Some Non-Classical Methods in(Epistemic) Modal Logic and Games:

A Proposal

Can Baskent

June 21st, 2011

Summer Solstice - 2011

Abstract

In this proposal, we discuss several non-classical frameworks, and theirapplications in epistemic modal logic. We largely consider topological se-mantics, as opposed to widely used Kripke semantics, paraconsistent sys-tems, as opposed to consistent systems, and non-well-founded sets, as op-posed to ZF(C) set theory. We discuss topological public announcement log-ics, introduce homotopies to modal logic, reason about topological seman-tics for paraconsistent logics, and introduce non-well-founded type spacesfor games. Finally, we discuss several popular game theoretical issues: theBrandenburger - Keisler paradox, and strategy logic whose primitives arestrategies. Then, we conclude with future research directions.

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Contents

1 Dynamic Epistemic Logic and Topologies 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Geometric Models . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Subset Space PAL . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Topological PAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Paraconsistent Topologies and Homotopies 162.1 What is Paraconsistency? . . . . . . . . . . . . . . . . . . . . . . . 162.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Topological Properties and Paraconsistency . . . . . . . . . . . . . 172.4 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 A Modal Logical Application . . . . . . . . . . . . . . . . . . . . . 212.6 An Epistemic Application . . . . . . . . . . . . . . . . . . . . . . . 23

3 A Paradox in Game Theory 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Non-well-founded Set Theoretical Approach . . . . . . . . . . . . 283.3 Paraconsistent Approach . . . . . . . . . . . . . . . . . . . . . . . 31

4 Strategies 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Strategy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Restricted strategy logic . . . . . . . . . . . . . . . . . . . . . . . 41

References 47

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1 Dynamic Epistemic Logic and Topologies

1.1 Introduction

Public announcement logic is a well-known example of dynamic epistemic log-ics (Plaza, 1989; van Ditmarsch et al., 2007). Dynamic epistemic logics set outto formalize knowledge and knowledge changes in (usually) multi-agent settingby defining different ways of model updates and interaction among the agents.The contribution of public announcement logic (PAL, henceforth) to the field ofknowledge representation is mostly due to its succinctness and clarity in reflect-ing the simple intuition as to how epistemic updates work in some situation.Moreover, as it will be clear in the next section, PAL is not more expressivenessthan the basic epistemic logic. PAL updates the model by the announcementsmade by a truthful agent who can be one of the agents/knowers or “God”. Af-ter the truthful announcement, the model is updated by eliminating the statesthat do not satisfy the announcement. Public announcement logic has manyapplications in the fields of formal approaches to social interaction, dynamiclogics, knowledge representation and updates (Balbiani et al., 2008; Baltag &Moss, 2004; van Benthem, 2006; van Benthem et al., 2005). Extensive appli-cation of PAL to different fields and frameworks has made PAL a rather familiarframework to many researchers. Especially, analyzing games such as “muddychildren” made PAL a rather popular framework.

Virtually almost all applications of PAL make use of Kripke models for knowl-edge representation. However, as it is very well known, Kripke models are notthe only representational tool for modal and epistemic logics.

In this section, we will consider PAL in two different geometrical frame-works: topological modal logic and subset space logic. Topological models arenot new to modal logics, indeed they are the first models for modalities (McK-insey & Tarski, 1944; McKinsey & Tarski, 1946; Goldblatt, 2006). The pastdecades have witnessed a revival of academic interest towards the topologi-cal models for modal logics in many different frameworks (Aiello et al., 2003;van Benthem & Bezhanishvili, 2007; van Benthem et al., 2006; Bezhanishvili& Gehrke, 2005). However, to the best of our knowledge, topological modelshave not been applied to dynamic epistemic logics. But, there has been someinfluential work on common knowledge in topological models which motivatedthe current work (van Benthem & Sarenac, 2004). In the aforementioned refer-ence, it was shown that the different definitions of common knowledge divergein topological models even though these definitions are equivalent in Kripkestructures, based on Barwise’s earlier investigation (Barwise, 1988). Neverthe-less, the authors did not seem to take the next immediate step to discuss dy-namic epistemologies in that framework. This is one of our goals in this paper:to apply topological reasoning to dynamic epistemological cases and presentthe immediate completeness results. The second framework that we will dis-cuss, subset space logic, is a rather weak yet expressive geometrical structuredispensing with the topological structure (Moss & Parikh, 1992; Parikh et al.,2007). Subset space logic has been introduced to reason about the notion of

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closeness and effort in epistemic situations. In this paper, we will also definePAL in subset space logic with its axiomatization and present the completenessof PAL in subset space logics improving the results based on an earlier work(Baskent, 2007).

The present section is organized as follows. First, we will introduce the ge-ometrical frameworks that we will need: topological spaces and subset spaces.Then, after a brief interlude on PAL, we will give the axiomatizations of PAL insuch spaces, and their completeness. The completeness proofs are rather im-mediate - which is usually the case in PAL systems. Then, we will make someobservations on PAL in geometric models. Our observations will be about thestabilization of updated models, backward induction in games and persistency.

1.2 Geometric Models

In this section, we will briefly recall the geometric models for some modal logics.What we mean by geometric models is topological models and subset space logicmodels as they are inherently geometrical structures. We will first start withtopological models and their semantics, and then discuss subset space models.

1.2.1 Topological Semantics for Modal Logic

Topological interpretations for modal logic historically precede the relationalsemantics (McKinsey & Tarski, 1944; Goldblatt, 2006). Moreover, as we willobserve very soon, topological semantics is arithmetically more complex thanrelational semantics. Let us start by introducing the definitions.

Definition 1. A topological space S = 〈S, σ〉 is a structure with a set S and acollection σ of subsets of S satisfying the following axioms:

1. The empty set and S are in σ.

2. The union of any collection of sets in σ is also in σ.

3. The intersection of a finite collection of sets in σ is also in σ.

The collection σ is said to be a topology on S. The elements of S are calledpoints and the elements of σ are called opens. The complements of open setsare called closed sets. Our main operator in topological spaces is called interioroperator I which returns the interior of a given set. The interior of a set is thelargest open set contained in the given set. A topological model M is a triple〈S, σ, v〉 where S = 〈S, σ〉 is a topological space, and v is a valuation functionsending propositional letters to subsets of S, i.e. v : P → ℘(S) for a countableset of propositional letters P .

The basic modal language L has a countable set of proposition letters P , atruth constant >, the usual Boolean operators ¬ and ∧, and a modal operator�. The dual of � is denoted by ♦ and defined �ϕ ≡ ¬♦¬ϕ. When we are intopological models, we will use the symbol I for � after the interior operatorfor intuitive reasons, and to prevent any future confusion. Likewise, we will use

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the symbol C for ♦. The notation M, s |= ϕ will read the point s in the modelM makes the formula ϕ true. We will call the set of points that satisfy a givenformula ϕ in modelM the extension of ϕ, and denote as (ϕ)M.

In topological models, the extension of a Boolean formula is obtained in thefamiliar sense. The extension of a modal formula in model M, then, is givenas follows (Iϕ)M = I((ϕ)M) - namely, the extension of Iϕ is the interior of theextension of ϕ. Now, based on this framework, the model theoretical semanticsof modal logic in topological spaces is given as follows.

M, s |= p iff s ∈ v(p) for p ∈ PM, s |= ¬ϕ iff M, s 6|= ϕM, s |= ϕ ∧ ψ iff M, s |= ϕ andM, s |= ψM, s |= Iϕ iff ∃U ∈ σ(s ∈ U ∧ ∀t ∈ U, M, t |= ϕ)M, s |= Cϕ iff ∀U ∈ σ(s ∈ U → ∃t ∈ U, M, t |= ϕ)

A few words on the semantics are in order here. The necessity modality Iϕsays that there is an open set that contains the current state and the formula ϕis true everywhere in this open. Obviously, this is a rather complex statement,first, it requires us to determine an open set that would work, and then checkwhether each point in this open set satisfies the given formula or not. On theother hand, the possibility modality Cϕ manifests the idea that for every openset that includes the current state, there is point in the same set that satisfies ϕ.

It is been shown by McKinsey and Tarski that the modal logic of topologicalspaces is S4 (McKinsey & Tarski, 1944). Moreover, the logic of many other topo-logical spaces has also been investigated (Aiello et al., 2003; Cate et al., 2009;Bezhanishvili et al., 2005; van Benthem et al., 2006; van Benthem & Bezhan-ishvili, 2007). Moreover, recently, the topological properties of paraconsistentsystems have also been investigated (Baskent, 2011; Mortensen, 2000).

The proof theory of the topological models is as expected: we utilize modusponens and necessitation. The modal logic S4 is long known to be completewith respect to the given semantics.

1.2.2 Subset Space Logic

Subset space logic (SSL, henceforth) was presented in early 90s as a bimodallogic to formalize reasoning about sets and points with an underlying motiva-tion from epistemic logic (Moss & Parikh, 1992). One of the modal operators ofSSL is intended to quantify over the sets (�) whereas the other modal operatorwas intended to quantify in the current set (K). The underlying motivation forthe introduction of these two modalities is to be able to speak about the notionof closeness. In this context, the K operator is intended to be the knowledgeoperator (for one agent only, as SSL is originally presented for single-agent),and the � modality is intended for the effort modality. Effort can correspond tovarious things: computation, observation, approximation - the procedures thatcan result in knowledge increase.

The language of subset space logic LS has a countable set P of propositionletters, a truth constant >, the usual Boolean operators ¬ and ∧, and two modal

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operators K and �. A subset space model is a triple S = 〈S, σ, v〉 where S is anon-empty set, σ ⊆ ℘(S) is a collection of subsets (not necessarily a topology),v : P → ℘(S) is a valuation function. Semantics of SSL, then is given inductivelyas follows.

s, U |= p iff s ∈ v(p)s, U |= ϕ ∧ ψ iff s, U |= ϕ and s, U |= ψs, U |= ¬ϕ iff s, U 6|= ϕs, U |= Kϕ iff t, U |= ϕ for all t ∈ Us, U |= �ϕ iff s, V |= ϕ for all V ∈ σ such that s ∈ V ⊆ U

The duals of � and K are ♦ and L respectively and defined as usual. The tuple(s, U) is called a neighborhood situation if U is a neighborhood of s, i.e. ifs ∈ U ∈ σ. The axioms of SSL reflect the fact that the K modality is S5-like whereas the � modality is S4-like. Moreover, we will need an additionalaxiom to state the interaction between those two modalities: K�ϕ → �Kϕ.The rules of inference are as expected: modus ponens and necessitation forboth modalities. Therefore, subset space logic is complete and decidable (Moss& Parikh, 1992; Georgatos, 1994; Georgatos, 1997).

Note that SSL is originally proposed as a single-agent system. There havebeen some attempts in the literature to suggest a multi-agent version of it, butto the best of our knowledge, as of today, there is no intuitive and clear presen-tation of multi-agent version of SSL.

1.2.3 Public Announcement Logic

Public announcement logic is a way to represent changes in knowledge. Theway PAL updates the epistemic states of the knower is by “state-elimination”. Atruthful announcement ϕ is made, and consequently, the agents updates theirepistemic states by eliminating the possible states where ϕ is false (Balbianiet al., 2007; Balbiani et al., 2008; van Ditmarsch et al., 2007).

Public announcement logic is typically interpreted on multi-modal (or multi-agent) Kripke structures (Plaza, 1989). Notationwise, the formula [ϕ]ψ is in-tended to mean that after the public announcement of ϕ, ψ holds. As usual, Ki isthe epistemic modality for the agent i. Likewise, Ri is the epistemic accessibilityrelation for the agent i. The language of PAL will be that of multi-agent (multi-modal) epistemic logic with an additional public announcement operator [∗]where ∗ can be replaced with any well-formed formula in the language of basicepistemic logic. To see the semantics of PAL, take a modelM = 〈W, {R}i∈I , V 〉where i denotes the agents and varies over a finite set I . For atomic proposi-tions, negations and conjunction the semantics is as usual. For modal operators,we have the following semantics.

M, w |= Kiϕ iffM, v |= ϕ for each v such that (w, v) ∈ RiM, w |= [ϕ]ψ iffM, w |= ϕ impliesM|ϕ,w |= ψ

Here, the updated modelM|ϕ = 〈W ′, {R′i}i∈I , V ′〉 is defined by restrictingM to those states where ϕ holds. Hence, W ′ = W ∩(ϕ)M; R′i = Ri∩(W ′×W ′),

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and finally V ′(p) = V (p)∩W ′. The axiomatization of PAL is the axiomatizationof S5n with the following axioms.

1. [ϕ]p↔ (ϕ→ p)

2. [ϕ]¬ψ ↔ (ϕ→ ¬[ϕ]ψ)

3. [ϕ](ψ ∧ χ)↔ ([ϕ]ψ ∧ [ϕ]χ)

4. [ϕ]Kiψ ↔ (ϕ→ Ki[ϕ]ψ)

The additional rule of inference which we will need for announcement modalityis called the announcement generalization and is described as expected: From` ψ, derive ` [ϕ]ψ.

PAL is complete and decidable. The completeness proof is quite straightfor-ward. Once the soundness of the given axiomatization is proved, then it meansthat every complex formula in the language of PAL can be reduced to a formulain the basic language of (multi-agent) epistemic logic. Since S5n epistemic logicis long known to be complete, we obtain the completeness of PAL.

Notice again that in this section, we have defined PAL in Kripke structures byfollowing the literature. In the next section, we will see how PAL is defined ingeometrical models. We will start with SSL and proceed to topological modelswith further observations.

1.3 Subset Space PAL

In SSL, we depend on neighborhoods instead of the epistemic accessibility rela-tions. Therefore, if we want to adopt public announcement logic to the contextof subset space logic, we first need to focus on the fact that the public announce-ments shrink the observation sets for each agent. Here, we model how a singleagent in SSL update his model after a public announcement.

Let us set a piece of notation. For a formula ϕ, let us (ϕ)S be the extensionof ϕ. In SSL, (ϕ)S = {(s, U) ∈ S × σ : s ∈ U, (s, U) |= ϕ} where S = 〈S, σ v〉is a subset model. Define (ϕ)S1 := {s : s, U ∈ (ϕ)S for some U 3 s}, and(ϕ)S2 := {U : s, U ∈ (ϕ)S for some s ∈ U} be the projection of (ϕ)S to thesecond coordinate. We will drop the superscript when it is obvious.

Now, assume that we are in a subset space model S = 〈S, σ, v〉. Then, afterpublic announcement ϕ, we will move to another subset space model Sϕ =〈S|ϕ, σϕ, vϕ〉 where S|ϕ = (ϕ)1 and σϕ is the reduced collection of subsets afterthe public announcement ϕ, and vϕ is the reduct of v on S|ϕ. The crucial pointis to construct σϕ. As we need to get rid of the refutative states, we eliminate thepoints which do not satisfy ϕ for each observation set U in σ. We will disregardthe empty set as no neighborhood situations can be formed with empty set.Hence, σϕ = {Uϕ : Uϕ = U ∩ (ϕ)2 6= ∅, for each U ∈ σ}. In other words, werelativize the topologic σ with respect to the announcement by keeping it inmind that the truth in SSL is defined with respect to tuples of points and sets.

