Paraconsistency and Topological Semantics

P araconsistency and T opological Semantics Can Bas ¸ kent Department of Computer Science, The Graduate Center The City University of New Y ork cbaskent@g c.cuny.edu www.canbas kent.net November 20, 2018 1 Introduction and Motivation 1.1 What is P araconsistency? The well-studied notion of deductive ex plosion describes the situation where any formula can be de duced from an inconsistent set of formulas. In other words, in deductively explosive logics, we have { ϕ, ¬ ϕ } ⊢ ψ for all formulas ϕ, ψ where ⊢ is a logica l consequence relation. In this respect, both “classical ” and intuitionistic logics are known to be deductively explosive. P araconsistent logic, on the other hand, is t he umbrella term for logical systems wher e the log- ical consequence relation is not ex p losive. V ariety of philosophical and logical objections can be raised against paraconsistency , and almost all of the se objec- tions can be defended in a rigorous f ashion. Here, we wil l not be concer ned about the philosophical implications of it, yet we refer the reader to the follow- ing references for comprehensive defenses of paraconsistency with a variety of well-structured applications ch o sen fro m mathe mat ics and philosophy with a rigorous h istory of t he subject ( ? ; Priest, 2002 ; Priest, 1998; ? ). 1.2 Why T opologie s? In this work, we investigate the relationship between paraconsistency a nd some topological space s. As it is widely known, par aconsistency has many occur- rences in mat hematics spanning a wide range from mod el theory t o set the ory (Priest, 2006, Chapters 2 and 3). In this paper , we present some fur t her app li- cations o f paraconsistency in modal logic with topological semantics. Nevertheless, th e use of topological semantics for p araconsistent logic is not new . T o our knowledge, the earliest work discussi ng the connection between inconsistency and to p ology goes back to Goodman (G o odman, 1981) 1 . In his 1 Thanks to Chris Mortensen for pointing this work out. Even if the paper ap peared i n 1981, the work h ad been carried out ar ound 1978. In his paper , Goodman indicted that the results were based on an early work that appeared in 1978 only as an abstract. 1 paper , Goo d m an discussed “pseudo-complements” in a lattice theor e tical set- ting and called the topological system he obtains “an ti-intuitionistic logic”. In a recent work, Priest discuss ed the dual of the intuitionistic negation ope r ator and considered that o perator in topological framework (Priest, 2009). Simi- larly , Mortensen discussed topological separation pr inciples from a paraconsis- tent and paracomplete point of view and investigated the th eories in such spaces (Mortensen, 2000). Similar approache s from modal perspective was discussed by B´ eziau, t o o (B´ eziau, 2005 ). The organization of the paper is as follows. First, we will present the topo- logical basics of our subject in a nutshell. Then, we will point out the connec- tions between topological modal semantics and paraconsistency . A ft e rwards, we will make some further observations be tween different types of to pologies and p araconsistency . Finally , we will conclude with po ssible resear ch directions for f uture work under lining the fact that the field is rather unexplored. 2 Basics 2.1 Definitions The history of the t o pological semantics o f (mod al) logics can be traced back to early 1920s making it the fir st semantics for variet y of modal logics (Goldblatt, 2006 ). The major revival of the to p ological semantics of modal logics and its co nnec- tions with algebras, however , is due to McKinsey and T ar ski (McKinsey & T arski, 1946; McKinsey & T ar ski, 1944) . In this section, we will briefly me ntion t h e basics of topological semantics in order to be able build our future constructions. W e give two equivalent d efinitions of to pological spaces here f or our p urposes. Definition 2.1 (T opological Space) . The structure h S, σ i is called a topological space if it satisfies the f ollowing co nditions. 1. S ∈ σ and ∅ ∈ σ . 2. σ is closed under ar bitrary unions and under finite intersections. Definition 2.2 (T opological Space) . T he structure h S, τ i is call ed a top ological space if it satisfies the f ollowing co nditions. 1. S ∈ τ and ∅ ∈ τ . 