Homotopies in Classical and Paraconsistent Modal Logics

Homotopies in Classical and P araconsistent Modal Logics Can Bas ¸ kent 1 Introduction Bisimulations and van Benthem’s celebrated theorem pr o vide a direct insight how truth preserving operations in modal logics work. Apart from bisimula- tions, there are several other op e rations in basic modal logic that preserve the truth (Blackburn et al. , 2001). However , there is a pro blem. Given a modal mode l, and several bisimil ar copies of it, there is no method to compare or measure the diffe rences between bisimilar models apart fr o m the basic model theoretical met h ods (i.e. the y ar e submodels of each other , for instance). A rath er negative slogan f o r this issue is the following: Modal language cannot distinguis h bisimila r models . Since the modal language cannot co unt o r mea s ure , several ex tensions of the languag e has been proposed to tackle this issue such as h ybrid logics and majority logics (Blackburn, 2000 ; F ine, 1972). However , such extensions are language based and introduce a non-natural, and sometimes counter-intuitive operators to the language, and ofte n criticized as being ad-hoc . In this work, we will focus o n the truth preserving operations r at her than ex- tending the modal language. In other wor ds, we will ask the following question: Is there a truth preserving natural mo dal o peration that ca n also distinguish the models it generat es , and even compares them? W e argue t h at such an op eration exist, and the answer to that question is positive. However , to conceptualize our concerns and questions, we nee d to be care- ful at picking the cor rect modal logical framework. Kripkean models, in this respect, are criticized as they ar e overly simplistic and can overshadow some mathematical pro p erties that can be appare nt in some other modal models. Therefor e, in this paper , we will concentr ate on topological models for modal logics. On t h e other hand, note th at topological models historically precede Kripke mo dels, and are mathematically more complex allowing us to expre ss variety of ideas within modal l ogic. Therefore, they can provide us with much stronger and richer structure of which we can take advantage. In this paper , we utilize a rathe r elementary concept from topology . W e fir st intro duce h omem- orphisms, and then homo t opies to the modal logical fr amework, and show the immediate invariance results. On the other hand, from an app lication orient e d point of view , we also have some applications t o illustrate how our constructions 1 can be useful. In a previous work, ho meomorphisms and homotopies were introduced to the co ntext of nonclassical modal logics (Bas ¸ kent, 2011) . In this work, our goal is to exte nd such results to both nonclassi cal and classical cases and see how homotopies can be d e fined in classical modal case. Before moving on, let us make it clear that what we mean b y nonclassical is either paraconsistent or paracomplete logical systems. 1.1 What is a T opology? The history of the top o logical semantics of (modal) logics can be trace d back to early 1920s making it the first semantics f or variety of modal logics (Goldblatt, 2006). The major revival of the topological semantics of mo dal logics and its connec- tions with algebras, howeve r , is due to McKinsey and T arski (McKinsey & T arski, 1946; McKinsey & T ar ski, 1944). I n t his sect ion, we will b riefly mention the b asics of topological semantics in o rder to be able build our future constr uctions. W e will give two equivalent definitions of top ological spaces he re f or our purposes. Definition 1.1. T he structure h S, τ i is called a topological space if it satisfies the following conditions. 1. S ∈ τ and ∅ ∈ τ 2. τ is closed under arbitrary unions and finite intersections Definition 1.2. T he structure h S, σ i is called a topological space if it satisfies the following conditions. 