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But then, how would the neighborhood situations be affected by the publicannouncement? The corresponding semantics can be suggested as follows:

s, U |= [ϕ]ψ iff s, U |= ϕ implies s, Uϕ |= ψ

Intuitively, remembering the fact that the truth is defined with respect totuples in SSL, we revise the subset component of the tuple by shrinking it. Forexample, consider that a policeman makes an observation and measures thespeed of the passing cars with a device which has an error range of +/− 5 mph(Taken from (?)). In one instance, he measures the speed of a car, and readsthat the speed lies in the interval [45, 55]. Yet, the policeman does not exactlyknow the speed. Therefore, he is at the neighborhood situation (v, [45, 55]).Then, let us assume that he hears an announcement, say, a message he receivesvia the police radio, saying that “the speed of that particular car is faster than orequal to 48 mph”. In other words, “v ∈ [48,∞)”. Then, the policeman updateshis situation to (v, [48, 55]) since the announcements are assumed to be truthful.

However, if there is an announcement such as “no car is slower than 52mph”, then he cannot update his situation to (50, [52, 55]) as 50 /∈ [52, 55]. There-fore, the latter announcement is not truthful in his situation.

Before checking whether this semantics satisfies the axioms of public an-nouncement logic, let us give the language and semantics of the topologic PAL.The language of the topologic public announcement logic interpreted in subsetspaces is given as follows:

p | ⊥ | ¬ϕ | ϕ ∧ ψ | �ϕ | Kϕ | [ϕ]ψ

Now, let us consider the soundness of the following axioms of basic PAL thatwe discussed earlier.

1. [ϕ]p↔ (ϕ→ p)

2. [ϕ]¬ψ ↔ (ϕ→ ¬[ϕ]ψ)

3. [ϕ](ψ ∧ χ)↔ ([ϕ]ψ ∧ [ϕ]χ)

4. [ϕ]Kψ ↔ (ϕ→ K[ϕ]ψ)

Theorem 2. The axioms of the basic PAL are sound in subset space logic.

Proof. As the atomic propositions do not depend on the neighborhood, the firstaxiom is satisfied by the subset space semantics of public announcement modal-ity. To see this, assume s, U |= [ϕ]p. So, by the semantics s, U |= ϕ impliess, Uϕ |= p. So, s ∈ v(p). So for any set V where s ∈ V , we have s, V |= p.Hence, s, U |= ϕ implies s, U |= p, that is s, U |= ϕ → p. Conversely, assumes, U |= ϕ→ p. So, s, U |= ϕ implies s ∈ v(p). As s, U |= ϕ, s will lie in Uϕ, thus(s, Uϕ) will be a neighborhood situation. Thus, s, Uϕ |= p. Then, we concludes, U |= [ϕ]p.

The axioms for negation and conjunction are also straightforward formulamanipulations and hence skipped.

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The important reduction axiom is the knowledge announcement axiom. As-sume, s, U |= [ϕ]Kψ. Suppose further that s, U |= ϕ. Then we have the follow-ing.

s, U |= [ϕ]Kψ iff s, Uϕ |= Kψiff for each tϕ ∈ Uϕ, we have tϕ, Uϕ |= ψiff for each t ∈ U , t, U |= ϕ

implies t, U |= [ϕ]ψiff s, U |= K(ϕ→ [ϕ]ψ)iff s, U |= K[ϕ]ψ

Hence, the above axioms are sound for the subset space semantics of publicannouncement logic. �

Now, recall that SSL has an indispensable modal operator �. One can won-der whether we can have a reduction axiom for it as well. We start by consid-ering the statement [ϕ]�ψ ↔ (ϕ → �[ϕ]ψ). Assume, s, U |= [ϕ]�ψ. Supposefurther that s, U |= ϕ. Then, we deduce the following.

s, U |= [ϕ]�ψ iff s, Uϕ |= �ψiff for each Vϕ ⊆ Uϕ we have s, Vϕ |= ψiff for each V ⊆ U , s, V |= ϕ

implies s, V |= [ϕ]ψiff s, U |= �(ϕ→ [ϕ]ψ)iff s, U |= �[ϕ]ψ

Now, it is easy to see that the following axiomatize the SSL-PAL together withthe axiomatization of SSL:

1. [ϕ]p↔ (ϕ→ p)

2. [ϕ]¬ψ ↔ (ϕ→ ¬[ϕ]ψ)

3. [ϕ](ψ ∧ χ)↔ ([ϕ]ψ ∧ [ϕ]χ)

4. [ϕ]Kψ ↔ (ϕ→ K[ϕ]ψ)

5. [ϕ]�ψ ↔ (ϕ→ �[ϕ]ψ)

Referring to the above discussions, the completeness of topologic PAL followseasily.

Theorem 3. Topologic PAL is complete with respect to the axiom system givenabove.

Proof. By reduction axioms we can reduce each formula in the language of topo-logic PAL to a formula in the language of SSL as we have shown above. As SSLis complete, so is PAL in subset space models. �

By the same idea, we can import the decidability result.

Theorem 4. PAL in subset space models is decidable.

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1.4 Topological PAL

1.4.1 Single Agent Topological PAL

We can use similar ideas to give an account of PAL in topological spaces. LetT = 〈T, τ, v〉 be a topological model and ϕ be a public announcement. We nowneed to obtain the topological model Tϕ which is the updated model after theannouncement. Recall that we denote the extension of a formula ϕ in modelMby (ϕ)M, so (ϕ)M = {w :M, w |= ϕ}.

Define Tϕ = 〈Tϕ, τϕ, vϕ〉 where Tϕ = T ∩ (ϕ), τϕ = {O ∩ Tϕ : O ∈ τ} andvϕ = v ∩ Tϕ. We now need to verify that τϕ is indeed a topology. However, notethat τϕ need not be a subtopology of τ .

The intuition as to how PAL works in topological spaces is similar to how itworks in Kripke spaces. A truthful announcement is made by an external agent,and the agent(s) update their topology by reducing their topological model rel-ative to the announcement. Now, we need to check whether the newly obtainedmodel is a topology or not.

Proposition 1. If τ is a topology, then τϕ = {O ∩ Tϕ : O ∈ τ} is a topology aswell.

Note that τϕ is the induced topology, and hence we skip the proof.Now, we can give a semantics for the public announcements in topological

models.T , s |= [ϕ]ψ iff T , s |= ϕ implies Tϕ, s |= ψ

In a similar fashion, we can expect the reduction axioms to work in topo-logical spaces. The reduction axioms for atoms and Booleans are quite straight-forward. So, consider the reduction axiom for the interior modality [ϕ]Iψ ↔(ϕ→ I[ϕ]ψ).

Let T , s |= [ϕ]Iψ which, by definition means T , s |= ϕ implies Tϕ, s |= Iψ.If we spell out the topological interior modality, we get ∃Uϕ 3 s ∈ τϕ s.t.∀t ∈ Uϕ, Tϕ, t |= ψ. By definition, since Uϕ ∈ τϕ, it means that there is anopen U ∈ τ such that Uϕ = U ∩ (ϕ). Under the assumption that T , s |= ϕ,we observe that ∃U 3 s ∈ τ (as we just constructed it), such that after theannouncement ϕ, the non-eliminated points in U (namely, the ones in Uϕ) willsatisfy ψ. Thus, we get T , s |= ϕ→ I[ϕ]ψ.

The other direction is very similar and hence we leave it to the reader. There-fore, the reduction axioms for PAL in topological spaces are given as follows.

1. [ϕ]p↔ (ϕ→ p)

2. [ϕ]¬ψ ↔ (ϕ→ ¬[ϕ]ψ)

3. [ϕ](ψ ∧ χ)↔ ([ϕ]ψ ∧ [ϕ]χ)

4. [ϕ]Iψ ↔ (ϕ→ I[ϕ]ψ)

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As a result, all the complex formulas involving the PAL operator can be reducedto a simpler one. This algorithm directly shows the completeness of PAL intopological spaces by reducing each formula in the language of topological PALto the language of basic topological modal logic. Thus, we conjecture that thefollowing result follows.

Conjecture 1. PAL in topological spaces is complete with respect to the axiomati-zation given.

By the same idea, we can import the decidability result.

Conjecture 2. PAL in topological models is decidable.

1.4.2 Multi-agent Topological PAL

It can be noted that the framework we have given was for a single agent. Wecan now discuss a multi-agent topological epistemic logic and how PAL works inthat framework. Topological frameworks for multi-agent epistemic logics havebeen discussed already in the literature (van Benthem et al., 2006; van Benthem& Sarenac, 2004). Therefore, our treatment of the subject will be based on theseworks.

Let T = 〈T, τ〉 and T ′ = 〈T ′, τ ′〉 be two topological spaces. Let X ⊆ T × T ′.We call X horizontally open (h-open) if for any (x, y) ∈ X, there is a U ∈ τsuch that x ∈ U , and U × {y} ⊆ X. In a similar fashion, we call X verticallyopen (v-open) if or any (x, y) ∈ X, there is a U ′ ∈ τ ′ such that y ∈ U ′, and{x} × U ′ ⊆ X.

Now, given two topological spaces T = 〈T, τ〉 and T ′ = 〈T ′, τ ′〉, let us asso-ciate two modal operators I and I′ respectively to these models. Then, we canobtain a product topology on a language with those two modalities. The prod-uct model will be of the form 〈T × T ′, τ, τ ′〉 on a language with two modalitiesI and I′.

The semantics of those modalities are given as such.

(x, y) |= Iϕ iff ∃U ∈ τ , x ∈ U and ∀u ∈ U , (u, y) |= ϕ(x, y) |= I′ϕ iff ∃U ′ ∈ τ ′, y ∈ U ′ and ∀u′ ∈ U ′, (x, u′) |= ϕ

It has been shown that the fusion logic S4⊕S4 is complete with respect toproducts of arbitrary topological spaces (van Benthem & Sarenac, 2004).

The language of multi-agent topological PAL is as follows. We specify it fortwo-agents for simplicity, but it can easily be generalized to n-agents.

p | ¬ϕ | ϕ ∧ ϕ | K1ϕ | K2ϕ | [ϕ]ϕ

For given two topological models T = 〈T, τ, v〉 and T ′ = 〈T ′, τ ′, v〉, theproduct topological model M = 〈T × T ′, τ, τ ′, v〉 has the following semantics.

M, (x, y) |= K1ϕ iff ∃U ∈ τ , x ∈ U and ∀u ∈ U , (u, y) |= ϕM, (x, y) |= K2ϕ iff ∃U ′ ∈ τ ′, y ∈ U ′ and ∀u′ ∈ U ′, (x, u′) |= ϕM, (x, y) |= [ϕ]ψ iff M, (x, y) |= ϕ implies Mϕ, (x, y) |= ψ

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where Mϕ = 〈Tϕ × T ′ϕ, τϕ, τ ′ϕ, vϕ〉 is the updated model. We define all Tϕ, T ′ϕ,τϕ, τ ′ϕ, and vϕ as before.

Therefore, the following axioms axiomatize the product topological PAL to-gether with the axioms of S4⊕S4.

1. [ϕ]p↔ (ϕ→ p)

2. [ϕ]¬ψ ↔ (ϕ→ ¬[ϕ]ψ)

3. [ϕ](ψ ∧ χ)↔ ([ϕ]ψ ∧ [ϕ]χ)

4. [ϕ]Kiψ ↔ (ϕ→ Ki[ϕ]ψ)

Conjecture 3. Product topological PAL is complete and decidable with respect tothe given axiomatization.

1.5 Applications

Now, we will briefly apply the previous discussions to some issues in PAL, foun-dational game theory and SSL. The purpose of such applications is to give thereader a sense how topological framework might affect the aforementioned is-sues, and in general how dynamic epistemic situations can be represented topo-logically.

1.5.1 Announcement Stabilization

Muddy Children presents an interesting case for PAL (Fagin et al., 1995). In thatgame, the model gets updated after each children says that she does not knowif she had mud on her forehead. The model keeps updated until the announce-ment is negated, and then becomes common knowledge (van Benthem, 2007).Therefore, after each update, we get smaller and smaller models up until themoment that the model gets stabilized in the sense that the same announcementdoes not update the model any longer.

As van Benthem pointed out, this is closely related to several issues (vanBenthem, 2007). First, PAL behaves like a fixed-point operator where the fixedpoint is the model which is stabilized. Second, there seems to be a close relationbetween game theoretical strategy eliminations, and solution methods based onsuch approaches. Therefore, it is rather important to analyze announcementstabilization. Here, we will approach the issue from a topological angle.

For a model M and a formula ϕ, we define the announcement limit limϕMas the first model which is reached by successive announcements of ϕ that nolonger changes after the last announcement is made. Announcement limits existin both finite and infinite models (van Benthem & Gheerbrant, 2010). For in-stance, for any model M , limpM = M |p for propositional variable p. Therefore,the limit model is the first updated model when the announcement is a groundBoolean formula. In muddy children, the announcement shrinks the model stepby step, round by round (van Benthem, 2007). However, sometimes in dialoguegames it may take too long to solve such puzzles until the model gets stabilized

12

Figure 1: A model for muddy children played with 3 children a, b, c taken from(van Ditmarsch et al., 2007). The state nanbnc for na, nb, nc ∈ {0, 1} representthat child i has mud on her forehead iff ni = 1 for i ∈ {a, b, c}. The propositionmi means that the child i ∈ {a, b, c} has mud on her forehead. The current stateis underlined.

as shown by Parikh (Parikh, 1991). Similarly, even Zermelo considered simi-lar approaches to understand as to how long it takes for the game to stabilize(Schwalbe & Walker, 2001).

Similar to the discussions of the aforementioned authors, we now analyzehow the models stabilize in topological PAL. We know that topological modelsdo present some differences in epistemic logical structures. For instance, intopological models, the stabilization of the fixed-point definition1 version ofcommon knowledge may occur later than ordinal stage ω. However, it stabilizesin ≤ ω steps in Kripke models (van Benthem & Sarenac, 2004).

We also know that there are two possibilities for the limit models. Eitherit is empty or nonempty. If it is empty, it means that the negation of the an-

1Formula ϕ is common knowledge among two-agents 1 and 2 C1,2ϕ is represented with the(largest) fixed-point definition as follows: C1,2ϕ := νp.ϕ∧K1p∧K2p where Ki, for i = 1, 2 is thefamiliar knowledge operator (Barwise, 1988).

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nouncement has become common knowledge, thus the announcement refuteditself. On the other hand, if the limit model is not empty, it means that theannouncement has become common knowledge (van Benthem & Gheerbrant,2010).

Conjecture 4. For some formula ϕ and some topological model M , it may takemore than ω stage to reach the limit model limϕM .

Proof. (Sketch) Note that it was shown that in multi-agent topological models,stabilization of common knowledge with fixed-point definition may occur laterthan ω stage. However, in Kripke models it occurs before ω stage (van Benthem& Sarenac, 2004).

Also note that it was shown that if the limit model is not empty, the an-nouncement has become common knowledge (van Benthem & Gheerbrant, 2010).