2. τ is closed under finite unions and under arbitrary intersections. Collections σ and τ are called top o logies. The e lements of σ are called open sets whereas the elements of τ are called clo sed sets. A set is called o pen if its complement in the same topology is a closed set and vice versa. Functions can easily b e defined on top o logical spaces. Recall th at a function is called co ntinuous if t he inverse image of an open (respectively , closed) set is op e n ( respectively , closed), and a function is called open if the image of an open (respect ively , closed) set is open (r espectively , closed). Moreover , two 2 topological spaces are called homeomorphic if there is function from o ne t o the other which is a continuous bijection with a co ntinuous inverse. T wo continuous functions are called homo topic if t h ere is a continuous defo rmation between t he two. Homotopy is then an equivalence relation and gives rise to homo topy groups wh ich is a foundational subject in algebraic topology . 2.2 Semantics In o ur setting, we will denote the set of pro positional variables with P . W e will use the language o f pro positional modal logic with the mod ality  , and we will define the dual ♦ in the usual sense. Therefo r e, we will construct t he language of th e basic unimodal logic recursively in the standard fashion. In t opological semantics, the modal operator for necessitation corresponds to the topo logical interio r operator Int where Int ( O ) is the largest o pen set con- tained in O . F urthermore, one can d ually associate the topological cl osure op- erator Clo with the p ossibility modal o perator ♦ where the closure Clo ( O ) of a given set O is the smallest closed set that contains O . Before connecting topology and modal logi c, let us set a piece of notation and term inology . The extension, i.e. the points at which t he f ormula is satisfied, of a formula ϕ in the model M will be denot ed as [ ϕ ] M . W e will omit the superscript if t h e model we are working with is obvious. Moreover , b y a t heory , we will mean a d e ductively closed set o f formulae. The extensions of Boolean cases are obvious. However , the extension of a modal formula  ϕ will then be associated with an open set in the topo logical system. Thus, we will have [  ϕ ] = Int ([ ϕ ]) . Simil arly , we will put [ ♦ ϕ ] = Clo ([ ϕ ]) . This means that in the basi c setting, t o pological entities such as op en or closed sets appe ar only with mo dalities. However , we can take one step further and suggest that extension of any propositional variable will be an open set (Morte nsen, 2000). In that setting, conjunction and disjunction works fine for finite intersections and unions. Nev- ertheless, the negation can be difficult as the complement o f an open set is generally not an open set, thus may not be the extension of a formula in the language. F or this reason, we will need to use a new negation symbo l ˙ ∼ that returns the open co m p lement (interior of the comp lement) of a given set. W e call such systems pa ra complete to pological models . A similar idea can also be applied to closed set topologies wher e we stip- ulate that the extension of any pro positional variable will be a closed set. In order to be able to avoid a similar problem with the negation, we stipulate yet another negation operator which returns the closed complement (closure of the complement) of a given set. In this setting, we will use the symbol ∼ that re- turns the closed complement of a given set. W e call such systems paraco ns i stent topologic al m odels . Therefor e, we generat e two classes of logics with different (yet quite similar) syntax and semantics. P aracomplete topological models are generated with ˙ ∼ , ∧ and  from the set of prop o sitional variables P . P araconsistent topological 3 models ar e generated with ∼ , ∧ and  from the set of propo sitional variables P . Let us now make th e notation clear . W e will call the open set topologies which are the basis of par acomplete topo logical mo dels as σ . Moreover , we will call the closed set topo logy which is the b asis of paraconsistent t opological model as τ . There f ore, we make a clear cut distinction between paracomplete and paraconsistent logics and their models. P araconsistent topo logical logic, in t his sense, is the logic with the negation symbol ∼ whereas paracomplete topological logic is the logic with the