1. S ∈ σ and ∅ ∈ σ 2. σ is closed under finite unions and arbitr ar y intersect ions Collections τ and σ ar e called t opologies. The e lements of τ are called open sets where as the eleme nt s of σ ar e called closed sets. Therefore , a set is open if its complement in the same topology is a closed set and vice ve rsa. A function is called continuous if the inverse image of an open (re spe ctively , closed) set is open (respectively , closed), and a function is called open if the image of an open (respe ct ively , closed) set is op en (re spectively , closed). Moreover , t wo topological spaces are called homeomo rp h ic if there is a continuous bijection from one to the ot her with a continuous inverse. Moreove r , two continuous functions are called homoto p ic if t here is a continuous defor mation between the two. Homotopy is then an equiva lence relation and, it gives rise to homo t opy groups which is a foundational subject in algebraic topology . 1.2 What is P araconsistency? An easy and immediate semantics of paraconsistent/paracomplete logics can be given by using topo logies. F or this reason, it is helpf ul t o remember some basics of paraconsistency . 2 First, note th at deductive explosion describes the situation where any for- mula can be deduced fr om an inconsistent set of formulae, i.e. for all fo r mulae ϕ and ψ , we have { ϕ, ¬ ϕ } ⊢ ψ , where ⊢ denotes logical consequence relation. In this respect, both “classical” and intuitionistic logics are known to be explosive. P araconsistent logic, on t he other hand, is the umbrella term for logical systems where th e logical consequence r elation ⊢ is not e x plosive (Priest , 2002). V ar iet y of philosophical and logical objections can be raised against paraconsistency , and almost all of these objections can be defended in a rigorous fashion. W e will not here be concerned about the philosophical implications of it, yet we re- fer the reader to the following for a compre hensive defe nse of paraco nsistency with a variety of well-structured applications (Priest, 1998). Use o f topo logical semantics for paraconsistent logics is not new . T o our knowledge, the earliest work discussing the connection b etween inconsistency and topology goes back t o Goo dman (Good m an, 1981) 1 . In h is paper , Good- man discussed “pseudo-complements” in a lattice the oretical setting and called the topological system he obtains “anti-intuitionistic logic” . In a recent work, Priest discussed the dual of the intuitionistic negation operator and considere d that o perator in topological framewo rk ( Priest, 2009). Similarly , Mort ensen discussed t opological separation principles f rom a paraconsistent and paracom- plete po int of view and investigated the theories in such spaces (Mor tensen, 2000 ) . Similar appro aches from modal pe rspective was discussed b y B´ eziau, too (B´ eziau, 2005). 1.3 Semantics In our setting, we denote set of pr opositional variables with P . W e use the language of propositional modal logic with the mo d ality  , and we will de fine the dual ♦ in the usual sense, and construct the language of t he basic unimodal logic recursively in the standard fashion. In topological semantics, the modal operator fo r necessitation corresponds to the to pological interior oper at or Int where Int ( O ) is the largest open set con- tained in O . Furthermore, one can duall y associa te the topological closure op- erator Clo with the po ssibility modal o perator ♦ wh ere th e closure Clo ( O ) of a given set O is the smallest closed set that contains O . Before connecting t o pology and modal logic, let us set a piece of notation and te rminology . The extension, i.e. the points at wh ich the f ormula is satisfied, of a f ormula ϕ in the model M will be denoted as [ ϕ ] M . W e omit the super script if the model we are wor king with is obvious. Mo r eover , by a theo ry , we mean a deductively closed set of formulae. The extensions of Boolean cases are o bvious. However , the extension of a modal fo rmula  ϕ is the n associated with an open se t in the topological system. Thus, we have [  ϕ ] = In t ([ ϕ ]) . Similarly , we put [ ♦ ϕ ] = Clo ([ ϕ ]) . This means that in the basic set ting, topological entities such as open or closed sets appear only with modalities. 1 Thanks to Chris M ortensen for this remark. Even if Goodman’s paper app eared in 1981, th e work had been carried o u t arou n d 1978. In his p aper , Goo d man indi cted that the results were based o n an early that app eared in 19 7 8 o n ly as an abstract. 