Therefore, combining these two observations, we conclude that in sometopological models with non-empty limit models, the number of stage for theannouncement to be common knowledge may take more than ω steps. �

Even if the stabilization takes longer, we can still obtain stable models bytaking intersections at the limit ordinals as a general rule (van Benthem &Gheerbrant, 2010). Therefore, we guarantee that the update procedure willterminate. Thus, the following result is now self-evident.

Conjecture 5. Limit models exist in topological models.

1.5.2 Backward Induction

The fact that limit models can be attained in more than ω steps can create someproblems in games. Consider the backward induction solution where playerstrace back their moves to develop a winning strategy. Notice that the Aumann’sbackward induction solution assumes common knowledge of rationality (Au-mann, 1995; Halpern, 2001)2. Granted, there can be several philosophical andepistemic issues about the centipede game and its relationship with rationality,but we will not pursue this direction here (Artemov, 2009a; Artemov, 2009b).

This issue can also be approached from a dynamic epistemic perspective.Recently, it has been shown that in any game tree model M taken as a PALmodel, limrationalM is the actual subtree computed by the backward inductionprocedure where the proposition rational means that “at the current node, noplayer has chosen a strictly dominated move in the past coming here” (van Ben-them & Gheerbrant, 2010). Therefore, the announcement of node-rationalityproduces the same result as the backward induction procedure. Each backwardstep in the backward induction procedure can then be obtained by the publicannouncement of node rationality. This result is quite impressive in the sensethat it establishes a closer connection between communication and rationality,and furthermore leads to several more intriguing discussions about rationality.

2Although according to Halpern, Stalnaker proved otherwise (Halpern, 2001; Stalnaker, 1998;Stalnaker, 1994; Stalnaker, 1996).

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In this work, we refrain ourselves from pursuing this line of thought for the timebeing.

However, there seems to be a problem in topological models. The admissi-bility of limit models can take more than ω steps in topological models as wehave conjectured earlier. Therefore, the BI procedure can take ω steps or more.

Conjecture 6. In topological models of games, under the assumption of rational-ity, the backward induction procedure can take more than ω steps.

This is indeed a problem about the attainability in infinite games: how can aplayer continue playing the game when she hit the limit ordinal ω-th step in thebackward induction procedure? In order not to diverge from our current focus,we leave this question open.

Notice that a similar result was obtained by Parikh in the context of finiteand infinite dialog games (Parikh, 1991). He showed that in a game whereplayers make consecutive announcements, one can obtain a model for transfi-nite ordinals.

1.5.3 Persistence

A similar notion has already been defined in SSL. Define persistent formula in amodel M as the formula ϕ whose truth is independent from the subsets in M(Dabrowski et al., 1996). In other words, ϕ is persistent if for all states s andsubsets V ⊆ U , we have s, U |= ϕ implies s, V |= ϕ. Clearly, Boolean formulasare persistent in every model.

Intuitively, persistent formulas are those formulas whose truth does not de-pend on the observational, computational etc. factors. For instance, the formula♦ is not persistent.

Now, we can apply this idea to the context of PAL. The punch-line is the factthat in PAL, we obtain a subset of the current observation set we are occupying.Therefore, persistent formulas do not get affected from announcements, andremain true.

Conjecture 7. Let M be a model and ϕ be persistent in M . Then, for any formulaχ and neighborhood situation (s, U), if s, U |= ϕ, then s, U |= [χ]ϕ. In otherwords, true persistent formulas are immune to the public announcements.

Proof. (Sketch) Proof follows directly from the definitions and the fact that afterthe public announcement of χ, we always have Uχ ⊆ U . �

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2 Paraconsistent Topologies and Homotopies

Let us now combine our geometric motivation with a non-classical logical sys-tem. In this section, we will discuss the connection between inconsistent logicalsystems and topological models.

2.1 What is Paraconsistency?

The well-studied notion of deductive explosion describes the situation whereevery formula can be deduced from an inconsistent set of formulae, i.e. for allϕ and ψ, we have {ϕ,¬ϕ} ` ψ, where ` denotes logical consequence relation. Inthis respect, both “classical” and intuitionistic logics are known to be explosive.Paraconsistent logic, on the other hand, is the umbrella term for logical systemswhere the logical consequence relation ` is not explosive (Priest, 2002). Varietyof philosophical and logical objections can be raised against paraconsistency,and almost all of these objections can be defended in a rigorous fashion. We willnot here be concerned about the philosophical implications of it, yet we refer thereader to the following for a comprehensive defense of paraconsistency with avariety of well-structured applications chosen from mathematics and philosophy(Priest, 2006, Chapters 2 and 3). In this chapter, we present some furtherapplications of paraconsistency in basic modal logic with topological semantics.

The use of topological semantics for paraconsistent logic is not new. To thebest of our knowledge, the earliest work discussing the connection betweeninconsistency and topology goes back to Goodman (Goodman, 1981)3. In hispaper, Goodman discussed “pseudo-complements” in a lattice theoretical set-ting and called the topological system he obtains “anti-intuitionistic logic”. Ina recent work, Priest discussed the dual of the intuitionistic negation operatorand considered that operator in topological framework (Priest, 2009). Simi-larly, Mortensen discussed topological separation principles from a paraconsis-tent and paracomplete point of view and investigated the theories in such spaces(Mortensen, 2000). Similar approaches from modal perspective was discussedby Beziau, too (Beziau, 2005).

2.2 Semantics

Recall that in topological models, modal formulas produce open (or dually,closed) sets due to the semantics of the modal operators in such models. We cantake one step further and suggest that extension of any propositional variablewill be an open set (Mortensen, 2000). In that setting, conjunction and dis-junction works fine for finite intersections and unions. Nevertheless, negationcan be difficult as the complement of an open set is not generally an open set,thus may not be the extension of a formula in the language. For this reason, we

3Thanks to Chris Mortensen for pointing this work out. Even if the paper appeared in 1981,the work had been carried out around 1978. In his paper, Goodman indicted that the results werebased on an early that appeared in 1978 only as an abstract.

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will need to use a new negation symbol ∼ that returns the open complement(interior of the complement) of a given set.

A similar idea can also be applied to closed sets where we assume that theextension of any propositional variable will be a closed set. In order to avoida similar problem with negation, we stipulate yet another negation operatorwhich returns the closed complement (closure of the complement) of a givenset. In this setting, we will use the symbol∼ that returns the closed complementof a given set.

So far, we have set up a lot of notational conventions. Let us make them clearonce more. Whenever we refer to a topology of open sets with the negation ∼,we will call it τ and, whenever we refer to a topology of closed sets with thenegation ∼, we will call it σ. We reserve the classical negation symbol ¬ for theset theoretical complement.

Now, let us consider the boundary ∂(·) of a set X where ∂(X) is defined as∂(X) := Clo(X)− Int(X). Consider now, for a given formula ϕ, the boundary ofits extension ∂((ϕ)) in τ . Let x ∈ ∂((ϕ)). Since (ϕ) is open, x /∈ (ϕ). Similarly,x /∈ (∼ϕ) as the open complement is also open by definition, and does notinclude the boundary. Thus, neither ϕ nor ∼ϕ is true at x. The point x wasselected arbitrarily from the boundary, thus, in τ , any theory that includes thetheory of the propositions that are true at the boundary is incomplete.

We can make a similar observation about the boundary points in σ. Now,take x ∈ ∂((ϕ)) where (ϕ) is a closed set in σ. By the above definition, since wehave x ∈ ∂((ϕ)), we obtain x ∈ (ϕ) as (ϕ) is closed. Yet, ∂((ϕ)) is also includedin (∼ ϕ) which we have defined as a closed set. Thus, by the same reasoning, weconclude x ∈ (∼ ϕ). Thus, x ∈ (ϕ∧ ∼ ϕ) yielding that x |= ϕ∧ ∼ ϕ. Therefore,in σ, any theory that includes the boundary points will be inconsistent. In thisrespect, the model M = 〈S, τ, V 〉 with the negation symbol ∼ will be called aparacomplete topological model, and similarly, the model M ′ = 〈S, σ, V 〉 withthe negation symbol ∼ will be called a paraconsistent topological model whereV is a valuation function.

So far, we have recalled how paracomplete and paraconsistent logics can beobtained in a topological setting. However, an immediate observation yieldsthat since extensions of all formulae in σ (respectively in τ) are closed (re-spectively, open), the topologies which are obtained in both paraconsistent andparacomplete logics are discrete. This observation may trivialize the matter as,for instance, discrete spaces with the same cardinality are homeomorphic.

Conjecture 8. Let M1 and M2 be paraconsistent and paracomplete topologicalmodels respectively. If |M1| = |M2|, then there is a homeomorphism from theparaconsistent topological model to the paracomplete one, and vice versa.

2.3 Topological Properties and Paraconsistency

In this section, we investigate the relation between some basic topological prop-erties and paraconsistency. Mostly, we will consider the closed set topology σwith its negation operator ∼ as it is the natural candidate for paraconsistent

17

topological models. Our work can be seen as an extension of Mortensen’s ear-lier work (Mortensen, 2000). Here we extend his approach to some other topo-logical properties and discuss the behavior of such spaces under some specialfunctions.

2.3.1 Connectedness

In the above section, we observed that boundary points play a central role inparaconsistent theories defined in topological spaces. There is also a close con-nection between boundary and connectedness which motivates this section.

A topological space is called connected if it is not the union of two disjointnon-empty open (closed) sets. Formally, a set X is then called connected if fortwo non-empty open (respectively closed) subsets A,B, if X = A ∪ B; thenA ∩ B 6= ∅. Moreover, in any connected topological space, the only subsetswith empty boundary are the space itself and the empty set (Bourbaki, 1966).Finally, connectedness is not definable in the basic modal language (Cate et al.,2009).

Definition 5. A formula ϕ in the language of propositional modal logic is calledconnected if for any two formulae α1 and α2 with non-empty extensions, ifϕ ≡ α1 ∨ α2, then we have (α1 ∧ α2) 6= ∅. We will call a theory T connected, ifit is generated by a set of connected formulae.

Based on this definition, we observe the following.

Conjecture 9. Every connected formula is satisfiable in some connected (classical)topological space.

Connected theories may be inconsistent or incomplete.

Conjecture 10. Every non-empty connected theory in closed set topology σ is in-consistent. Moreover, every connected theory in open set topology τ is incomplete.

The converse direction is a bit more interesting. Do connected spaces satisfyonly connected formulae?

Conjecture 11. Let X be a connected topological space of closed sets. Then, theonly subtheories that are not inconsistent are the trivial ones (i.e. empty theoryand X itself).

2.3.2 Continuity

A recent research program that considers topological modal logics with contin-uous functions were discussed in an early work of Artemov et al., and later byKremer and Mints (Artemov et al., 1997; Kremer & Mints, 2005). In these afore-mentioned works, they associated continuous functions with temporal modaloperator and discussed the orbits of such functions. In that setting, ©p =

18

f−1(p) where© is the temporal next time operator and f is a continuous func-tion. This framework allows us to discuss the orbits of continuous functions andexpress continuity modally.

An immediate theorem, which was stated and proved in variety of differentwork, would also work for paraconsistent logics. Now, let us take two closedset topologies σ and σ′ on a given set S and a homeomorphism f : 〈S, σ〉 →〈S, σ′〉. We have a simple way to associate the respective valuations betweentwo models M and M ′ which respectively depend on σ and σ′ so that we canhave a truth preservation result. Therefore, define V ′(p) = f(V (p)).

Conjecture 12. Let M = 〈S, σ, V 〉 and M ′ = 〈S, σ′, V ′〉 be two paraconsistenttopological models with a homeomorphism f from 〈S, σ〉 to 〈S, σ′〉. Define V ′(p) =f(V (p)). Let w ∈ S,w′ = f(w), then for all ϕ, we have M,w |= ϕ iff M ′, w′ |= ϕ.

Proof. (Sketch) By induction on the complexity of the formulae.�

Notice that the above theorem also works in paracomplete topological mod-els, and we leave the details to the reader.

Assuming that f is a homeomorphism may seem a bit strong. We can thenseparate it into two. One direction of the biconditional can be satisfied by con-tinuity whereas the other direction is satisfied by the openness of f .

Corollary 1. Let M = 〈S, σ, V 〉 and M ′ = 〈S, σ′, V ′〉 be two paraconsistent topo-logical models with a continuous f from 〈S, σ〉 to 〈S, σ′〉. Define V ′(p) = f(V (p)).Then M,w |= ϕ implies M ′, w′ |= ϕ for all ϕ where w′ = f(w).

Corollary 2. Let M = 〈S, σ, V 〉 and M ′ = 〈S, σ′, V ′〉 be two paraconsistent topo-logical models with an open f from 〈S, σ〉 to 〈S, σ′〉. Define V ′(p) = f(V (p)).Then M ′, w′ |= ϕ implies M,w |= ϕ for all ϕ where w′ = f(w).

Proofs of both corrolaries depend on the fact that Clo operator commuteswith continuous functions in one direction, and it commutes with open func-tions in the other direction. Furthermore, similar corrolories can be given forparacomplete frameworks as the Int operator also communtes in one directionunder similar assumptions, and we leave it to the reader as well.

Furthermore, any topological operator that commutes with continuous, openand homeomorphic functions will reflect the same idea and preserve the truth4.Therefore, these results can easily be generalized.

2.4 Homotopies

2.4.1 Paraconsistent Case

We can now take one step further to discuss homotopies in paraconsistent topo-logical modal models. A homotopy is a description of how two continuous func-tions from a topological space to another can be deformed to each other.

4Thanks to Chris Mortensen for pointing this out.

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Definition 6. Let S and S′ be two topological spaces with continuous functionsf, f ′ : S → S′. A homotopy 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).

An alternative definition can also be given: a homotopy between f and f ′

is a family of continuous functions Ht : S → S′ such that for t ∈ [0, 1] we haveH0 = f and H1 = g and the map t 7→ Ht is continuous from [0, 1] to the space ofall continuous functions from S to S′. It is easy to show that homotopy relationis an equivalence relation. Thus, if f and f ′ are homotopic, we denote it withf ≈ f ′.

Definition 7. Given a model M = 〈S, σ, V 〉, we call the family of models{Mt = 〈St ⊆ S, σt, Vt〉}t∈[0,1] generated by M and homotopic functions ho-motopic models. In the generation, we put Vt = ft(V ).

Conjecture 13. Given two topological paraconsistent models M = 〈S, σ, V 〉 andM ′ = 〈S′, σ′, V ′〉 with two continuous functions f, f ′ : S → S′ both of whichrespect the valuation: V ′ = f(V ) = f ′(V ). If there is a homotopy H between fand f ′, then M and M ′ satisfy the same modal formulae.

M

Mt

Mt′

ft

ft′

�������3

QQQQQQQs

H

Figure 2: Homotopic Models

Notice that we have discussed truth in the image sets that are obtained underf, f ′, ft, . . . . Nevertheless, the converse can also be true, once the continuousfunctions have continuous inverses: this is exactly what is guaranteed by home-omorphisms. The corresponding notion at the level of homotopies is an isotopy.An isotopy is a continuous transformation between homeomorphic functions.Thus, we have the following.