negation symbo l ˙ ∼ . The p rior uses closed set to pologies [extension of every p roposition is closed] in its semantics, t he latter uses open set topologies [extension o f every proposition is open]. Now , let us consider the bo undary ∂ ( · ) of a set X where ∂ ( X ) is defined as ∂ ( X ) := Clo ( X ) − Int ( X ) . Consider now , for a given formuala ϕ , th e boundary of its extension ∂ ([ ϕ ]) in τ . Let x ∈ ∂ ([ ϕ ]) . Since [ ϕ ] is open, x / ∈ [ ϕ ] . Similarly , x / ∈ [ ˙ ∼ ϕ ] as the open co mplement is also o pen by definition. Thus, neither ϕ nor ˙ ∼ ϕ is true at the boundary . Thus, in τ , any theory that inc ludes the theory of th e propositions that ar e true at the boundary is incomplete. Consequently , we can make a similar o b servation abo ut the boundary points in σ . Now , take x ∈ ∂ ([ ϕ ]) where [ ϕ ] is a closed set in σ . By the above defini- tion, since we have x ∈ ∂ ([ ϕ ]) , we obtain x ∈ [ ϕ ] as [ ϕ ] is closed. Y et, ∂ [( ϕ )] is also included in [ ∼ ϕ ] which we have defined as a closed set. T hus, by the same reasoning, we conclude x ∈ [ ∼ ϕ ] . Thus, x ∈ [ ϕ ∧ ∼ ϕ ] yielding that x | = ϕ ∧ ∼ ϕ . Therefor e , in σ , any theory that includes the boundary points will be inconsis tent. In this respect, a paracomplete topological model M is a tuple M = h S, σ , V i where h S, σ i is an open set t opology . W e associate such model with a syntax that uses th e negation symbol ˙ ∼ . Similarly , the model M ′ = h T , τ , V i where h T , τ i is a closed set t opology is associated with a syn- tax t h at uses the negation symbol ∼ , and theref ore will be called paraconsistent topological model. In each cases, we call V is a valuation f unction taking pr opo- sitional variables f rom P and ret urns subsets of S or T respectively . So far , we h ave recalled how paracomplete and paraconsistent logics can be obtained in a topo logical setting. However , an immediate ob servation yields that since exte nsions of every formulae in σ (r espectively in τ ) are open (re- spectively , closed), the topologies which are obtained in both paraconsistent and paracomplete logics are discrete. This observation may trivialize the matter as, fo r instance, discre t e spaces with the sa me cardinality are homeomorphic. Proposition 2.3. Let M = h T , τ , V i and M ′ = h S, σ , V i be parac onsistent and paracompl ete t opologica l models respectively . If | S | = | T | , then there is a homeo- morphism from a pa raconsistent topologic al model to the para c omplete one, and vice versa . Moreov er , M amd M ′ satisfy t he same p o sitive formula. Proof . Since M and M ′ are paraconsistent and par aco m p lete respe ct ively , they have discrete topo logies. Since th eir space have t he same cardinality , h S, σ i and h T , τ i are homeomorphic. Call the homeomorphism f . Then, M , w | = ϕ if and only if M , f ( w ) | = ϕ for negation free ϕ . The proof of this claim is a standard induction on th e leng th of the formula. 4 Moreover , we also obse r ve that M , w | = ∼ ψ iff M ′ , f ( w ) | = ˙ ∼ ψ where ψ is negation-free.  3 T op ological Properties and P araconsistency In this section, we investigate the relation bet we en some basic topological prop- erties and paraconsistency . Mostly , we will consider the closed set topology τ with its negation operator ∼ as it is the natural candidate for the semantics for the par aconsistent t opological mode ls. Our work can b e seen as an extension of Mortensen’s earlier work (Mortensen, 2000). Here we extend his appr oach to some other topological prop erties and discus s the be havior of such spaces under some special functions. 3.1 Connectedness In the above section, we obser ved that boundary points play a cent r al ro le in paraconsistent th eories defined in t o pological spaces. One of t he immediate topological p roperties that comes to mind when one deals with bo undary is connectedness . A topological space is called connected if it is not the union o f two disjoint non-empty open sets. The same de fi nition works if we replace “open sets” with “closed sets”. F ormally , a set X is called connected if for two non-empty open ( r espectively closed) subsets A, B , we have X = A ∪ B ; then consequently we have A ∩ B 6 = ∅ . Moreover , in any connect ed top ological space, the only subsets with empt y boundary are the space itself and the empty set ( Bo ur b aki, 