3 However , we can take one step further and sugg est that ex t ension of a ny propositional variable be an open set (Mortensen, 2000; Mints, 2000). In that setting, conjunction and disjunction works fine f o r finite inte r sections and unions. Nevertheless, th e negation can be difficult as the complement of an open set is not generally an open set, thus may no t be the extension of a formula in the language. F or this reason, we need to use a new negation symbol ˙ ∼ that re t urns the ope n complement (interior of the compleme nt ) of a given set. A similar ide a can also be applied to closed sets where we assume that the extension of any propositional variable will be a closed set. In o r der to be able to avoid a similar problem with the negation, we stipulate yet anoth e r negation operator which returns the closed complement (closure of the complement) of a given set. In this setting, we use the symbol ∼ that returns the closed complement of a given set. Under these assumptions and recalling Definitions 1.1 & 1.2 , it is easy to observe the following (Morte nsen, 2000). • In the topo logy of open sets τ , any theory th at includes the theo r y of th e propositions that are true at th e boundary is incomplete. • In the topology of closed sets σ , any theory that includes the boundary points will be inconsistent. An immediate ob ser vation yields t hat since extensions of all for mulae in σ (respectively in τ ) are closed (respectively , open), the topologies which are obtained in both p araconsistent (and paracomplete) logics are discrete. W e already made the f ollowing simple connection (Bas ¸ kent, 2011) . F or a given model M , let | M | denot e t he size of M ’s carr ier set. Proposition 1.3. Let M 1 and M 2 be paraconsist ent a nd paracomplete topologi- cal models respecti vely . If | M 1 | = | M 2 | , then there is a homeomorphis m from a paraconsis t ent t opologica l model to the pa racomplete one, and vice versa. 2 Homotopies Let us clarify an important point. W e stipulate t hat the ext ension of any formula is closed to obtain a paraconsistent system, and stipulate that to be open to obtain a paracomplete set. In o ther words, we do not mix such systems. On the other hand, note that in th e classical case, open or closed sets appear only under the presence of modal ope r ators, and they may appear together . Now , in this section, we will start off with the easier case and consider para- consistent topological spaces. Second, we will exte nd o ur re sults to classi cal case. 2.1 P araconsistent Case A recent research p rogram that considers topo logical modal logics with continu- ous functions were discussed in an early wor k (Artemo v et a l. , 1997; Kreme r & Mints, 2005). 4 An immediate theore m, which was stated and pro ved in variety of diffe r ent work, would also work for paraconsistent logics (Kremer & Mints, 2005). Now , let us take t wo closed set top ologies σ and σ ′ on a given set S and a h o meomor- phism f : h S, σ i → h S, σ ′ i . Akin to a previous theor e m of Kremer and Mints, we have a simple way to associate the respective valuations betwe e n two models M and M ′ which respectively depe nd on σ and σ ′ so that we can have a truth preservation result. T herefore, define V ′ ( p ) = f ( V ( p )) . Then, we have M | = ϕ iff M ′ | = ϕ . Theorem 2.1. Let M = h S, σ , V i and M ′ = h S, σ ′ , V ′ i be t wo paraconsist ent topologic al models with a ho m eomorphism f from h S, σ i to h S, σ ′ i . Define V ′ ( p ) = f ( V ( p )) . Then M | = ϕ i ff M ′ | = ϕ for all ϕ . Proof . The pro of is by induction on the complexity of the f ormulae. See (Bas ¸ kent, 2011) for the proof in non-classical case.  