Conjecture 14. Given two topological paraconsistent models M = 〈S, σ, V 〉 andM ′ = 〈S′, σ′, V ′〉 with two homeomorphism f, f ′ : S → S′ both of which respectthe valuation: V ′ = f(V ) = f ′(V ). If there is an isotopy H between f and f ′,then, for all ϕ, we have

Mt |= ϕ iff M |= ϕ iff M ′ |= ϕ

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What makes the non-classical case easy is the fact that the extension of eachformula is an open or a closed set. Furthermore, a similar theorem can be statedfor paracomplete cases with a similar proof.

Conjecture 15. Given two topological paracomplete models M = 〈S, σ, V 〉 andM ′ = 〈S′, σ′, V ′〉 with two continuous functions f, f ′ : S → S′ both of whichrespect the valuation: V ′ = f(V ) = f ′(V ). If there is a homotopy H between fand f , then M and M ′ satisfy the same formulae.

2.4.2 Classical Case

Let us now observe if we can have similar results in classical modal logics.Let N = 〈T, η, V 〉 and N ′ = 〈T ′, η′, V ′〉 be classical topological modal mod-

els. Define a homotopy H : T × [0, 1] → T ′. Therefore, as before, for eacht ∈ [0, 1], we obtain models Nt = 〈Tt, ηt, Vt〉.

Conjecture 16. Given two classical topological modal models N = 〈T, η, V 〉 andN ′ = 〈T ′, η′, V ′〉 with two continuous functions f, f ′ : T → T ′ both of whichrespect the valuation: V ′ = f(V ) = f(V ′). If there is a homotopy H between fand f ′, then N and N ′ satisfy the same formula.

2.5 A Modal Logical Application

Consider the following two bisimilar Kripke models M and M ′. Assume thatw,w′ and u, u′, u′′ and v, v′, v′′ do satisfy the same propositional letters. Then itis easy to see that w and w′ are bisimilar, and therefore satisfy the same modelformulae. As it is well-known, modal logic cannot distinguish bisimilar models.However, the difference between bisimilar models may raise some philosophicaland mathematical issues.

w

v

u

�������3

QQQQQQQs

M

w′

v′′

u′

u′′

v′

�������3

QQQQQQQs

����

���1

PPPPPPPq

M ′

Figure 3: Two Bisimular Models

We can still pose a conceptual question about the relation between M andM ′. Even if these two models are bisimilar, they are clearly different models.Moreover, it is plausible to contract M ′ to M in a validity preserving fashion.

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Therefore, we may need to transform one model to its bisimilar copy. Further-more, given a model, we may need to measure the level of change from the fixedmodel to another model which is bisimilar to the given one.

Especially, in epistemic logic, such concerns do make sense. Given an epis-temic situation, we can model it upto bisimulation. In other words, from anagent’s perspective whose language is propositional modal logic, bisimilar mod-els are indistinguishable. But, from a (modal) model theoretical perspective,they are distinguishable. Therefore, there can be several ways to model thegiven epistemic situation. We will now define how these models are related anddifferent from each other by using the constructions we have presented earlier.Before proceeding further, let us give the definition of topological bisimulations(Aiello & van Benthem, 2002).

Definition 8. Let M = 〈S, σ, v〉 and M ′ = 〈S′, σ′, v′〉 be two topological models.A topo-bisimulation is a nonempty relation�⊆ S × S′ such that if s� s′, thenwe have the following:

1. BASE CONDITION

s ∈ v(p) if and only if s ∈ v′(p) for any propositional variable p.

2. FORTH CONDITION

s ∈ U ∈ σ implies that there exists U ′ ∈ σ′ such that s′ ∈ U ′ and for allt′ ∈ U ′ there exists t ∈ U with t� t′

3. BACK CONDITION s′ ∈ U ′ ∈ σ′ implies that there exists U ∈ σ such thats ∈ U and for all t ∈ U there exists t′ ∈ U ′ with t� t′

We can take one step further and define a homoemorphism that respectbisimulations. Let M = 〈S, σ, v〉 and M ′ = 〈S′, σ′, v′〉 be two topo-bisimularmodels. If there is a homoemorphism f from 〈S, σ〉 to 〈S, σ〉 that respect thevaluation, we call M and M homeo-topo-bisimilar models. We give the precisedefinition as follows.

Definition 9. Let M = 〈S, σ, v〉 and M ′ = 〈S′, σ′, v′〉 be two topological mod-els. A homeo-topo-bisimulation is a nonempty relation �f⊆ S × S′ based ona homeomorphism f from S into S′ such that if s �f s′, then we have thefollowing:

1. BASE CONDITION

s ∈ v(p) if and only if s ∈ v′(p) for any propositional variable p.

2. FORTH CONDITION

s ∈ U ∈ σ implies that there exists f(U) ∈ σ′ such that s′ ∈ f(U) and forall t′ ∈ f(U) there exists t ∈ U with t�f t

′

3. BACK CONDITION s′ ∈ f(U) ∈ σ′ implies that there exists U ∈ σ suchthat s ∈ U and for all t ∈ U there exists t′ ∈ f(U) with t�f t

′

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Based on this definition, we immediately observe the following.

Conjecture 17. Homeo-topo-bisimulation preserve the truth.

Now, we can discuss the homotopy of homeo-topo-bisimilar models. Whatwe aim is the following. Given a topological model (either classical, intuition-istic or paraconsistent), we will construct two homeomorphic image of it re-specting homeo-topo-bisimulation where these two homeomorphisms are ho-motopic. Then, by using homotopy, we will measure the level of change of theintermediate homeomorphic models with respect to these two functions.

Let M be a given topological model. Construct Mf and Mg as the home-omorphic image of M respecting the valuation where f and g are homeomor-phism. For simplicity, assume that M �f Mf and M �g Mg. Now, if f and gare homotopic, then we have functions hx for x continuous on [0, 1] with h0 = fand h1 = g.

Therefore, given x ∈ [0, 1] the model Mx will be obtained by applying hx toM respecting the valuation. Hence, M0 = Mf and M1 = Mg. Therefore, givenM , the distance of any homeo-topo-bisimilar model Mx to M will be x, and itwill be the measure of non-modal change in the model. In other words, even ifM �h(x) Mx, we will say M and Mx are x-different than each other.

The procedure we described offers a well-defined method of indexing thehomeo-topo-bisimular models. But, indexing is not random. It is continuous onthe closed unit interval.

2.6 An Epistemic Application

Consider two believers Ann and Bob where Ann is an ordinary believer whileBob is a religious cleric of the religion that Ann is following. Therefore, theybelieve in the same religion and the same rules of the religion. For the sakeof our example, let us assume that the religion in question is really simple andthere is nothing that Ann does not know about it. In other words, even if Bobis a clergyman, Ann believes in the religion as much as Bob does. However,we feel that Bob believes it more than Ann even though they believe in exactlythe same propositions. In other words, there is still a difference between theirbelief. Then, what is this difference? The reason for this is the fact that theextent of Bob’s knowledge is wider than that of Ann’s. Namely, Bob believesbetter. What does this mean?

In this context, we can ask the following two questions.

1. How much wider is Bob’s belief?

2. How is Ann’s belief transformed to Bob’s?

These two questions are meaningful. Even if their language cannot tell uswhich one has wider knowledge, ontologically, we know that Bob has moreknowledge in some sense even if they agree on every proposition. Clearly, thereason for that is the fact that Bob considers more possible worlds for a given

23

proposition which makes his belief more robust than Ann’s5. Let us set up apiece of notation first. By [w], we denote the set of accessible states from w, i.e.[w] = {v : wRv}.

Definition 10. Given two agents i and j with their respective bisimilar modelsM and N . We say i’s knowledge more robust than j’s at w if (ϕ)N ∩ [w] ⊆(ϕ)M ∩ [w] for all formula ϕ in the language.

Moreover, from a dynamic epistemic angle, homeomorphisms and homo-topies can explain this transformation from Ann’s beliefs to Bob’s belief withrespect to their models.

The timestamp subscript in the definition of homotopies can easily be con-sidered as a temporal parameter. In our simple example, this reading of thehomotopy index is especially helpful. It help us to give a step by step accountof the transformation between Ann’s and Bob’s belief.

Therefore, focusing on the transformations reveals more information on theontology of the agents when the epistemics of the agents are already known.Robust knowledge/belief captures this notion by focusing on the ontology ofthe model and connects the epistemics with ontology.

5We borrowed the term “robust” from Artemov.

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3 A Paradox in Game Theory

Can we use such non-classical methods and even more to analyze a well-knownparadox in epistemic game theory? With some help from category theory, weobtain some intriguing results using paraconsistency and non-well-founded settheory.

3.1 Introduction

The Brandenburg-Keisler paradox (‘BK paradox’, henceforth) is a two-personself-referential paradox in epistemic game theory (Brandenburger & Keisler,2006). Due to its considerable impact on the various branches of game the-ory and logic, it has gained increasing interest in the literature.

In short, for players Ann and Bob, the BK paradox arises when we considerthe following statement “Ann believes that Bob assumes that Ann believes thatBob’s assumption is wrong” and ask the question if “Ann believes that Bob’sassumption is wrong”.

There can be considered two main reasons why the Brandenburger-Keislerargument turns out to be a paradox. First, the limitations of set theory presentssome restrictions on the mathematical model which is used to describe self-referantiality and circularity in the formal language. Second, Boolean logiccomes with its own Aristotelian meta-logical assumptions about consistency.Namely, Aristotelian principle about consistency, principium contradictionis, main-tains that contradictions are impossible. In this paper, we will consider some al-ternatives to such assumptions, and investigate their impact on the BK paradox.

The BK paradox is based on the ZF(C) set theory. The ZFC set theory comeswith its own restrictions one of which is the axiom of foundation. It can bededuced from this axiom that no set can be an element of itself. In non-well-founded set theory, on the other hand, the axiom of foundation is replaced bythe anti-foundation axiom. In non-well-founded set theory, among many otherthings, it is possible to have sets to be their own member (Mirimanoff, 1917;Aczel, 1988). Therefore, we claim that switching to non-well-founded set theorysuggests a new approach to the paradox, and game theory in general. Thepower of non-well-founded set theory comes from its genuine methods to dealwith circularity (Barwise & Moss, 1996; Moss, 2009).

Second, what makes the BK paradox a paradox is the principium contra-dictionis. Paraconsistent logics challenge this assumption (Priest, 1998; Priest,2006). Therefore, we will also investigate the BK paradox in paraconsistentsystems. This line of research, as we shall see, is rather fruitful. The reasonfor this is the following. The BK paradox is essentially a self-referential para-dox, and for any other paradox of the same kind, it can be analyzed from acategory theoretical or algebraic point of view (Yanofsky, 2003; Abramsky &Zvesper, 2010). Moreover, paraconsistent logics also present an algebraic andcategory theoretical structure. In this work, we will make the connection be-tween self-referantiality and paraconsistency clearer and see whether we cansolve the paradox if we embrace a paraconsistent framework.

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What is the significance of adopting non-classical frameworks then? Thereare many situations where circularity and inconsistency are integral parts ofthe game. A game when some players can reset the game can be thought ofa situation where the phenomenon of circularity appears. Moreover, inconsis-tencies occur in games quite often as well. Situations where information setsof some players become inconsistent after receiving some information in a di-alogue without a consequent belief revision are such examples where inconsis-tencies occur (Lebbink et al., 2004; Rahman & Carnielli, 2000).

The Brandenburger - Keisler paradox was presented in its final form in arelatively recent paper in 2006 (Brandenburger & Keisler, 2006). A generalframework for self-referential paradoxes was discussed earlier by Yanofsky in2003 (Yanofsky, 2003). In his paper, Yanofsky used Lawvere’s category theoret-ical arguments in well-known mathematical arguments such as Cantor’s diago-nalization, Russell’s paradox, and Godel’s Incompleteness arguments. Lawvere,on the other hand, discussed self-referential paradoxes in cartesian closed cat-egories in his early paper which appeared in 1969 (Lawvere, 1969). Abramskyand Zvesper used Lawvere’s arguments to analyze the BK paradox in a categorytheoretical framework (Abramsky & Zvesper, 2010).

Pacuit approached the paradox from a rather modal logical perspective andpresented a detailed investigation of the paradox in neighborhood models andin hybrid systems (Pacuit, 2007). Neighborhood models are used to repre-sent modal logics weaker than K, and can be considered as weak versions oftopological semantics (Chellas, 1980). This argument later was extended toassumption-incompleteness in modal logics (Zvesper & Pacuit, 2010).

Paraconsistent games in the form of dialogical games were largely discussedby Rahman and his co-authors (Rahman & Carnielli, 2000). Co-Heyting al-gebras, on the other hand, have gained interest due to their use in “regionbased theories of space” within the field of mereotopology (Stell & Worboys,1997). Mereotopology discusses the qualitative topological relations betweenthe wholes, parts, contacts and boundaries and so on.

Now, we can discuss the paradox briefly. The BK paradox can be consideredas a game theoretical two-person version of Russell’s paradox where playersinteract in a self-referential fashion. Let us call the players Ann and Bob withassociated type spaces Ua and U b respectively. Now, consider the followingstatement which we call the BK sentence:

Ann believes that Bob assumes that Ann believes that Bob’s assumptionis wrong.

A Russell-like paradox arises if one asks the question whether Ann believesthat Bob’s assumption is wrong. In both cases, we get a contradiction, hencethe paradox. Because, if the answer to the question is “Yes”, then it turns outthat it is not the case that Ann believes that Bob’s assumption is wrong, whichcontradicts with the “Yes” answer. Similarly, if the answer to that question is“No”, we get a contradiction as well. Thus, the BK sentence is impossible.

Brandenburger and Keisler use belief sets to represent the players’ beliefs.The model (Ua, U b, Ra, Rb) that they consider is called a belief structure where

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Ra ⊆ Ua × U b and Rb ⊆ U b × Ua. The expression Ra(x, y) represents thatin state x, Ann believes that the state y is possible for Bob, and similarly forRb(y, x). We will put Ra(x) = {y : Ra(x, y)}, and similarly for Rb(y). At a statex, we say Ann believes P ⊆ U b ifRa(x) ⊆ P . Now, a modal logical semantics forthe interactive belief structures can be given. We use two different modalities� and ♥ which stand for the belief and assumption operators respectively withthe following semantics.

x |= �abϕ iff ∀y ∈ U b.Ra(x, y) implies y |= ϕx |= ♥abϕ iff ∀y ∈ U b.Ra(x, y) iff y |= ϕ

A belief structure (Ua, U b, Ra, Rb) is called assumption complete with respectto a set of predicates Π on Ua and U b if for every predicate P ∈ Π on U b, thereis a state x ∈ Ua such that x assumes P , and for every predicate Q ∈ Π on Ua,there is a state y ∈ U b such that y assumes Q. We will use special propositionsUa and Ub with the following meaning: w |= Ua if w ∈ Ua, and similarly forUb. Namely, Ua is true at each state for player Ann, and Ub for player Bob.

Brandenburger and Keisler showed that no belief model is complete for itsfirst-order language. Therefore, “not every description of belief can be rep-resented” with belief structures (Brandenburger & Keisler, 2006). The incom-pleteness of the belief structures is due to the holes in the model. A model, then,has a hole at ϕ if either Ub ∧ ϕ is satisfiable but ♥abϕ is not, or Ua ∧ ϕ is satis-fiable but ♥baϕ is not. A big hole is then defined by using the belief modality �instead of the assumption modality ♥.