1966 ) . Moreover , yet another notion in geometric t opology is connected component which is a maximal connected subspace of a given space. In t his respect, we can separate to pological spaces into their connect ed compo- nents. Also, not e that connectedness is not definable in the (classi cal) modal language (Cate et a l. , 2009). Based o n t his de fi nition, now e st ab lish a relation between connected sp ace s and t heories. F o r this reason, we now define co nnect ed formulas as follows. Definition 3.1. A formula ϕ is called connect e d if for any two fo r mulae α 1 and α 2 with non-empty open (or dually , closed) extensions, if ϕ ≡ α 1 ∨ α 2 , t hen we have [ α 1 ∧ α 2 ] 6 = ∅ . W e will call a th eory T connected, if it is generated by a set of connect e d formulas. This definition identifies formulas with their exte nsions. Th erefore, a con- nected fo rmula ϕ is actually considered as the set [ ϕ ] at wh ich it is true. Based on the abo ve definition, we ob serve th e following . Proposition 3.2. Every connected formula is satisfia ble i n some connected (clas - sical) to pological space. Proof . Let ϕ be a connected formula and M = h W, ν, V i a (classical) top o logical space wher e for some w ∈ W , w | = ϕ . T hen, defi ne a connected subspace 5 M | ϕ = h W ϕ , ν ϕ , V ϕ i as follows. Let W ϕ = W ∩ [ ϕ ] M so that W ϕ = [ ϕ ] M | ϕ . Notice that W ϕ 6 = ∅ as w ∈ W ϕ . The to p ology ν ϕ then is de fined as follows ν ϕ = { O ∩ W ϕ : O ∈ ν } . It is easy to verify that ν ϕ is indeed a to pology ( in fact the induced topology), so we skip it. V aluation V is restricted in the usual sense. Now , we need to show that ν ϕ is connected. Now , take any two formulae α 1 and α 2 with non-empty open e xtensions in M | ϕ . Observe that if ϕ ≡ α 1 ∨ α 2 , then [ α 1 ∧ α 2 ] 6 = ∅ . Since W ϕ = [ ϕ ] , and the ex tensions [ α 1 ] and [ α 2 ] are nonempty by the condition, th is shows that the space W ϕ is connecte d with respect to t h e topology ν ϕ .  Note that the way we obtained a topological submodel is a rather standart method in modal log ics. A simila r theorem wit h in the context of d ynamic epis- temic logic showing the completeness of that logic in topological spaces also used t he same construction (Bas ¸ kent, 2011; ? ). Corollary 3.3. Every connected theory i s satisfiable in some connected (classical ) topologic al s pace. So far , we have made ob ser vations in classical t opological spaces. Neverthe- less, connected theories may be inconsistent or incomplete in some situations. Proposition 3.4. Every connected theory in paraco nsistent topo logical logic is i n- consistent. Moreover , every connected theory in paracomplete topol o gical logic is incomplete. Proof . Let T be a connected theo ry generated by a set of connected formulas { ϕ i } i , so ϕ i ∈ T for each i in a closed set to pology . By the earlier corollary , T is satisfiable in some connect e d space, say h W, σ i . Consider an arbitrary ϕ i from the basis o f T . Si nce it is a connected fo r mula, assume that we can write it as ϕ i ≡ α ∨ β f or [ α ∧ β ] 6 = ∅ . Le t x ∈ ∂ [ α ∧ β ] ⊆ [ ϕ i ] as we are in a closed set topology and therefore [ ϕ ] is closed. Thus, T includes ϕ i which in turn includes the theories at x . By our earlier remarks, th is makes T inconsistent in σ . As a special case, in paraconsistent topological logic, observe that if ⊤ ∈ T where [ ⊤ ] = W , t h en T is inconsistent as well. T ake ⊤ ≡ p ∨ ∼ p for some propositional variable p . Then, [ p ∧ ∼ p ] 6 = ∅ . Second part of the coro llary about t he incomplete theo ries and paracomplet e models can be proved similarly .  The converse direct ion is a bit more inte resting. Do connecte d spaces satisfy only th e connected formulas? Proposition 3.5. Let X be a connected topo logical spa ce of closed sets with a para - consistent topo l ogical model on it. Th en, th e only subtheory th at is no t inconsist ent is th e emp t y t heory . Proof . As we mentioned earlier , in any connected topo logical space, the only subsets with empty b oundary are the space itself and t h e empty set. Thus, all other subsets will have a boundary , and their the o ries will be inconsistent by 6 the ear lier observations. By the earlier proposition, the space itself produce an inconsistent theory . Th erefore, the only theory which is not inconsi stent is the empty th eory .  