Assuming that f is a homeomorphism may seem a bit strong. W e can t hen separate it into two chunks. One direction of the biconditional can be satisfied by continuity whereas the other direct ion is satisfied by the openness of f . Corollary 2.2. Let M = h S, σ, V i and M ′ = h S, σ ′ , V ′ i be two paraconsistent topologic al models with a cont i nuous f from h S, σ i to h S, σ ′ i . Define V ′ ( p ) = f ( V ( p )) . Then M | = ϕ i mplies M ′ | = ϕ for all ϕ . Corollary 2.3. Let M = h S, σ, V i and M ′ = h S, σ ′ , V ′ i be two paraconsistent topologic al models with an open f from h S, σ i to h S, σ ′ i . Define V ′ ( p ) = f ( V ( p )) . Then M ′ | = ϕ impli es M | = ϕ for all ϕ . Proofs of both corollaries depend on the fact th at Clo operator co mmutes with continuous functions in one direction, and it commutes with open func- tions in the other direction. F urthermore, similar corollaries can be given for paracomplete framewo rks as the Int operator also commutes in one direction under similar assumptions, and we leave it to the reader as well. Furthermore , any topological oper ator that commutes with continuous, o pen and homeomorp hic functions will reflect t h e same idea and preser ve the truth 2 . Therefor e, t hese results can easily be generalized. W e can now take one step f urther to discuss homotopies in paraconsistent topological modal models. T o best of our knowledge, homotopies first intro- duced to (non-classical) modal logic in (Bas ¸ kent, 2011). Now , recall that a homotopy is a de scr ipt ion of how two continuous function fro m a topological space to another can be deforme d to each other . W e can now state the formal definition. Definition 2.4. Let S and S ′ be two t opological spaces with continuous func- tions f , f ′ : S → S ′ . A homoto py between f and f ′ is a continuous function H : S × [0 , 1] → S ′ such that if s ∈ S , then H ( s, 0) = f ( s ) and H ( s, 1) = g ( s ) . 2 Thanks to Chris Mortens en for pointing this out. 5 In othe r wo r ds, a homot opy b etween f and f ′ is a family o f co nt inuous functions H t : S → S ′ such t hat for t ∈ [0 , 1] we have H 0 = f and H 1 = g and the map t 7→ H t is continuous from [0 , 1] t o the space of all continuous functions from S to S ′ . Notice t h at homotopy relation is an equivalence r e lation. Thus, if f and f ′ are h omotopic, we denote it with f ≈ f ′ . W e will now use h omotopies to obtain a generalization of Theo rem 2.1. M M t M t ′ f t f t ′ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✸ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s H Figure 1: Homotopic M o dels Definition 2.5. Given a mo del M = h S, σ, V i , we call t h e f amily of models { M t = h S t ⊆ S, σ t , V t i} t ∈ [0 , 1] generated by M and homotopic functions homo- topic models. In the generation, we put V t = f t ( V ) . Theorem 2.6. G iven t wo topo logical paraco ns istent models M = h S, σ, V i and M ′ = h S ′ , σ ′ , V ′ i with two continuo u s functions f , f ′ : S → S ′ both of which respect the valua tion: V ′ = f ( V ) = f ′ ( V ) . If there is a homotopy H between f and f ′ , then homot opic models sat isfy the same formulae. Proof . T o make the pr oof a bit more readable, note that the object whose names have a prime ′ , are in th e range of the functions. T o make the proof go easil y , we will assume that continuous functions are onto. If not, we can easily rearrange the range in such a way that it will be. Let M t = h S ′ t , σ t , V t i and M t ′ = h S ′ t ′ , σ t ′ , V t ′ i be homotopic models with continuous f t , f t ′ : S → S ′ . Observe that M t = h S ′ t , σ t , f t ( V ) i , or equally M t = h S ′ t , σ t , H t ( V ) i for homot o py H . T ake a point s t such that f t ( s ) for some s ∈ S . Then, t ake h S ′ t , σ t , H t ( V ) i , f t ( s ) | = ϕ f o r arbitrary ϕ . Since, H is continuous on t by some h , and by Cor o llary 2.2, we ob serve M t ′ = h S t ′ , σ t ′ , H t ′ ( V ) i , s t ′ | = ϕ . In this case s t ′ exists as H t is continuos o n t and s t ′ = h ( f t ( s )) = h ( s t ) . Therefor e, M t | = ϕ implies M t ′ | = ϕ . Notice that in th is case, we did not need t o present a proo f on the complex ity of ϕ . The r eason for that is the f act t hat the ex t ension o f each formula is a closed set (since we are in a p ar aconsistent sett ing).  Corollary 2.7. M | = ϕ impli es M t | = ϕ , but not the other way around. 