(Γ).According to the definition ofD(Γ),it implies thatBB¬α/∈Γ.Then,by axiom¬α →BB¬α,we have¬α/∈Γ,which meansα ∈Γ.Becauseαis arbitrary,we haveD(Γ′)⊆Γ,and this meansRΓ′Γ orR0ΓΓ′′.Thus,RΓ′Γ.

For 4.Assume thatRwΓΓ′andRΓΓ′′.Then,D(Γ)⊆Γ′andD(Γ)⊆Γ′′.Letαbe an arbitrary formula that is not in Γ′′butB¬α ∈Γ′.By Theorem 2 (4),B¬α →B(¬ α ∧¬β),and RE,we obtainB¬α →B¬(α ∨β).ByB¬α ∈Γ′,B¬(α∨β)∈Γ′.Because Γ is not identical to Γ′′,there must exist someβ,β/∈Γ′′butβ ∈Γ.Hence,α∨β/∈Γ′′.BecauseRΓΓ′′,B¬(α∨β)/∈Γ by which¬B¬(α∨β)∈Γ.Because ofβ ∈Γ,(α∨β)∈Γ.Therefore,(α∨β)∧¬B¬(α∨β)∈Γ.According to¬φ ∧¬Bφ →BBφ,BB¬(α ∨β)∈Γ.¬B¬(α ∨β)∈Γ′.However,we haveB¬(α ∨β)∈Γ′.A contradiction arises.

For 5.Assume thatRwΓΓ′andRwΓΓ′′.Then,D(Γ)⊆Γ′andD(Γ)⊆Γ′′.Letαbe an arbitrary formula that is not in Γ′′butB¬α ∈Γ′.Then,byRΓΓ′′,B¬α/∈Γ.According to axiom¬B¬α →BB¬α,we haveBB¬α ∈Γ.Therefore,B¬¬B¬α ∈Γ,which means that¬B¬α ∈D(Γ).BecauseD(Γ)⊆Γ′,we have¬B¬α ∈Γ′,which contradictsB¬α ∈Γ′.Becauseαis arbitrary,we haveD(Γ′)⊆Γ′′,and this means thatRwΓ′Γ′′.□

4 Conclusions and Remarks

This paper presents formal systems for modally real events and modally nonreal events with respect to different structures or frames of possible worlds,demonstrates their soundness,and establishes their completeness.Four additional remarks are pre sented as follows:

First,formally,compared with normal modal logics,the expressivity of the log ics presented in this paper is relatively weak.The logics cannot distinguish reflexive models.Nevertheless,the logics provide us with all formulae that we want.

Second,we regard an event to be modally real or modally nonreal with respect to a world in a model.Some events that are modally real(or nonreal)with respect to a world in a model could be modally nonreal(or real)with respect to other world(s)in the model.We,in the Introduction section,designate a place whereallmodally nonreal events of a world occur as a modally black hole.Themodally black holeof a world,here,is not a possible world but a place where all modal nonreal events of the world are“stored”.It can be seen that given a model,each world hasonlya modally black hole,and the number of modally black holes in a model is equivalent to the number of the worlds of the model.

Third,although our world consists of infinite events,there exist also infinite events that do not occur in our world.Hence,if possible worlds of our world are not infinite,there exist infinite events that do not exist in our world or in its possible worlds;alternatively,even if possible worlds of our world are infinite,any contradic tion does not exist in any worlds.Consequently,our world has its modally black hole.

Fourth,as we know,the term“black hole”in physics refers to a special space time region that has sufficiently compact mass and allows nothing to escape from the inside.We cannot observe any event in a physically black hole.How about an event in the modally black hole of our world? If the conceivable is possible and exists in possible worlds([1,8]),any event in the modally black hole of our world is inconceiv able,even unimagined.For instance,we can describe a contradiction,but we cannot conceive a contradictory event.Nevertheless,events in the modally black hole have their own logical structure,as revealed by the aforementioned logics.