Based o n t his observation, we can show a more general result. Proposition 3.6. Let X be a connected topological s pace of closed sets. Then, for a collection of non-empty theories T 1 , . . . , T n with non-empty intersection T i T i , then we c onclude S i T i is inco nsistent. Proof . Each theory T i will have closed set of points X i that satisfie s it in the given topology . Since, T i T i 6 = ∅ , we observe T i X i 6 = ∅ . Therefor e, S i X i is connected and not equal to X . Thus, S i X i has a non-empty bo undary and the theories generated at t he boundary po ints will be inconsistent.  These observations hint out t h at boundary points play a significant ro le in paraconsistent topologies. A basic property of boundary gives us the following observation. Proposition 3.7. Let X be a n arbitra ry c onnected topol ogical s pace of c l osed sets. Define X = { C : C = B c for some B in X } . Then, X and X have t he sam e inconsistent boundary theories. Proof . R ecall that f or any set S , we have ∂ S = ∂ ( S c ) . Therefor e, the subsets in X and X will have the same boundary , thus the same boundary theories.  A simil ar result can be shown for paracomplete theories, and we leave it to the re ader . 3.2 Continuity A recent re search pro gram t h at considers topological mo dal logics with contin- uous functions were discussed in some early works ( ? ; Kreme r & Mints, 2005). In these work , they associated the modalities with continuous functions as such:  p = f − 1 ( p ) where  is the temporal ne x t time oper at or and f is a continuous function. In our work, we tend to diverge from the classical modal logi cal approach. Our focus will rather be the connection betwe e n cont inuous o r homeomorphic functions and modal logics with an hidden agenda of applying such approaches to paraco nsistent e p istemic logics in future wo rks. An immediate t heorem, which was st at ed and p roved in variety of diff e rent work, would also wor k for p araconsistent logics (Kre mer & Mints, 2005). Now , let us take two closed set topologies τ and τ ′ on a given set T and a homeomor - phism f : h T , τ i 7→ h T , τ ′ i . Akin to a previous theorem of Kremer and Mints, we have a simple way to associate the respective valuations betwe en two models M and M ′ which respectively dep end on σ and σ ′ so t hat we can have a truth preservation result. The r efore, define V ′ ( p ) = f ( V ( p )) . Then, we have M | = ϕ iff M ′ | = ϕ . 7 Theorem 3.8. Let M = h T , τ , V i and M ′ = h T , τ ′ , V ′ i be two parac onsistent topologic al mo dels ( where τ , τ ′ are closed set topo logies) with a homeomorphism f from h T , τ i to h T , τ ′ i . Define V ′ ( p ) = f ( V ( p )) . Then M | = ϕ iff M ′ | = ϕ for all ϕ . Proof . The proof is by induction on the complexity of the fo rmulae. Let M , w | = p for some pr o postional variable p . Then, w ∈ V ( p ) . Since we are in a paraconsistent topo logical model, V ( p ) is a closed set and since f is a homeo mo rphism f ( V ( p )) is closed as well, and f ( w ) ∈ f ( V ( p )) . Thus, M ′ , f ( w ) | = p . Converse direction is similar and b ased on th e fact that the inverse function is also continuous . Negation ∼ is less immediate. Let M , w | = ∼ ϕ . Ther efore, w is in t h e closure of the complement o f V ( ϕ ) . So, w ∈ Clo (( V ( ϕ )) c ) . Then, f ( w ) ∈ f ( Clo ( V ( ϕ )) c ) . Mo r eover , since f is bicontinuous as f is a home omorphism, we observe that f ( w ) ∈ Clo ( f (( V ( ϕ )) c )) . Then, by the induction hypot hesis, f ( w ) ∈ Clo (( V ′ ( ϕ )) c ) yielding M ′ , f ( w ) | = ∼ ϕ . Converse direct ion is also simi- lar . W e l eave the conjun ction case t o the r eader and pr o ceed to the mo dal case. Assume M , w | = ♦ ϕ . Thus, w ∈ V ( ♦ ϕ ) . Thus, w ∈ Clo ( V ( ϕ )) . Then, f ( w ) ∈ f ( Clo ( V ( ϕ ))) . Since f is a homo morphism, we have f ( w ) ∈ Clo ( f ( V ( ϕ ))) . By the induction hypot hesis, we then deduce that f ( w ) ∈ Clo ( V ′ ( ϕ )) which in turn yields that f ( w ) ∈ V ′ ( ♦ ϕ ) . T hus, we deduce M ′ , f ( w ) | = ♦ ϕ . Converse direction is as expecte d and we leave it to the reader .  