6 Corollary 2.8. In Theorem 2.6, if we tak e th e cases for t = 0 and t = 1 , we obta i n Corollary 2.2. Notice that we h ave dis cussed the t r uth in the image sets that are obtained under f , f ′ , f t , . . . . Nevertheless, the converse can also be true, once the con- tinuous functions have continuous inverses: th is is e xactly what is guaranteed by homeomorph isms. Th e corresponding notion at the level of homotop ies is an is o topy . An isotopy is a continuous tr ansformation bet we en homeomo rphic functions. Thus, we h ave the following. Theorem 2.9. G iven t wo topo logical paraco ns istent models M = h S, σ, V i and M ′ = h S ′ , σ ′ , V ′ i with two homeo m orphism f , f ′ : S → S ′ both of whic h respect the valuat ion: V ′ = f ( V ) = f ′ ( V ) . If there is an isot opy H between f and f ′ , then, for all ϕ , we hav e M t | = ϕ iff M | = ϕ iff M ′ | = ϕ Proof . Immediate , thus left to the reader .  What make s the non-classical case easy is the fact that the extension of each formula is an op en or a closed set. Furthermore, a similar t heorem can be state d for paracomplete cases with a similar pro o f . Theorem 2.10. Given two topolo gical paracomplete models M = h S, σ, V i and M ′ = h S ′ , σ ′ , V ′ i with two continuo u s functions f , f ′ : S → S ′ both of which respect the valua tion: V ′ = f ( V ) = f ′ ( V ) . If there is a homotopy H between f and f , then hom otopic models s atisfy the same formulae. 2.2 Classical Case The reason why homotopies work nicely in non-classical cases is immediate: because we stipulate t hat the extension of propositions to be open (or dually closed) sets. This is a strong assumption. In the topological semantics of b asic modal logic, extensions of only modal formulae are taken to be open ( or closed) . Can we t hen have re sults similar to those we had in non-classical case? This is t he prob lem we ar e going to address in this section. Let N = h T , η , V i and N ′ = h T ′ , η ′ , V ′ i b e classical topological modal mod- els. D efine a homotopy H : T × [0 , 1] → T ′ . The r efore, as bef ore, for each t ∈ [0 , 1] , we obtain mode ls N t = h T t , η t , V t i . Theorem 2.11. Given two clas sical topolo gical modal models N = h T , η , V i and N ′ = h T ′ , η ′ , V ′ i with two c o ntinuous functions f , f ′ : T → T ′ both of which respect the valuation: V ′ = f ( V ) = f ( V ′ ) . If th ere is a hom otopy H between f and f ′ , then homot opic models sat isfy the same formula. Proof . Let two classical t o pological modal models N = h T , η , V i and N ′ = h T ′ , η ′ , V ′ i with two continuous f unctions g , g ′ : T → T ′ both of which re- spect t h e valuation: V ′ = f ( V ) = f ( V ′ ) be given. Let H b e a homo topy 7 H : T × [0 , 1] → T ′ . W e will show t hat homoto p ic models satisfy the same formula. T ake two homot opic mod e ls N t and N t ′ for t, t ′ ∈ [0 , 1 ] . Let f t ( w t ) ∈ T t be a point in N t where f t : T → T t . Consider N t , f t ( w t ) | = ϕ . W e will show that for some f t ′ ( w t ′ ) ∈ T t ′ , we will have N t ′ , f t ′ ( w t ′ ) | = ϕ . Proof is by induction on the complexity of ϕ . F irst, let ϕ = p for a propo - sitional variable p . Then, let N t , f t ( w t ) | = p . By using H , we can rewrite as N t , H ( w t , t ) | = p . Therefore, H ( w t , t ) ∈ V t ( p ) . Since H is co nt inuous on t , we have a continuous function from i : t → t ′ . Now , since V t = f ( V ) we observe H ( w t , t ) ∈ f t ( V ( p )) which is equivalent to say H ( w t , t ) ∈ H ( V ( p ) , t ) . W e can compose both sides with i to get H ( w t ′ , t ′ ) ∈ H ( V ( p ) , t ′ ) for some w t ′ ∈ T . In short, we obtain N t ′ , f t ′ ( w t ′ ) | = p . Note that since H is a homotop y , the function i : t → t ′ exists and is con- tinuous. Similarly , another f unction j : t ′ → t exists and is cont inuous, and j is needed to pr o ve t he other direction. The cases for Boolean ϕ is similar and thus left to the reader . Let us now consider t he modal case ϕ =  ψ for some ψ . Now , let N t , f t ( w t ) | =  ψ . Thus, f t ( w t ) ∈ Int ([ ψ ]) wher e [ ψ ] denotes the extension of ψ . Since f t is continuous the inver se image of an open set is open, thus f − 1 t ( Int ([ ψ ])) is open. F or the previously constructed i , we observe i − 1 ◦ ( f − 1 t ( Int ([ ψ ]))) is also open as f t and i are both continuous. Thus, the inverse image of Int ([ ψ ]) is open under f t ′ . Therefor e, b y the similar reasoning, H ( w t ′ , t ′ ) ∈ Int ([ ψ ]) fo r some w t ′ in the neighborhood. Thus, N t ′ , f t ′ ( w t ′ ) | =  ψ . The reverse direct ion of the proof f rom N t ′ to N t is similar , and this con- cludes the proof.  