In the original paper, the authors make use of two lemmas before identifyingthe holes in the system. These lemmas are important for us as we will challengethem in the next section. First, let us define a special propositional symbol Dwith the following valuation D = {w ∈W : (∀z ∈W )[P (w, z)→ ¬P (z, w)]}.

Lemma 1 ((Brandenburger & Keisler, 2006)).

1. If ♥abUb is satisfiable, then �ab�ba�ab♥baUa → D is valid.

2. ¬�ab♥ba(Ua ∧D) is valid.

Based on these lemmas, the authors observe that there are no completebelief models. Here, we give the theorem in two forms.

Theorem 11 ((Brandenburger & Keisler, 2006)).

• First-Order Version: Every belief model M has either a hole at Ua, a hole atU b, a big hole at one of the formulas

(i) ∀x.P b(y, x)

(ii) x believes ∀x.P b(y, x)

(iii) y believes [x believes ∀x.P b(y, x)]

a hole at the formula

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(iv) D(x)

or a big hole at the formula

(v) y assumes D(x)

Thus, there is no belief model which is complete for a language L whichcontains the tautologically true formulas and formulas (i)-(v).

• Modal Version: There is either a hole at Ua, a hole at Ub, a big hole at oneof the formulas

♥baUa, �ab♥baUa, �ba�ab♥baUa

a hole at the formula Ua ∧D, or a big hole at the formula ♥ba(Ua ∧D).Thus, there is no complete interactive frame for the set of all modal formulasbuilt from Ua, Ub, and D.

3.2 Non-well-founded Set Theoretical Approach

Non-well-founded set theory is a theory of sets where the axiom of foundation isreplaced by the anti-foundation axiom which is due to Mirimanoff (Mirimanoff,1917). Then, decades later, it was formulated by Aczel within graph theory, andthis motivates our approach here (Aczel, 1988). In non-well-founded (NWF,henceforth) set theory, we can have true statements such as ‘x ∈ x’, and suchstatements present interesting properties in game theory. NWF theories arenatural candidates to represent circularity (Barwise & Moss, 1996).

On the other hand, NWF set theory is not immune to the problems that theclassical set theory suffers from. For example, note that Russell’s paradox isnot solved in NWF setting, and moreover the subset relation stays the same inNWF theory (Moss, 2009). Therefore, we may not expect the BK paradox todisappear in NWF setting. Yet, NWF set theory will give us many other tools ingame theory.

What we call a non-well-founded model is a tuple M = (W,V ) where Wis a non-empty non-well-founded set (hyperset, for short), and V is a valuationassigning propositional variables to the elements of W . We also relax the condi-tion that Ua ∩ U b = ∅. Namely, some states may belong to both of the players,and whoever gets there may make a move there.

Now, we give the semantics of (basic) modal logic in non-well-founded set-ting (Gerbrandy, 1999). We will use the symbol |=+ to represent the semanticalconsequence relation in a NWF model.

M,w |=+ ♦ϕ iff ∃v ∈ w. such that M, v |=+ ϕM,w |=+ �ϕ iff ∀v ∈ w. v ∈ w implies M,v |=+ ϕ

Now, we can give a non-standard semantics for the belief and assumption modal-ities �ij and ♥ij respectively for i, j ∈ {a, b}.

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M,w |=+ �ijϕ iff M,w |=+ Ui ∧ ∀v ∈ w(M, v |=+ Uj →M, v |=+ ϕ)M,w |=+ ♥ijϕ iff M,w |=+ Ui ∧ ∀v ∈ w(M, v |=+ Uj ↔M, v |=+ ϕ)

Note that the following formulas discussed in the original paper are still validas before.

�abUb ↔ Ua, �baUa ↔ Ub, �abUa ↔ ⊥, �baUb ↔ ⊥

Furthermore, the following formulas are still unsatisfiable as before.

�abUb → Ub, �abUb → �ba�abUb, �abUb → �ab�abUb

Now, we need to redefine the diagonal set in NWF case. Recall that, in thestandard case, diagonal set D is defined with respect to the accessibility relationP which we defined earlier. In NWF case, we will use membership relation forthat purpose.

Definition 12. Define D+ = {w ∈W : ∀v ∈W.(v ∈ w → w /∈ v)}.

We define the propositional variable D+ as the propositional variable withthe valuation set D+. We call a set A transitive whenever a ∈ A and b ∈ a, thenb ∈ A.

Now, we observe how the NWF models make a difference in the context ofthe BK paradox. Notice that BK argument relies on two lemmas which we havementioned earlier in Lemma 1. Now, we present counter-models in NWF theoryfor them.

Conjecture 18. In a NWF belief structure, if♥abUb is satisfiable, then the formula

�ab�ba�ab♥baUa ∧ ¬D+

is also satisfiable.

Conjecture 19. The formula �ab♥ba(Ua ∧D+) is satisfiable in some NWF beliefstructures.

Therefore, the BK Lemma 1 is refuted in NWF belief models. Notice thatLemma 1 is central in Brandenburger and Keisler’s proof of the incompletenessof belief structures. Thus, we ask whether the failure of Lemma 1 would meanthat there can be complete NWF belief structures.

Consider the following NWF counter-model M . Let W = {w, u, v, t, y}where Ua = {w, u}, and U b = {v, t, y}. Put w = {v, t}, v = {u,w}, u = {t},y = {u}.

Then, M satisfies the formulas given in Theorem 11. First, M has no holes atUa and Ub as the first is assumed at v, and the latter is assumed at w. Therefore,v |=+ ♥baUa. Moreover, it has no big holes, thus w believes ♥baUa givingw |=+ �ab♥baUa. Similarly, v believes �ab♥baUa yielding v |=+ �ba�ab♥baUa.The state u also satisfiesD+, and it is assumed by y, thus y assumesD+(u). Thiscounter-model shows that Theorem 11 does not hold in NWF belief structures.Yet, we have to be careful here. Our counter model does not establish the fact

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that NWF belief models are complete. It does establish the fact that they donot have the same holes as the standard belief models. We will get back to thisquestion later on, and give an answer from a category theoretical point of view.

Games with NWF Belief Structures In our analysis, we altered the set theorythat the belief structures and belief sets use. Then, the immediate question is thefollowing: What is the game theoretical meaning of it? Let us briefly elaborateon it.

Aczel’s Anti-Foundation Axiom states that for each connected rooted di-rected graph, there corresponds a unique set (Aczel, 1988). Since the set inquestion is the game theoretical belief set, it means that for every (labeled) di-rected graph we have NWF set that represent this game (up to the name andthe order of the players). Hence, the following theorem.

Conjecture 20. For every (labeled) rooted directed connected graph, there corre-sponds to a unique two-player NWF belief structure up to the permutation of typespaces, and the order of players.

Therefore, we can use any connected graphs (i.e. not only trees, but alsoconnected graphs with loops) to represent games in extensive form (up to thenatural conditions in the theorem). This extension of game theory and similarissues have been raised by Harsanyi (Harsanyi, 1967). Universal type spaces doemploy some circularity in definitions, and NWF theories are needed to give anexplanation of such structures. Harsanyi mentions the following.

It seems to me that the basic reason why the theory of games withincomplete information has made so little progress so far lies in thefact that these games give rise, or at least appear to give rise, to aninfinite regress in reciprocal expectations on the part of the players.(Harsanyi, 1967)

Moss, for example, takes this quote and considers universal type spaces fromNWF set theory point of view (Moss, 2009). The following example illustratesTheorem 20.

Example 13. Consider the following labeled, connected directed graph.

u

w

v

RL

The two-player NWF belief structure of this game is as follows. Put W ={w, v, u} where w = {u, v} and u = {w}. Assume that Ua = {w}, U b = {u, v}(or any other combination of type spaces). Therefore, this graph corresponds tothe game where Bob can reset the game if Alice plays L at w.

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We conclude that NWF belief sets allow us to consider a much larger collec-tion of belief sets for games which was implied by Harsanyi much earlier. Byconstruction, as the Example 13 demonstrates, NWF allows circularity in game.This is not surprising considering the motivation for NWF set theory.

3.3 Paraconsistent Approach

Paraconsistent logics can be captured by using several rather strong algebraic,topological and category theoretical structures. In this chapter, we will approachparaconsistency from such directions, and analyze the BK paradox within para-consistent logics interpreted in such systems.

3.3.1 Algebraic and Category Theoretical Approach

A recent article on the BK paradox shows the general pattern of such paradoxicalcases, and gives some positive results such as fixed-point theorems (Abramsky& Zvesper, 2010). In this section, we will generalize their fixed-point results tosome other mathematical structures that can represent paraconsistent logics.

First, let us recall some facts about paraconsistency. Paraconsistency is theumbrella term for logical systems where ex contradictione quodlibet fails. Namely,in paraconsistent logics, for some ϕ,ψ, we have ϕ,¬ϕ 6` ψ.

Note that the semantical issue behind the failure of ex contradictione quodli-bet in paraconsistent systems is the fact that in those logics, the intersection ofa formula and its negation may not be the empty set.

There are a variety of different logical and algebraic structures to representparaconsistent logics (Priest, 2002). Co-Heyting algebras are natural algebraiccandidates to represent paraconsistency.

Definition 14. Let L be a bounded distributive lattice. If there is a binaryoperation⇒: L× L→ L such that for all x, y, z ∈ L,

x ≤ (y ⇒ z) iff (x ∧ y) ≤ z,

then we call (L,⇒) a Heyting algebra.Dually, if we have a binary operation \ : L× L→ L such that

(y \ z) ≤ x iff y ≤ (x ∨ z),

then we call (L, \) a co-Heyting algebra. We call⇒ implication, \ subtraction.

An immediate example of a co-Heyting algebra is the closed subsets of agiven topological space and subtopoi of a given topos (Lawvere, 1991; Mortensen,2000; Baskent, 2011; Priest, 2002).

Both operations⇒ and \ give rise to two different negations. The intuition-istic negation ¬ is defined as ¬ϕ ≡ ϕ → 0 and paraconsistent negation ∼ isdefined as ∼ϕ ≡ 1 \ ϕ where 0 and 1 are the bottom and the top elementsof the lattice respectively. Therefore, ¬ϕ is the largest element disjoint from

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ϕ, and ∼ϕ is the smallest element whose join with ϕ gives the top element 1(Reyes & Zolgaghari, 1996). In a Boolean algebra the intuitionistic and para-consistent negations coincide, and give the usual Boolean negation where weinterpret ϕ ⇒ ψ as ¬ϕ ∨ ψ, and ϕ \ ψ as ϕ ∧ ¬ψ with the usual Boolean nega-tion ¬. What makes closed topologies a paraconsistent structure is the fact thattheories that are true at boundary points include formulas and their negation(Mortensen, 2000; Baskent, 2011). Because, a formula ϕ and its paraconsistentnegation ∼ϕ intersect at the boundary of their extensions. We will discuss theparaconsistent negation in the following sections as well.

On the other hand, the algebraic structures we have mentioned can be ap-proached from a category theory point of view. Before discussing Lawvere’s ar-gument, we need to define weakly point surjective maps. An arrow f : A×A→ Bis called weakly point surjective if for every p : A → B, there is an x : 1 → Asuch that for all y : 1→ A where 1 is the terminal object, we have

p ◦ y = f ◦ 〈x, y〉 : 1→ B

In this case, we say, p is represented by x. A category is called cartesian closed(CCC for short, henceforth), if it has a terminal object, and admits products andexponentiation. A set X is said to have the fixed-point property for a functionf , if there is an element x ∈ X such that f(x) = x. Category theoretically,an object X is said to have the fixed-point property if and only if for everyendomorphism f : X → X, there is x : 1→ X with xf = x (Lawvere, 1969).

Theorem 15 ((Lawvere, 1969)). In any cartesian closed category, if there existsan object A and a weakly point-surjective morphism g : A → Y A, then Y has thefixed-point property for g.

It was observed that CCC condition can be relaxed, and Lawvere’s Theo-rem works for categories that have only finite products (Abramsky & Zvesper,2010)6. These authors showed how to reduce Lawvere’s Lemma to the BK para-dox and, to reduce the BK paradox to Lawvere’s Lemma7. Now, our goal is to goone step further, and investigate some other cartesian closed categories whichrepresent the non-classical frameworks that we have investigated in this paper.Therefore, by Lawvere and Abramsky & Zvesper results, we will be able to showthe existence of fixed-points in our framework, which will give the BK paradoxin those frameworks.

Now, we observe the category theoretical properties of co-Heyting algebrasand the category of hypersets. Recall that the category of Heyting algebrasis a CCC. A canonical example of a Heyting algebra is the set of opens of atopological space (Awodey, 2006). The objects of such a category will be theopen sets. The unique morphisms in that category exists from O to O′ if O ⊆ O′.What about co-Heyting algebras?

6This point was already made by Lawvere and Schanuel in Conceptual Mathematics. Thanks toNoson Yanofsky for pointing this out.

7In order to be able to avoid the tecnicalities of categorical logic, we do not give the detalis oftheir construction and refer the reader to (Abramsky & Zvesper, 2010)

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Proposition 2. Co-Heyting algebras are Cartesian closed categories.

Proof. We skip the standard proof. �

Example 16. The co-Heyting algebra of the closed sets of a topology is a well-known example of a CCC. Given two objects C1, C2, we define the unique arrowfrom C1 to C2, if C1 ⊇ C2. The product is the union of C1 and C2 as the finiteunion of closed sets exists in a topology. The exponent CC2

1 is then defined asClo(C1 ∩ C2) where C1 is the complement of C1.

Now, we have the following corollary for Theorem 15.

Corollary 3. In a co-Heyting algebra, if there is an object A and a weakly point-surjective morphism g : A→ Y A, then Y has the fixed-point property.

This is interesting. In other words, even if we allow nontrivial inconsisten-cies and represent them as a co-Heyting algebra, we will still have fixed-points.This is our first step to establish the possibility of having the BK paradox inparaconsistent setting.

Corollary 4. In a co-Heyting algebraic model, there exists an impossible BK sen-tence.

The procedure is simple. Take a co-Heyting algebra (or co-Brouwer lattice)that represents a paraconsistent logic. Therefore, Corollary 3 applies. Togetherwith Abramsky & Zvesper’s construction reducing Lawvere’s Lemma to the BKparadox, the result follows. Yet, we need to be careful as the paraconsistent(or co-Heyting) negation is not the same as the Boolean negation. Therefore,we obtain the BK sentence in a structure with different negation. On the otherhand, the aforementioned theory does not mean that there cannot be somesatisfiable BK sentences. We will elaborate on this later on.

What about the category of non-well-founded sets? Consider the categoryAFA of hypersets with total maps between them8. Category AFA admits a finalobject 1 = {∅}. Moreover, it also admits exponentiation and products in theusual sense, making it a CCC.

Corollary 5. There exists an impossible BK sentence in all non-well-founded inter-active belief structures.

This complements our earlier result. Namely, non-well founded belief struc-tures with hypersets may have fixed-points, indeed different fixed-points.

3.3.2 Topological Approach

Now, we construct the BK argument in the paraconsistent topological setting,and call it paraconsistent topological belief structure. For the agents a and b, wehave a corresponding type space A and B and define closed set topologies τA

8Thanks to Florian Lengyel for pointing this out.