Notice that the abo ve theorem also works in paracomp lete to pological mod- els, and we leave t he details t o the reader . Assuming that f is a home omorphism may seem a b it strong. W e can then seperate it into two chunks. One direction of the biconditional can be satisfied by continuity whe reas the o ther d ir e ction is satisfied b y t he ope nness of f . Corollary 3.9. Let M = h T , τ , V i and M ′ = h T , τ ′ , V ′ i be two paraconsis tent topologic al models wit h a continuous f from h T , τ i to h T , τ ′ i . Define V ′ ( p ) = f ( V ( p )) . Then M | = ϕ im p lies M ′ | = ϕ for all ϕ . Corollary 3.10. Let M = h T , τ , V i and M ′ = h T , τ ′ , V ′ i be two pa ra consistent topologic al m odels with an open f from h T , τ i to h T , τ ′ i . Defi ne V ′ ( p ) = f ( V ( p )) . Then M ′ | = ϕ implies M | = ϕ for all ϕ . Proofs o f both corollaries depend on the fact t hat Clo operat o r commutes with continuous functions in one direction, and it commutes with op e n f unc- tions in t he other direction. Furthermore , similar corollaries can be given for paracomplete frameworks as the Int oper ator also commutes in one direction under similar assumptions, and we leave it to the reader as well. Furthermore , any topological operator that commutes with co ntinuous, open and homeomorphic functions will re fle ct th e same idea and prese r ve the truth 2 . Therefor e, the se results can easily be g eneralized. 2 Thanks to Chris Mortensen for pointing this ou t. 8 W e can now take one step f urther to discuss homotopies in paraconsistent topological modal mode ls. T o our knowledge, the role of h o motopies as tr ans- formations between truth preserving continuous isomorphisms or bisimulations under some restrictions has not yet be en discussed within the field of t o pologi- cal models of classical modal logic. The refore, we be lieve our treatment is the first introduction of homotopies in topological semantics of modal logics. The reason why we start from paraconsistent (paracomp lete) modal logics is the simple fact that th e extension of each pr opositional let t er is a closed ( open) se t which makes our task relatively easy and str aightf o rward. R ecall that a hom otopy is a description of how two continuous f unction f rom a topological space to anot her can b e deformed to e ach other . W e can now state the f o rmal de finition. Definition 3.11. Let S and S ′ be two topological spaces with co nt inuous func- tions f , f ′ : S → S ′ . A homo t opy between f and f ′ is a continuous f unction H : S × [0 , 1] → S ′ such th at if s ∈ S , then H ( s, 0) = f ( s ) and H ( s, 1) = g ( s ) In othe r word s, a ho motopy between f and f ′ is a family of continuous functions H t : S → S ′ such that for t ∈ [0 , 1] we have H 0 = f and H 1 = g and the map t → H t is co ntinuous fr om [0 , 1 ] to the space of all continuous functions f r om S to S ′ . Notice that homot opy relation is an equivalence relation. Thus, if f and f ′ are homoto p ic, we denote it with f ≈ f ′ . But, why d o we need homotopies? W e will now use homotop ies to obtain a generalization o f Theorem 3.8 . Assume t hat we are given t wo to pological spaces h S, σ i and h S, σ ′ i and a family o f continuous functions f t for t ∈ [0 , 1] . Define a model M as M = h S, σ, V i . Then, for each f t with t ∈ [0 , 1] , de fine M t = h S, σ , V t i whe re V t = f t ( V ) . T hen, by The orem 3.8, we observe that M | = ϕ iff M t | = ϕ . Now , what is the relation among M t s? The obvious answer is that their valuation for m a homotopy equivalance class. Let us now see how it works. Define H : S × [0 , 1 ] → S ′ such that if s ∈ S , then H ( s, 0) = f 0 ( s ) and H ( s, 1) = f 1 ( s ) . T hen, H is a homotopy . There fore, given a (paraconsistent) topological modal model M , we g enerate a f amily of models { M t } t ∈ [0 , 1] whose valuations are generate d by homotopic functions. Definition 3.12. Given a model M = h S, σ , V i , we call the family of models { M t = h S, σ, V t i} t ∈ [0 , 1] generated by homotopic functions and M homotopic models. In t h e generation, we put V t = f t ( V ) . Theorem 3.13. Homot opic pa raconsist ent (paraco mplete) t o pological models sa t- isfy t he same m odal formul ae. Proof . See the above discussion.  In the above discussions, we have focused on continu ous functions and the homotopies the y generate. W e can also d iscuss ho meomorphisms and their homotopies which generate homotop y equivalences between spaces. In that case, homoto p ic equivalent spaces can be continuously def ormed to each o ther . 