In conclusion, under a suitable valuation, isoto pic models are tr uth invariant both in classical and non-classical cases that we h ave investigated. 3 A M odal L ogical Application Consider t h e following two bisimilar Kripke models M and M ′ . Assume that w, w ′ and u , u ′ , y ′ and v , v ′ , x ′ do satisfy the same pro positional lett ers. T hen it is easy to see that w and w ′ are bisimilar , and therefo re satisfy th e same model formulae. W e can still pose a conceptual question abo ut the relation betwee n M and M ′ . Eve n if the se two models are bisimilar , they are diff erent models. Moreover , it is plausible to co nt ract M ′ to M in a validity preser ving f ashion. T h erefore, we may need to t r ansform one model to a b isimilar mode l of it. Furthermo r e, given a model, we may need to m ea sure the level of change from the fi xed model to another model which is bisimilar to th e given one. Especially , in epistemic logic, such concerns do make sense. Given an epis- temic situation, we can mo del it upto bis imulation . In other wor ds, from an agent’s p e rspective, bisimilar mod e ls are indistinguishable. But, from a model 8 w v u ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✸ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s M w ′ v ′ u ′ y ′ x ′ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✸ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✶ P P P P P P P q M ′ Figure 2: T wo Bisimular Models theoretical perspective, they are distinguishable. T herefore, t here can be sev- eral modal logical ways to mo del the given ep istemic situation. W e will now define how t hese models are r elated and d if ferent f r om each oth er by using the constructions we have presented earlier . Before proceeding f urther , let us give the definition of to p ological bisimula- tions (Aiello & van Benthem, 2002). Definition 3.1. Let M = h S, σ , v i and M ′ = h S ′ , σ ′ , v ′ i be two topological mod- els. A topo-bisimulation is a nonempty relation ⇄ ⊆ S × S ′ such t hat if s ⇄ s ′ , then we have the following: 1. BASE CONDIT ION s ∈ v ( p ) if and only if s ∈ v ′ ( p ) for any prop o sitional variable p . 2. FORTH CONDITION s ∈ U ∈ σ implies that there exists U ′ ∈ σ ′ such that s ′ ∈ U ′ and for all t ′ ∈ U ′ there exists t ∈ U with t ⇄ t ′ 3. BACK CONDITIO N s ′ ∈ U ′ ∈ σ ′ implies that there exists U ∈ σ such that s ∈ U and for all t ∈ U there exists t ′ ∈ U ′ with t ⇄ t ′ W e can t ake one step further and define a homo emorphism that respect bisimulations. Let M = h S, σ, v i and M ′ = h S ′ , σ ′ , v ′ i be t wo topo-bisimular models. If there is a homoemor phism f from h S, σ i to h S, σ i that respect the valuation, we call M and M homeo-top o -bisimilar models. W e give the precise definition as follows. Definition 3.2. Let M = h S, σ , v i and M ′ = h S ′ , σ ′ , v ′ i be two topological mod- els. A homeo-topo-bisimulation is a nonempty relation ⇄ f ⊆ S × S ′ based on a homeomorp hism f fr om S into S ′ such that if s ⇄ f s ′ , then we have t he following: 1. BASE CONDIT ION s ∈ v ( p ) if and only if s ∈ v ′ ( p ) for any prop o sitional variable p . 9 2. FORTH CONDITION s ∈ U ∈ σ implies that there exists f ( U ) ∈ σ ′ such t hat s ′ ∈ f ( U ) and fo r all t ′ ∈ f ( U ) there exists t ∈ U with t ⇄ f t ′ 3. BACK CONDITIO N s ′ ∈ f ( U ) ∈ σ ′ implies that t here exists U ∈ σ such that s ∈ U and for all t ∈ U there exists t ′ ∈ f ( U ) with t ⇄ f t ′ Based on this defi nition, we immediate ly ob serve the following. Theorem 3.3. Ho meo-topo-bisimula t ion preserve the vali dit y . Proof . Th e p roof is an induction on the complexity o f the for mulae and thus left to the reader .  Notice that we can define more than o ne homeomorphism between t opo- bisimilar models. Now , we can discuss the homotopy of homeo -topo-bisimilar mo dels. What we aim is the following. Given a t opological model (either classical, intuition- istic or paraconsistent) , we will construct two homeomorp hic image of it re - specting homeo-topo-bisimulation where t hese two homeomorphisms are ho- motopic. Then, by using homot opy , we will measure the level of change of the intermediate homeomorphic models with respect to th e se two functions. Let M be a given topological mode l. Construct M f and M g as th e home- omorphic image of M respecting the valuation