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and τB on A and B respectively. Furthermore, to connect them in order to rep-resent belief interaction for the players, we will have additional constructionstA ⊆ A×B, and tB ⊆ B ×A. We will then call F = (A,B, τA, τB , tA, tB) para-consistent topological belief structure. In this setting, the set A represent thepossible epistemic states of the player a in which she holds beliefs about playerb, or b’s beliefs etc, and vice versa for the set B and the player a. Moreover,the topological structure determines those beliefs. For instance, for player a atthe state x ∈ A, tA returns a closed set in Y ∈ τB ⊆ ℘(B). In this case, wewrite tA(x, Y ) which means that at state x, player a believes that the states inY ∈ τB are possible for the player b. Moreover, a state x ∈ A believes ϕ ⊆ B if{y : tA(x, y)} ⊆ ϕ. Furthermore, a state x ∈ A assumes ϕ if {y : tA(x, y)} = ϕ.Notice that in this definition, we identify logical formulas with their extensions.

The modal language we use will have two modalities representing the beliefsof each agent. Based on the previous modal logical structures, we will give atopological semantics for the BK argument in paraconsistent topological beliefstructures (Brandenburger & Keisler, 2006; Pacuit, 2007). Let us first give theformal language which we use.

ϕ := p | ∼ϕ | ϕ ∧ ϕ | �a | �b | �a | �b

where ∼ is the topological negation symbol which we defined earlier, and �iand �i are the belief and assumption operators for player i respectively.

We have discussed the semantics of the negation already. For x ∈ A, y ∈ B,the semantics of the modalities are given as follows with a modal valuationattached to F .

x |= �aϕ iff ∃Y ∈ τB with tA(x, Y )→ ∀y ∈ Y.y |= ϕx |= �aϕ iff ∃Y ∈ τB with tA(x, Y )↔ ∀y ∈ Y.y |= ϕy |= �bϕ iff ∃X ∈ τA with tB(y,X)→ ∀x ∈ X.x |= ϕy |= �bϕ iff ∃X ∈ τA with tB(y,X)↔ ∀x ∈ X.x |= ϕ

We define conjunction, and ♦a and ♦b as usual.Now, we have sufficient tools to represent the BK sentence in our paracon-

sistent topological belief structure with respect to a state x0:

x0 |= �a �b ϕ ∧ ♦a>

Let us analyze this formula in our structure. Notice that the second conjunctguarantees that for the given x0 ∈ A, there exists a corresponding set Y ∈ τBwith tA(x, Y ). On the other hand, the first conjunct deserves closer attention:

x0 |= �a �b ϕ iff ∃Y ∈ τB with tA(x0, Y )⇒ ∀y ∈ Y. y |= �bϕiff ∃Y ∈ τB with tA(x0, Y )⇒

[∀y ∈ Y,∃X ∈ τA with tB(y,X)⇔ ∀x ∈ X. x |= ϕ]

Notice that in our framework, some special x can satisfy falsehood ⊥ to givex0 |= �a �b ⊥ ∧ ♦a> for some x0. Let the extension of p be X0. Pick x0 ∈ ∂X0

where ∂(·) operator denotes the boundary of a set ∂(·) = Clo(·)− Int(·). By the

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assumptions of our framework X0 is closed. Moreover, by simple topology ∂X0

is closed as well. By the second conjunct of the formula in question, we knowthat some Y ∈ τB exists such that tA(x0, Y ). Now, for all y in Y , we make anadditional supposition and associate y with ∂X0 giving tB(y, ∂X0). We knowthat for all x ∈ ∂X0, we have x |= p as ∂X0 ⊆ X0 where X0 is the extension ofp. Moreover, x |= ∼p for all x ∈ ∂X0 as ∂X0 ⊆ (∼X0), too. Thus, we concludethat x0 |= �a �b ⊥ ∧ ♦a> for some carefully selected x0.

In this construction, we have several suppositions. First, we picked the cur-rent state from the boundary of the extension of some proposition (ground ormodal). Second, we associate the epistemic accessibility of the second player tothe same boundary set. Namely, a’s beliefs about b includes her current state.

Now, the BK paradox appears when one substitutes ϕ with the followingdiagonal formula (whose extension is a closed set by definition of the closed settopology), hence breaking the aforementioned circularity:

D(x) = ∀y.[tA(x, y)→ ∼tB(y, x)]

The BK impossibility theorem asserts that, under the seriality condition, there isno such x0 satisfying the following.

x0 |= �a �b D(x) iff ∃Y ∈ τB with tA(x0, Y )⇒[∀y ∈ Y, ∃X ∈ τA with tB(y,X)⇔∀x ∈ X. x |= ∀y′.(tA(x, y′)→ ∼tB(y′, x))]

Motivated by our earlier discussion, let us analyze the logical statement inquestion. Let X0 satisfy the statement tA(x, y′) for all y′ ∈ Y and x ∈ X0 forsome Y . Then, ∂X0 ⊆ X0 will satisfy the same formula. Similarly, let ∼X0

satisfy ∼tB(y′, x) for all y′ ∈ Y and x ∈ X0. Then, by the similar argument,∂(∼X0) satisfy the same formula. Since ∂(X0) = ∂(∼X0), we observe thatany x0 ∈ ∂X0 satisfy tA(x, y′) and ∼tB(y′, x) with the aforementioned quan-tification. Thus, such an x0 satisfies �a �b D(x). Therefore, the states at theboundary of some closed set satisfy the BK sentence in paraconsistent topologi-cal belief structures.

Conjecture 21. The BK sentence is satisfiable in paraconsistent topological beliefstructures.

3.3.3 Product Topologies

Now, we will use product topologies to represent belief interaction among theplayers. The novelty of this approach is not only to economize on the notationand the model, but also to present a more natural way to connect such beliefsets of respective agents. For our purposes here, we will only consider two-player games, and our results can easily be generalized to n-player. We makeuse of the constructions presented in some recent works (van Benthem et al.,2006; van Benthem & Sarenac, 2004).

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Definition 17. Let a, b be two players with corresponding type space A and B.Let τA and τB be the (paraconsistent) closed set topologies of respective typespaces. The product topological paraconsistent belief structure for two agentsis given as (A×B, τA × τB).

In this framework, we will assume that the topologies are full on their sets -namely

⋃τA = A, and likewise forB. Therefore, if player a believes proposition

P ⊆ B at state x ∈ A, we will stipulate that there is a closed set X ∈ τA suchthat x ∈ X and a closed set Y ∈ τB with Y ⊆ P , all implying X × Y ∈ τA × τB .Player a assumes P if Y = P , and likewise for player b. Similar to the previoussection, we will make use of paraconsistent topological structures with closedsets and paraconsistent negation.

Borrowing a standard definitions from topology, we say that given a set S ⊆A × B, we say that S is horizontally closed if for any (x, y) ∈ S, there exists aclosed set X with x ∈ X ∈ τA and X×{y} ⊆ S. Similarly, S is vertically closed iffor any (x, y) ∈ S, there exists a closed set Y with y ∈ Y ∈ τB , and {x}×Y ⊆ S(van Benthem et al., 2006; van Benthem & Sarenac, 2004). In this framework,we say player a at x ∈ A is said to believe a set Y ⊆ B if {x} × Y is verticallyclosed.

Now, we can define assumption-complete structures in product topologies.For a given language L for our belief model, let La and Lb be the familiesof all subsets of A and B respectively. Then, we observe that by assumption-completeness, we require every non-empty set Y ∈ Lb is assumed by somex ∈ A, and similarly, every non-empty set X ∈ La is assumed by some y ∈ B.

We can now characterize assumption-complete paraconsistent topologicalbelief models. Given type spaces A and B, we construct the coarsest topologieson respective type spaces τA and τB where each subset of A and B are in τA andτB . Therefore, it is easy to see that A × B is vertically and horizontally closedfor any S ⊆ A × B. Moreover, under these conditions, our belief structure inquestion is assumption-complete.

We can relax some these conditions. Assume that now τA and τB are notthe coarsest topologies on A and B respectively. Therefore, we define, we weakassumption-completeness for a topological belief structure if every set S ∈ A×Bis both horizontally and vertically closed. In other words, weak assumption-complete models focus only on the formula that are available in the given struc-ture. There can be some formulas expressible in L, but not available in La or inτA for some reasons. Epistemic game theory, indeed, is full of such cases whereplayers may or may not be allowed to send some particular signals, and someinformation may be unavailable to some certain players. The following theoremfollows directly from the definitions.

Conjecture 22. Let M = (A×B, τA×τB) be a product topological paraconsistentbelief model. If M is horizontally and vertically closed, then it is weak assumption-complete.

36

4 Strategies

In this section, we consider strategies as the primitive object of our inquiry.Considering a formal framework that treats them as “first-class citizens”, wewill give a dynamic logic that formalizes some types of strategy updates.

4.1 Introduction

In game theory, strategy for a player is defined as “a set of rules that describeexactly how (...) [a] player should choose, depending on how the [other] play-ers have chosen at earlier moves” (Hodkinson et al., 2000). Nevertheless, thisdefinition of strategies is static, and presumably is constructed before the gameis actually played.

For example, consider chess. According to Zermelo’s well-known theorem,chess is determined. Then, why would you play chess if you know you willlose the game? Similarly, why would you even play if you know you will win.Clearly, if we have logical omniscience (which we don’t), then it is pointless forthe player, who is going to lose, to even start playing the game as she knowsthe outcome already. If we are not logical omniscient, then chess is only aperfect information game for God, not for the players. Therefore, there seemsto be a problem. The static notion of strategies falls short of analyzing perfectinformation games. Because, we, people, do not strategize as such even inperfect information games - largely because we are not logically omniscient,and we have limited computational power and bounded memory.

While people play games, they observe, learn, recollect and update theirstrategies during the game as well as adopting deontological strategies and goalsbefore the game. Players update and revise their strategies, for instance, whentheir opponent makes an unexpected or irrational move. Similarly, external fac-tors may sometimes force the players not to make some certain moves. For in-stance, assume that you play a video game by using a gamepad or a keyboard,and in the middle of the game, one of the buttons on the gamepad breaks.Hence, from that moment on, you will not be able to make some moves in thegame that are controlled by that button on the gamepad. This is most certainly

(1, 0)s6

•

��

s0, P1 a

d

(0, 2)s7

•

��

s1, P2 a

d

//

(3, 1)s8

•

��

s2, P1 a

d

//

(2, 4)s9

•

��

s3, P2

d

a//

(5, 2)s10

•

��

s4, P1

d

a // (4, 6)s5//

Figure 4: Centipede game

37

not part of your strategy. Therefore, you will need to revise your strategy insuch a way that some moves will be excluded from your strategy from then on.However, for your opponent, that is not the case as she can still make all themoves available to her.

In some cases, assumptions about the game and the players may fail as well.For instance, consider the centipede game between two players P1 and P2.Under the assumption of common rationality, the usual backward inductionscheme produces the solution that P1 needs to make a d move at s0. Whathappens then, if P1 is prohibited from or prevented from making a d move at s0

and onwards right after the beginning of the game (or similarly, if the key thatis used to make a d move is broken)? It means that from a behavioral perspec-tive, the assumption of common rationality is violated since the player did notfollow the move which gives the highest pay-off, and P2 may need to updateher strategy during the game based on what she observed. There can be manyreasons why P1 may make such a move. The move can be prohibited, a taboo,or simply forbidden or restricted by nature/God, thus becomes unavailable.

Based on this short discussion, we underline that the game theory does notdistinguish the following two sorts of strategies. First is the class of strategiesconstructed before the game - which we call deontological. Second is the class ofstrategies which are constructed as you go in the game-play. For instance, whenyou are on your way back home after work, being hungry, you plan to cooksome pasta. You strategize how to cook it, and what to add, how to boil andso on so forth. Then, when you are home, you notice that you are out of pasta.This is certainly not part of your strategy, because you didn’t even know thatyou did not have any pasta. Therefore, you revise your strategy, and cook riceinstead. Clearly, there are many epistemic issues here that deal with awarenessand learning that we will not address at the present paper. Yet, our work falls inbetween these two different understandings of strategies, and aims at to drawattention to the differences.

4.2 Strategy logic

In this section, we give a short overview of the strategy logic as presented in(Ghosh et al., 2010) based on the framework of (Ramanujam & Simon, 2008).The focus is on games played between two players given by the set N = {1, 2}and a single admissible set of moves Σ for both. Let T = (S,⇒, s0) be a treerooted at s0, on the set of vertices S. A partial function⇒: S ×Σ→ S specifiesthe labeled edges of such a tree where labels represent the moves at the states.The extensive form game tree, then, is a pair T = (T, λ) where T is a tree asdefined before, and λ : S → N specifies whose turn it is at each state. A strategyµi for a player i ∈ N is a function µi : Si → Σ where Si = {s ∈ S : λ(s) = i}.For player i and strategy µi, the strategy tree Tµ = (Sµ,⇒µ, s0, λµ) is the leastsubtree of T satisfying the following two conditions:

1. s0 ∈ Sµ;

38

2. For any s ∈ Sµ, if λ(s) = i, then there exists a unique s′ ∈ Sµ and actiona such that s a⇒µ s

′. Otherwise, if λ(s) 6= i, then for all s′ with s a⇒ s′ forsome a, we have s a⇒µ s

′.

In other words, in the strategy tree, the root is included, and for the states thatbelong to the strategizing player, a unique move is assigned to the player, and forthe other player, all possible moves are considered. Notice that strategies returnunique moves in SL. Nevertheless, we’d still have a tree even if the strategiesare set-valued.

The most basic constructions in SL are the strategy specifications. First, for agiven countable set X, a set of formulas BF (X) is defined as follows, for a ∈ Σ:

BF (X) := x ∈ X | ¬ϕ | ϕ ∧ ϕ | 〈a〉ϕ

Let P i be a countable set of atomic observables for player i, with P = P 1 ∪P 2. The syntax of strategy specifications is given as follows for ϕ ∈ BF (P i):

Strati(P i) := [ϕ→ a]i | σ1 + σ2 | σ1 · σ2

The specification [ϕ→ a]i at player i’s position stands for “play awhenever ϕholds”. The specification σ1 +σ2 means that the strategy of the player conformsto the specification σ1 or σ2 and σ1 · σ2 means that the strategy of the playerconforms to the specifications σ1 and σ2. By the abuse of the notation, we willuse ↔ to denote the equivalence of specifications: namely, those specificationsreturn the same moves under the same preconditions.

Let M = (T, V ) where T = (S,⇒, s0, λ) is an extensive form game treeas defined before, and V is a valuation function (V : S → 2P ) for the set ofpropositional variables P . The truth of a formula ϕ ∈ BF (P ) is given as usualfor the propositional, Boolean and modal formulas. The notion “strategy µconforms to specification σ for player i at state s” (notation µ, s |=i σ) is definedas follows, where outµ(s) denotes the unique outgoing edge at s with respectto µ.