9 This would give us, under t he cor rect valuation, a stronger notion of bisim- ulation t h at we call co ntinuous top o-bisimulatio n . W e will first start with the definition of t o po-bisimulation before introducing continuous topo-bisimulation (Aiello & van Benthem, 2002 ). Definition 3.14. Let two (classical) topological models h S, σ , V i and h S ′ , σ ′ , V ′ i be given, a topological bisimulation is a relation on S × S ′ , and when two points s from S and s ′ from S ′ are topo-bisimular , they satisfy the following conditions. 1. The points s and s ′ satisfy t h e same propositional variables. 2. F or s ∈ O ∈ σ , there is O ′ ∈ σ ′ such that s ′ ∈ O ′ and ∀ t ′ ∈ O ′ , ∃ t ∈ O such that t and t ′ are topo-bisimular 3. F or s ′ ∈ O ′ ∈ σ ′ , t here is O ∈ σ such that s ∈ O and ∀ t ∈ O , ∃ t ′ ∈ O ′ such that t and t ′ are topo-bisimular Now we can extend it to continuity . Definition 3.15. Let M = h S, σ , V i and M ′ = h S ′ , σ ′ , V i be t wo paraconsistent (paracomplete) topological models. W e say M , w and M ′ , w ′ are continuously topo-bisimular (denoted M , w ⇋ M ′ , w ′ ) if M , w and M ′ , w ′ are top o-bisimular and ther e is a h omeomorphism f betwee n h S, σ i and h S ′ , σ ′ i such that V ′ = f ( V ) . Note that in the above definition, we need a stronger notion of homeomor- phism r ather than just co ntinuity as the bisimulation is a symmetric relation. Theorem 3.16. Continuously bisimu l ar states sati s fy the same modal formulae. Proof . The pr o of is an induction on the complexity of the formulas in the stan- dard sense, and uses T heorem 3.8.  What about the conver se ? Can we have a p r operty akin to Hennesy- Millner property so that for some topologies that satisfy exactly the same formulae, we can construct a homeomorphism in be t ween? Clearly , answer to this ques- tion is positive if we are in finite spaces, and the construction is essentially the same as in the classical case. W e r efer the inter ested re ader to a textb ook treat- ment of classical modal logic to see how Hennesy-Millner proper t y is treated (Blackburn et al. , 2001). Now , mathematically oriented r eader might anticipate a second move to- wards homot opy groups and the ir use in modal logic. Note that homotop y groups essentially classifies the spaces with regard to the ir continuous d e forma- bility to each other , and it seems feasible to import such a concept to modal logics. Nevertheless, in order not t o diverge our focus here, we will not pursue that pat h here, and leave if for a future work. 10 3.3 Modal Direction This section of the pape r wil l briefly review the modal approach e s to the p ara- consistency in order to make our work more self-contained. One possible modal inter p retation of paraconsistency focuses on the nega- tion o perator (B´ eziau, 200 5). Under the usual alethic reading of  and ♦ modalities, one can define an additional operato r ∼ as ¬  , or equivalently ∼ ϕ ≡ ♦ ¬ ϕ . Notice that this definition corresponds to our earlier definition of negation being the closed comp lement. F or th is interpretat ion, re call t hat ♦ operator nee d s t o be takes as t he Clo operator . The Kripkean semantics of t he new paraco nsistent negation operato r ∼ is as follows (B´ eziau, 2005). Let us take a modal model M = h W, R, V i where R is a binary r elation on t he non- empty set of worlds W and V is valuation. T ake an arbitrary stat e w ∈ W . ∼ ϕ is false at w if and only if ϕ is true at e ve ry v with wR v More t echnically , we have the following reasoning. w 6| = ∼ ϕ iff w 6| = ¬  ϕ w | =  ϕ ∀ v . ( wR v → v | = ϕ ) w | = ϕ Furthermore , as it was obser ved, ∼ modality is indeed S5, and fur t hermore an S5 logic can be given by taking ∼ as the p r imitive negation symbol with the intended interpret ation. Nevertheless, for our current purpo ses, S4-character of that modality is sufficient, and we will not go into the details of such an S5 construction. W e refer the interested reader to the following references for a further investigation of this subect (B´ eziau, 2002; B´ eziau, 2005). Moreover , it is easy to notice the similarity of modal negation we presented here and the top o logical negation th at we used throughout his paper . Therefore, it is a nice exercise to impor t our to p ological results from topological semantics to Kripke semantics with the modal negation at hand. T h erefore, one can de fine a modal negation in Kripke models that r e flect the exact same