where f and g are homeomor- phism. F or simplicity , assume that M ⇄ f M f and M ⇄ g M g . Now , if f and g are homotopic, t hen we h ave f unctions h x for x co ntinuous o n [0 , 1] with h 0 = f and h 1 = g . Therefor e, given x ∈ [0 , 1] th e model M x will be o b tained by applying h x to M respect ing the valuation. Hence, M 0 = M f and M 1 = M g . Theref ore, given M , the distance of any home o-topo-bisimilar model M x to M will be x , and it will be t h e measure of non-modal change in the model. In other words, even if M ⇄ h ( x ) M x , we will say M and M x are x -different than each oth e r . The procedure we described offers a we ll-defined met h od of indexing the homeo-topo-bisimular models. But, indexing is not random. It is continuously on the closed unit interval. Note that invariance results are usually used to pro ve undefinability r e sults in modal logic ( Blackburn et al. , 2001). F or ex amp le, in order to show irreflex- ivity is not modally definable, one needs to come up with two bisimilar mo dels - one is irreflexive, the other is not. Homeo-topo-bisimulations can also be used to show some topo logical prop- erties are not modally definable. F or instance, in th is respect, dimensions o f spaces is not modally definable in topological modal logic. Similarly , as t rifoil and circle are homeomor phic, knots are also not definable. 10 4 An Epistemic L ogical Application Consider two believers Ann and Bob where Ann is an ordinary believer while Bob is a religious cleric of the religion th at Ann is following. The refore, they believe in the same religion and the same rules of t he religion. F or the sake of o ur example, let us ass ume that the religion in question is really simple and there is nothing that Ann does not know about it. I n other words, even if Bob is a clergyman, Ann believes in the religion as much as Bob does. However , we feel that Bob believes it more th an Ann even though they believe in exactly the same propositions. In othe r words, there is still a difference bet we en their belief . The n, what is this difference? Th e immediate reason f or this is the f act that th e extend of Bo b ’s knowledge is wider than that of Ann’s. Namely , Bo b believes better . What does this mean? In this context, we can ask t he following two questions. 1. How wider is Bob’s b e lief? 2. How is Ann’s belief transformed to Bob’s? These two questions are meaningful. Even if their languag e cannot tell us which one h as wider knowledge, ontologically , we know th at Bob has more knowledge in some sense even if they agree on every proposition. Clearly , th e reason for that is the f act that Bob considers more possible worlds for a given proposition which makes his belief more rob ust than Ann’s. Definition 4.1. Given t wo agents i and j with th e ir respective bisimilar mode ls M and N . W e say i ’s knowledge more ro bust than j ’s if [ ϕ ] N ⊆ [ ϕ ] M for all formula ϕ in the language. What matters is kno wledge, and in basic modal logic with topological se- mantics, we know that t he ex t ension of modal formulas is open (or , closed dually). Ther efore, in bisimilar models f or Ann and Bob , the extension of Bob’s robust belief is a superset o f that of Ann’s. Homeo morphisms and homotopies can explain this transformation from Ann’s be liefs t o Bob ’s belief . The timestamp in the definition of the homotopy can easily be considered as a temporal parameter . In our simple example, this r eading of the homotopy parameter is esp e cially helpful. It h elp us to give a step by step account of the transformation between Ann’s and Bob’s belief. Therefor e, focusing on the transformations reveal mo re informat ion on the ontology of the agents when the epistemics of t he agents is already kno wn. R obust knowledge/belief captures this notion by fo cusing on the ontology of the model and connects the ep istemics with onto logy . Acknowledgement I am grateful t o Chris Mortensen, Graham Priest, Melvin Fitting for their encouragament and comme nt s. 11 R efe rence s A I E L L O , M A R C O , & VA N B E N T H E M , J O H A N . 2002. A Modal W alk Thr ough Space. Journal of Applied Non-Class ical Logic s , 12 (3-4), 319–363. 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