µ, s |=i [ϕ→ a]i iff M, s |= ϕ implies outµ(s) = a for a ∈ Σµ, s |=i σ1 + σ2 iff µ, s |=i σ1 or µ, s |=i σ2

µ, s |=i σ1 · σ2 iff µ, s |=i σ1 and µ, s |=i σ2

Now, based on the strategy specifications, the syntax of the strategy logic SLis given as follows:

p | ¬ϕ | ϕ1 ∧ ϕ2 | 〈a〉ϕ | (σ)i : a | σ i ψ

for propositional variable p ∈ P , action a ∈ Σ, strategy σ ∈ Strati(P i), andBoolean formula ψ over P i. The intuitive reading of (σ)i : a is that at thecurrent state the strategy specification σ for player i suggests that the movea can be played. The intuitive meaning of σ i ψ is that following strategyσ player i can ensure ψ. The other Boolean connectives and modalities aredefined as usual.

39

We now define the set of available moves moves(s) := {a ∈ Σ : ∃s′ ∈S with s a⇒ s′}. Then, based on moves, we construct the set of enabled movesat state s in strategy σ is defined by induction as follows.

• [ψ → a]i(s) =

{a} : λ(s) = i;M, s |= ψ, a ∈ moves(s)∅ : λ(s) = i;M, s |= ψ, a /∈ moves(s)Σ : otherwise

• (σ1 + σ2)(s) = σ1(s) ∪ σ2(s)

• (σ1 · σ2)(s) = σ1(s) ∩ σ2(s)

The truth definition for the strategy formulas are as follows:

M, s |= 〈a〉ϕ iff ∃s such that s a⇒ s′ and M, s′ |= ϕM, s |= (σ)i : a iff a ∈ σ(s)M, s |= σ i ψ iff ∀s′ such that s⇒∗σ s′ in Ts|σ,

we have M, s′ |= ψ ∧ (turni → enabledσ)

where σ(s) denotes the set of the enabled moves at state s in strategy σ, and⇒∗σdenotes the reflexive transitive closure of ⇒σ. Furthermore, Ts is the tree thatconsists of the unique path from the root (s0) to s and the subtree rooted at s,and Ts|σ is the least subtree of Ts that contains a unique path from s0 to s andfrom s onwards, for each player i node, all the moves enabled by σ, and for eachnode of the opponent player, all possible moves . The proposition turni denotesthat it is i’s turn to play. Finally, define enabledσ =

∨a∈Σ(〈a〉> ∧ (σ)i : a). The

axioms of SL are given as follows:

• All the substitutional instances of the tautologies of propositional calculus

• [a](ϕ→ ψ)→ ([a]ϕ→ [a]ψ)

• 〈a〉ϕ→ [a]ϕ

• 〈a〉> → ([ψ → a]i : a)i

• [turni ∧ ψ ∧ ([ψ → a]i)i : a)]→ 〈a〉>

• turni ∧ ([ψ → a]i)i : c↔ ¬ψ for all a 6= c

• (σ + σ′)i : a↔ (σ : a)i ∨ (σ′ : a)i

• (σ · σ′)i : a↔ (σ : a)i ∧ (σ′ : a)i

• σ i ψ → [ψ ∧ invσi (a, ψ) ∧ invσ−i(ψ) ∧ enabledσ]

Here, invσi (a, ψ) = (turni ∧ (σ)i : a) → [a](σ i ψ) which expresses the factthat after an a move by i which conforms to σ, the statement σ i ψ continuesto hold, and invσ−i(ψ) = turni → �(σ i ψ) states that after any move of −i,σ i ψ continues to hold. Here,©ϕ ≡

∨a∈Σ〈a〉ϕ and �ϕ ≡ ¬©¬ϕ.

40

Now, we discuss the inference rules that SL employs. Modus ponens is fa-miliar as well as generalization for [a] for each a ∈ Σ. The induction rule is a bitmore complex: From the formulas ϕ ∧ (turni ∧ (σ)i : a)→ [a]ϕ,ϕ ∧ turn−i →�ϕ,ϕ → ψ ∧ enabledσ derive ϕ → σ i ψ. The axiom system of SL is soundand complete with respect to the given semantics (Ramanujam & Simon, 2008).

4.3 Restricted strategy logic

4.3.1 Basics

Let us now extend SL to restricted strategy logic, henceforth RSL, by allowingmove restrictions during the game. Recall that our motivation can be illustratedwith the example of a gamepad/keyboard which gets broken during the gameplay disallowing the player to make some certain moves from that moment on.

We denote the move restriction by [σ!a]i for a strategy specification σ andaction a for player i. Informally, after the move restriction of σ by a, playeri will not be able to make an a move. We incorporate restrictions in RSL atthe level of strategy specifications. In SL, recall that strategies are functions.Therefore, they only produce one move per state. However, our dynamic takein strategies cover more general cases where strategies can offer a set of movesto the player. Thus, in RSL, we define strategy µi as µi : Si → 2Σ. By outrµi(s)we will denote the set of moves returned by µi at s. Then, the extended syntaxof strategy specifications for player i is given as follows.

Strati(P i) := [ψ → a]i | σ + σ | σ · σ | [σ!a]i

Notice that the restrictions affect only the player who gets a move restriction.In other words, if a is prohibited to player i, it does not mean that some otherplayer j cannot make an a move. In other words, if my gamepad/keyboard isbroken, it doesn’t mean that yours is broken as well. Thus, we do not allowconstructs like [[ψ → a]1!c]2.

Once a move is restricted at a state, we will prone the strategy tree removingthe prohibited move from that state on. Therefore, given µi : Si → 2Σ, wedefine the updated strategy relation µi!a : Si → 2Σ−{a}. We are now ready todefine confirmation of restricted specifications to strategies. Note that we skipthe cases for · and + as they are exactly the same.

µ, s |=i [ϕ→ a]i iff M, s |= ϕ implies a ∈ outrµ(s)µi, s |=i [σ!a]i iff a /∈ outrµi(s) and µ!a, s |=i σ

In the sequel, we omit the superscript that indicates the agents, thus, we writeS for Si, and σ!a for [σ!a]i when it is obvious. Given a strategy µ and its strategytree Tµ = (Sµ,⇒µ, s0, λµ), we define the restricted strategy structure Tµ!a withrespect to an action a. Once we removed the restricted moves, the updatedstructure may not be a tree (it may be a forest). For this reason, we take (µ!a, s)as the connected component of Tµ!a that includes s. Therefore, for a fixed strat-egy, restrictions may yield different restricted strategy trees at different states.

41

This is perfectly fine for our intuition, because the state of the game where therestriction is made is important. In other words, it matters where may gamepadis broken during the game play. Now, for player i and strategy µi and move a,the restricted strategy tree Tµ!a = (Sµ!a,⇒µ!a, s

′, λµ!a) is the least subtree of Tsatisfying the following two conditions:

1. s′ ∈ Sµ!a;

2. For any s ∈ Sµ!a, if λ(s) = i, then there exists a unique t ∈ Sµ!a and actionb 6= a ∈ µ!a(s) such that s a⇒µ!a t. Otherwise, if λ(s) 6= i, then for all t

with s b⇒ t for b 6= a, we have s b⇒µ!a t.

Note that we introduce such conditions so that an RSL model can easily beconsidered as a submodel of a SL model with some additional assumptions. Wecan now make some observations.

Conjecture 23. For any strategy µ, state s, specification σ, and formula ψ, wehave µ, s 6|= [[ψ → a]!a].

Conjecture 24. For strategy specifications σ and σ′, and action a, we have (σ ·σ′)!a↔ (σ!a) · (σ′!a) and (σ + σ′)!a↔ (σ!a) + (σ′!a).

Restrictions stabilize immediately. For n ≥ 1, we use notation σ!na to denoten ≥ 1-consecutive restrictions of σ by move a. Similarly, we put µ!na for thecorresponding strategy tree µ.

Conjecture 25. For arbitrary strategy specification σ, action a and state s, wehave (σ!a)!a↔ σ!a. Moreover, we have σ!na↔ σ!a.

Since restrictions are local and operate by elimination, the order of the re-strictions does not matter.

Conjecture 26. For any actions a, b,we have (σ!a)!b↔ (σ!b)!a.

4.3.2 A Case Study: The Centipede Game

Let us consider the centipede game (see Figure 4) and see how RSL can for-malize it when a restricted strategy specification can change the game after anunexpected move. Let us call the players P1 and P2. The set of actions inthe centipede game is Σ = {d, a} where d, a mean that the player moves downor across, respectively. Utilities for individual players are indicated by a tuple(x, y) where x is the utility for P1, and y is the utility for P2. For the sake ofgenerality, we will not impose any further conditions on the strategies, such asrationality or max min etc.

In a recent work, Artemov approached the centipede game from a rationalityand epistemology based perspective (Artemov, 2009b). Now, similar to his ap-proach, we will use symbols r1 and r2 to denote the propositions “P1 is rational”and “P2 is rational”, respectively. Let us now construct rational strategies µ andν for P1 and P2 respectively following the backward induction scheme. At s4,

42

•s0, P1 a

(0, 2)s7

•

��

s1, P2 a

d

// •s2, P1 a//

(2, 4)s9

•

��

s3, P2

d

a// •s4, P1 a //

(4, 6)s5

//

Figure 5: Restricted centipede game-tree

P1 makes a d move, if she is rational. Therefore, we have µ, s4 |= [r1 → d]P1.However, at s3, P2 would be aware of P1’s possible move at s4 and the factthat P1 is rational as well, thus makes a d move if she herself is rational. So,we have ν, s3 |= [(r2 ∧ (〈a〉r1 ∨ 〈d〉r1))→ d]P2. Following the same strategy, weobtain the following.

µ, s2 |= [r1 ∧ (〈a〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1) ∨ 〈d〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1)))→ d]P1

ν, s1 |= [r2∧(〈a〉(r1∧(〈a〉(r2∧(〈a〉r1∨〈d〉r1)∨〈d〉(r2∧(〈a〉r1∨〈d〉r1))))〈d〉(r1∧(〈a〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1) ∨ 〈d〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1)))))→ d]P2

µ, s0 |= [(r1 ∧ (〈a〉(r2 ∧ (〈a〉(r1 ∧ (〈a〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1) ∨ 〈d〉(r2 ∧ (〈a〉r1 ∨〈d〉r1))))〈d〉(r1∧ (〈a〉(r2∧ (〈a〉r1∨〈d〉r1)∨〈d〉(r2∧ (〈a〉r1∨〈d〉r1))))))∨〈d〉(r2∧(〈a〉(r1 ∧ (〈a〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1) ∨ 〈d〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1))))〈d〉(r1 ∧ (〈a〉(r2 ∧(〈a〉r1 ∨ 〈d〉r1) ∨ 〈d〉(r2 ∧ (〈a〉r1 ∨ 〈d〉r1)))))))→ d]P1

Let ♦ϕ := 〈a〉ϕ ∨ 〈d〉ϕ. Then, we have the following statements.µ, s4 |= [r1 → d]P1

ν, s3 |= [r2 ∧ ♦r1 → d]P2

µ, s2 |= [r1 ∧ ♦(r2 ∧ ♦r1)→ d]P1

ν, s1 |= [r2 ∧ ♦(r1 ∧ ♦(r2 ∧ ♦r1))→ d]P2

µ, s0 |= [r1 ∧ ♦(r2 ∧ ♦(r1 ∧ ♦(r2 ∧ ♦r1)))→ d]P1

Therefore, backward inductively, under the assumption of common rational-ity, we observe that P1 should make a d move at s0. Furthermore, this can begeneralized to many other games.

Conjecture 27. Assuming common rationality in RSL framework, backward in-duction scheme produces a unique solution in games with ordinal pay-offs.

Argument for the proof of the theorem is quite straight-forward. Even ifthe strategies may be set valued, and hence return a set of moves per state,the assumption of common rationality forces the player to choose the movewhich returns the highest pay-off. Since this fact is known among players, byinduction, we can show that the solution is unique.

Let us follow the backward induction scheme again with the updated gametree given above. Notice that the specification [r1 → d]P1 does not conform withµ!d (as µ!d, s 6|= [[r1 → d]P1!d]P1 for all s)9. Therefore, the move that rationality

9We slightly abuse the formal language here. For a specification σ, we put µ, s |= ¬σ if it is notthe case that µ, s |= σ.

43

implies is not admissible for P1 from that point on. Thus, µ!d conforms to thosespecifications that implies an a move. Thus, µ!d, s4 |= [> → a]P1 · ¬[r1 → d]P1.Therefore, at s3, being rational, P2 choses the move with the highest pay-off,and makes an a move. Thus, ν, s3 |= r2 ∧ (♦¬[r1 → d])→ a]P2. Similarly, at s1,we have ν, s1 |= r2 ∧ (♦¬[r1 → d] ∧ ♦(r2 ∧ (♦¬[r1 → d]))) → a]P2. Finally, atthe root, we have µ!d, s0 |= [> → a]P1. Thus, in the restricted centipede game,P1 makes an a move.

4.3.3 Axiomatization, Completeness, and Complexity

Before we discuss the axiomatization of RSL and its completeness with respectto the intended semantics, we first define the set of enabled moves for the newconstruct as ([σ!a]i)(s) = σ(s) − {a} reflecting our intuition. We now give thesyntax of RSL, which is the same as that of SL:

p | (σ)i : a | ¬ϕ | ϕ1 ∧ ϕ2 | 〈a〉ϕ | σ i ψ

The semantics and the truth definitions of the formulas are defined as earlierwith the exception of strategy specifications for restrictions (cf. Section 2). Theaxiom system of RSL consists of the axioms and rules of SL together with thefollowing axioms for the added specification construct:

• (σ!a)i : c↔ (¬δσi (a)) ∧ (σ : c), c 6= a

• (σ!a)i : a↔ ⊥

Here, δσi (a) = turni ∧ (σ)i : a, meaning it is i’s turn and the move a isenabled by strategy σ for the player i. Now, observe that the satisfaction of([σ!a]i)i : c gives c ∈ ([σ!a]i)(s). For this reason, we have ([σ!a]i)i : a false. Thesoundness of the given axioms are straightforward and hence skipped. The newspecification we introduced does not bring along an extra derivation rule sincethe new operator is at the level of specifications, not the formulas.

Conjecture 28. The axiom system of RSL is complete with respect to the givensemantics.

Now we discuss a reduction10, of first SL, and then of RSL to (multi-modal)Computational Tree Logic CTL*. We refer the readers who are not familiar withCTL* to (Emerson, 1990; Emerson & Halpern, 1986). First, we discuss howwe construct a CTL* model based on a given SL/RSL model, then describe atranslation from SL to CTL*. Let us take a strategy model M = (T, V ) whereT = (S,⇒, s0, λ). Notice that the function⇒ was defined from S × Σ to S. Wecan redefine it by currying. Given⇒: S×Σ→ S, we can define the transition a⇒:S → S for each move a. Therefore, we can curry⇒ to get⇒=

⋃a∈Σ

a⇒. We canthink of M as a pointed multi-modal CTL* tree model M∗ = (S, s0, {

a⇒}a∈Σ, V )that has next time modalities for each action. In this case, corresponding to

10Thanks to Sujata Ghosh for suggesting this reduction.