negation that we used f or par aco nsistent t o pologies. F or this reason, we can offe r a transformation from topological models to Kripke models which is similar to the standard translation bet we en classical topological models and Kripke models (Aiello & van Benthem, 2002). Given a topological paraconsistent model M = h S, σ, V i , we put sR σ t when s ∈ Clo ( t ) to get a Kr ipke mo d e l M σ = h S, R σ , V i . This tr ansfo rmation is truth preserving. Theorem 3.17. Given a topo logical paraco nsistent model M , if M , w | = ϕ t hen M σ , w | = ϕ where M σ is obtained from M by the transforma t ion that w R σ v when w ∈ Clo ( v ) . Proof . Induction on t he complexity o f the formulae, and the proof is a careful interplay between different negations. W e will only show it for negation then. 11 Note that we use ∼ f or bot h paraconsistent Kripkean negation and paraconsis- tent closed set negation; nevertheless, the contex t will make it clear which one we mean. Let M = h T , τ , V i be given. Assume M , w | = ∼ ϕ . Since, the topo logical negation ∼ is the closure of the set theoretical complement , we o b serve that M , w | = ♦ ¬ ϕ . T herefore , for every closed set U ∈ τ , ther e is a point v ∈ U such that M , w | = ¬ ϕ . Observe that si nce v ∈ U for closed U , we observe that w ∈ Clo ( v ) . Then, put w R τ v . So, in the model M τ = h T , R τ , V i , we h ave M τ , w | = ∃ v ( w R τ v and M τ , v | = ¬ ϕ ) . Then, by the usual semantics of modal logic, we observe M τ , w | = ♦ ¬ ϕ which is not hing but M τ , w | = ¬  ϕ . F inally , by definition, we conclude M τ , w | = ∼ ϕ .  A well-known tr ansfo rmation from Kripke fr ames generat e a topological space: in that case, op ens are downward (or upward) closed sets ( subtrees) in the Kripke model. It is also e asy to prove that this transformation respects the tr uth of t he fo r mulae. Theorem 3.18. Given a parac onsistent Kripke model M , if M , w | = ϕ then M R , w | = ϕ where M R is obtained from M by the t ransformatio n that the clo sed sets a re downward closed subsets with respect t o the access i bility relation R . This establishes the connection bet ween paraconsistent t o pological models and p araconsistent Kripke mo dels. 4 Conclusion and F uture W ork In this work, we focused on the connection between top ological spaces and paraconsistent logic. There are many o pen questions that we h ave left for fur- ther wo rk. Some of them can be su mmarized as follows. • How can we logically define ho motopy and cohomot o py groups in para- consistent or paracomplete topological modal models? • How would p araconsistency be affect ed under to pological prod ucts? • What is the (paraconsistent) logic of r egular sets? Aforementioned questions pose yet another research program in which al- gebraic t opological and alg ebraic geometrical ideas are utilized in non-classical modal logics. T he interaction between truth and in such f r ameworks ex hibits a nove l line of research . More o ver , region based modal logi cs have presented variety of results abo ut the logic of space (Pratt-Hartman, 2007). Considering their use of regular sets within the fr amework of region based modal logics, it is not difficult to see a connection between region based modal logics and paraconsistent logics. Furthermore , the strong algebraic connection bet ween variety of topological models pose a very interesting approach. Considering the dual relation be tween 12 intuitionistic and paraconsistent logics and their r e spective algebraic st r uctures being Heyting and Brouwer algebras, t h eir connection in the modal fr ame wo rk was also investigated (Rauszer , 1977 ). Theref ore, connection topological ideas with the existing algebraic work is yet anot h er research dire ction for future work. Y et another p ossible app lications of such systems is epistemic logi cs where the knowers or agents can have inconsistent or incomplete belief basis. T he in- tuitive connection be tween AG M update and paraconsistency within this frame- work is yet to be established. Moreover , within the domain of dynamic e pistemic logic, paraconsistent announcements can be considered wh e re agents may have inconsistent knowledge set, and ye t maintain a sensible way to make deduc- tions. W e leave such stimulating dis cussions to future work. 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