44

each binary relation a⇒, we will have a dynamic next-time modality Xa whichquantifies over the sub-path on the same branch (note that state formulas arealso path formulas (Emerson & Halpern, 1986)). For the reflexive-transitiveclosure of a⇒, we will use � for all accessible future times in the same branch.Before giving the translation of SL specifications and formulas into CTL*, let usintroduce some special propositions. We label the states that are returned bya strategy µ with the proposition strategyµ, stipulating that strategyµ holdsonly at those points. Notice that given two points in the domain of µ, there is aunique path between these two (which satisfies strategyµ). Moreover, we useturni as a proposition that denotes that it is i’s turn to play, i.e. s |= turni iffs ∈ Si. We give two translations. First, tr translates strategy specifications toCTL* formulas while the second Tr translates SL formulas to CTL* formulas.Given a strategy µ, conformation to µ is translated as follows.

• tr([ψ → a]i) = Tr(ψ)→ E(strategyµ ∧ Xa>)

• tr(σ1 + σ2) = tr(σ1) ∨ tr(σ2)

• tr(σ1 · σ2) = tr(σ1) ∧ tr(σ2)

Notice that in the first case the translation makes use of the formula trans-lation Tr for formula ψ defined below. Assuming the correctness of Tr (whichwe will show next in Theorem 30), correctness of the translation tr is straight-forward.

Conjecture 29. Let µ be a strategy and σ a strategy specification in SL, and let µ∗

be the corresponding (sub)tree in a CTL* model. Then, µ, s |= σ iff µ∗, s |= tr(σ).

Here follows the translation Tr of formulas from SL to CTL* skipping theBoolean cases. Note that the translation is very similar to the Kripke semanticsfor Propositional Dynamic Logic where for each action a, a relation Ra and amodality associated with Ra are introduced.

• Tr(〈a〉ϕ) = XaTr(ϕ)

• Tr((σ)i : c) = Xc> for c ∈ σ(s)

• Tr((σ)i : c) = ⊥ if c /∈ σ(s)

• Tr(σ i ψ) = E�(strategyσ ∧ (Tr(ψ) ∧ (turni → enabledσ)))

Here, the atom enabledσ is true at a state s in CTL* if and only if for at leastone a ∈ Σ we have Xa> and a ∈ σ(s). Finally, the CTL* correspondence of σ(s)to the set of enabled moves at s by the strategy specification σ is defined exactlyas before with one small arrangement in the definition of admissible moves at agiven state s, namely moves(s) = {a : s |=CTL∗ Xa>}. As an example, considerthe translation of the proposition outµ = a. Recall that it means that a is theunique outgoing edge according to strategy µ at the state where the formula isinterpreted. Therefore, there is a branch that is followed by strategy µ, and at

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that branch, at the current state, a is an admissible move. This corresponds tothe translation E(strategyµ ∧ Xa>). The following theorem summarizes ourefforts here.

Conjecture 30. Let M be a SL model and let M∗ be its CTL* correspondent. Then,M, s |= ϕ iff M∗, s |= Tr(ϕ) for any state s ∈ S.

Notice that the translation we suggest is model-dependent. For example,the predicate strategyµ, depends on the strategy µ, thus the model. For thisreason, the suggested translation is not entirely syntactic and does not give usan immediate decidability result (using the fact that CTL* is decidable). How-ever, model checking problem for CTL* is PSPACE-complete (Emerson, 1990).Therefore, we have an upper bound for the complexity of the model checkingproblem for SL. We next show that the model checking problem for both SL andRSL is in PSPACE.

Conjecture 31. The model checking problem for SL is in PSPACE.

Corollary 6. The model checking problem for RSL is in PSPACE.

46

References

ABRAMSKY, SAMSON, & ZVESPER, JONATHAN. 2010. From Lawvere toBrandenburger-Keisler: Interactive forms of diagonalization and self-reference. CoRR, abs/1006.0992.

ACZEL, PETER. 1988. Non-well-founded Sets. Vol. 14. CSLI Lecture Notes.

AIELLO, MARCO, & VAN BENTHEM, JOHAN. 2002. A Modal Walk Through Space.Journal of Applied Non-Classical Logics, 12(3-4), 319–363.

AIELLO, MARCO, VAN BENTHEM, JOHAN, & BEZHANISHVILI, GURAM. 2003. Rea-soning About Space: the Modal Way. Journal of Logic and Computation,13(6), 889–920.

ARTEMOV, S., DAVOREN, J. M., & NERODE, A. 1997 (June). Modal Logics andTopological Semantics for Hybrid Systems. Tech. rept. Mathematical SciencesInstitute, Cornell University.

ARTEMOV, SERGEI. 2009a. Intelligent Players. Tech. rept. Department of Com-puter Science, The Graduate Center, The City University of New York.

ARTEMOV, SERGEI. 2009b. Rational Decisions in Non-probablistic Setting. Tech.rept. TR-2009012. Department of Computer Science, The Graduate Center,The City University of New York.

AUMANN, ROBERT J. 1995. Backward Induction and Common Knowledge ofRationality. Games and Economic Behavior, 8(1), 6–19.

AWODEY, STEVE. 2006. Category Theory. Oxford University Press.

BALBIANI, PHILIPPE, BALTAG, ALEXANDRU, VAN DITMARSCH, HANS, HERZIG, AN-DREAS, HOSHI, TOMOHIRO, & DE LIMA, TIAGO. 2007. What Can We Achieveby Arbitrary Announcements? A Dynamic Take on Fitch’s Knowability. In:SAMET, DOV (ed), Procedings of the 11th Conference on Theoretical Aspectsof Rationality and Knowledge (TARK-2007).

BALBIANI, PHILIPPE, BALTAG, ALEXANDRU, VAN DITMARSCH, HANS, HERZIG, AN-DREAS, & DE LIMA, TIAGO. 2008. ‘Knowable’ as ‘known after an announce-ment’. Review of Symbolic Logic, 1(3), 305–334.

BALTAG, ALEXANDRU, & MOSS, LAWRENCE S. 2004. Logics for Epistemic Pro-grams. Synthese, 139(2), 165–224.

BARWISE, JON. 1988. Three Views of Common Knowledge. Pages 365–379 of:PUBLISHERS, MORGAN KAUFMANN (ed), Proceedings of the 2nd conferenceon Theoretical aspects of reasoning about knowledge.

BARWISE, JON, & MOSS, LAWRENCE S. 1996. Vicious Circles. Center for Studyof Language and Information.

47

BASKENT, CAN. 2007 (July). Topics in Subset Space Logic. M.Phil. thesis, Institutefor Logic, Language and Computation, Universiteit van Amsterdam.

BASKENT, CAN. 2011. Paraconsistency and Topological Semantics. under sub-mission, available at canbaskent.net.

BEZHANISHVILI, GURAM, & GEHRKE, MAI. 2005. Completeness of S4 with re-spect to the Real Line: Revisited. Annals of Pure and Applied Logic, 131(??),287–301.

BEZHANISHVILI, GURAM, ESAKIA, LEO, & GABELAIA, DAVID. 2005. Some Resultson Modal Axiomatization and Definability for Topological Spaces. StudiaLogica, 81(3), 325–55.

BEZIAU, JEAN-YVES. 2005. Paraconsistent Logic from a Modal Viewpoint. Jour-nal of Applied Logic, 3(1), 7–14.

BOURBAKI, NICOLAS. 1966. General Topology. Elements of Mathematics, vol. 1.Addison-Wesley.

BRANDENBURGER, ADAM, & KEISLER, H. JEROME. 2006. An Impossibility Theo-rem on Beliefs in Games. Studia Logica, 84, 211–240.

CATE, BALDER TEN, GABELAIA, DAVID, & SUSTRETOV, DMITRY. 2009. ModalLanguages for Topology: Expressivity and Definability. Annals of Pure andApplied Logic, 159(1-2), 146–170.

CHELLAS, BRIAN. 1980. Modal Logic. Cambridge University Press.

DABROWSKI, ANDREW, MOSS, LAWRENCE S., & PARIKH, ROHIT. 1996. Topo-logical Reasoning and the Logic of Knowledge. Annals of Pure and AppliedLogic, 78(1-3), 73–110.

EMERSON, E. ALLEN. 1990. Temporal and Modal Logic. Pages 995–1072 of:VAN LEEUWEN, JAN (ed), Handbook of Theoretical Computer Science, vol. B.Amsterdam, the Netherlands: Elsevier.

EMERSON, E. ALLEN, & HALPERN, JOSEPH Y. 1986. ‘Sometimes’ and ‘Not Never’Revisited: On Branching versus Linear Time Temporal Logic. Journal of theAssociation for Computing Machinery, 33(1), 151–178.

FAGIN, RONALD, HALPERN, JOSEPH Y., MOSES, YORAM, & VARDI, MOSHE Y.1995. Reasoning About Knowledge. MIT Press.

GEORGATOS, KONSTANTINOS. 1994. Knowledge Theoretic Properties of Topolog-ical Spaces. Pages 147–159 of: MASUCH, M., & POLOS, L. (eds), KnowledgeRepresentation and Uncertainty. Lecture Notes in Artificial Intelligence, vol.808. Springer-Verlag.

GEORGATOS, KONSTANTINOS. 1997. Knowledge on Treelike Spaces. Studia Log-ica, 59(2), 271–301.

48

GERBRANDY, JELLE. 1999. Bisimulations on Planet Kripke. Ph.D. thesis, Instituteof Logic, Language and Computation; Universiteit van Amsterdam.

GHOSH, SUJATA, RAMANUJAM, RAMASWAMY, & SIMON, SUNIL. 2010. On Strat-egy Composition and Game Composition.

GOLDBLATT, ROBERT. 2006. Mathematical Modal Logic: A View of Its Evolution.In: GABBAY, DOV M., & WOODS, JOHN (eds), Handbook of History of Logic,vol. 6. Elsevier.

GOODMAN, NICOLAS D. 1981. The Logic of Contradiction. Zeitschrift fur Math-ematische Logik und Grundlagen der Mathematik, 27(8-10), 119–126.

HALPERN, JOSEPH Y. 2001. Substantive Rationality and Backward Induction.Games and Economic Behavior, 37(2), 425–435.

HARSANYI, J. C. 1967. Games with incomplete information played by ‘Bayesian’players: I. The basic model. Management Science, 14(3), 159–182.

HODKINSON, IAN, WOLTER, FRANK, & ZAKHARYASCHEV, MICHAEL. 2000. Decid-able Fragments of First-Order Temporal Logics. Annals of Pure and AppliedLogic, 106(??), 85–134.

KREMER, PHILIP, & MINTS, GRIGORI. 2005. Dynamic Topological Logic. Annalsof Pure and Applied Logic, 131(1-3), 133–58.

LAWVERE, F. WILLIAM. 1969. Diagonal Arguments and Cartesian Closed Cat-egories. Pages 134–145 of: DOLD, A., & ECKMANN, B. (eds), CategoryTheory, Homology Theory and their Applications II. Lecture Notes in Mathe-matics, vol. 92. Springer.

LAWVERE, F. WILLIAM. 1991. Intrinsic Co-Heyting Boundaries and the LeibnizRule in Certain Toposes. Pages 279–281 of: CARBONI, AURELIO, PEDICCHIO,MARIA, & ROSOLINI, GUISEPPE (eds), Category Theory. Lecture Notes inMathematics, vol. 1488. Springer.

LEBBINK, HENK-JAN, WITTEMAN, CILIA L. M., & MEYER, JOHN-JULES. 2004.Dialogue Games for Inconsistent and Biased Information. Electronic Notesin Theoretical Computer Science, 85(2).

MCKINSEY, J. C. C., & TARSKI, ALFRED. 1944. The Algebra of Topology. TheAnnals of Mathematics, 45(1), 141–191.

MCKINSEY, J. C. C., & TARSKI, ALFRED. 1946. On Closed Elements in ClosureAlgebras. The Annals of Mathematics, 47(1), 122–162.

MIRIMANOFF, DMITRY. 1917. Les antimonies de Russell et de Burali-Forti etle probleme fondamental de la theorie des ensembles. L’EnseignementMathematique, 19(1), 37–52.

49

MORTENSEN, CHRIS. 2000. Topological Seperation Principles and Logical The-ories. Synthese, 125(1-2), 169–178.

MOSS, LAWRENCE S. 2009. Non-wellfounded Set Theory. In: ZALTA, EDWARD N.(ed), The Stanford Encyclopedia of Philosophy, fall 2009 edn.

MOSS, LAWRENCE S., & PARIKH, ROHIT. 1992. Topological Reasoning and theLogic of Knowledge. Pages 95–105 of: MOSES, YORAM (ed), Proceedings ofTARK IV.

PACUIT, ERIC. 2007. Understanding the Brandenburger-Keisler Belief Paradox.Studia Logica, 86(3), 435–454.

PARIKH, ROHIT. 1991. Finite and Infinite Dialogues. Pages 481–498 of:MOSCHOVAKIS, Y. (ed), Proceedings of a Workshop on Logic from ComputerScience. Springer.

PARIKH, ROHIT, MOSS, LAWRENCE S., & STEINSVOLD, CHRIS. 2007. Topologyand Epistemic Logic. In: AIELLO, MARCO, PRATT-HARTMAN, IAN E., & VAN

BENTHEM, JOHAN (eds), Handbook of Spatial Logics. Springer.

PLAZA, JAN A. 1989. Logic of Public Communication. Pages 201–216 of: EM-RICH, M. L., PFEIFER, M. S., HADZIKADIC, M., & RAS, Z. W. (eds), 4thInternational Symposium on Methodologies for Intelligent Systems.

PRIEST, GRAHAM. 1998. What Is So Bad About Contradictions? Journal ofPhilosophy, 95(8), 410–426.

PRIEST, GRAHAM. 2002. Paraconsistent Logic. Pages 287–393 of: GABBAY, DOV,& GUENTHNER, F. (eds), Handbook of Philosophical Logic, vol. 6. Kluwer.

PRIEST, GRAHAM. 2006. In Contradiction. 2. edn. Oxford University Press.

PRIEST, GRAHAM. 2009. Dualising Intuitionistic Negation. Principia, 13(3),165–84.

RAHMAN, SHAHID, & CARNIELLI, WALTER A. 2000. The Dialogical Approach toParaconsistency. Synthese, 125, 201–231.

RAMANUJAM, RAMASWAMY, & SIMON, SUNIL. 2008. A Logical Structure forStrategies. Pages 183–208 of: Logic and the Foundations of Game and Deci-sion Theory (LOFT 7). Amsterdam University Press.

REYES, GONZALO E., & ZOLGAGHARI, HOUMAN. 1996. Bi-Heyting Algebras,Toposes and Modalities. Journal of Philosophical Logic, 25(1), 25–43.

SCHWALBE, ULRICH, & WALKER, PAUL. 2001. Zermelo and the Early History ofGame Theory. Games and Economic Behavior, 34(1), 123–137.

STALNAKER, ROBERT. 1994. On the Evaluation of Solution Concepts. Theoryand Decision, 37(1), 49–73.

50

STALNAKER, ROBERT. 1996. Knowledge, Belief and Counterfactual Reasoning inGames. Economics and Philosophy, 12(2), 133–163.

STALNAKER, ROBERT. 1998. Belief Revision in Games: Forward and BackwardInduction. Mathematical Social Sciences, 36(1), 31–56.

STELL, J. G., & WORBOYS, M. F. 1997. The Algebraic Structure of Sets ofRegions. Pages 163–174 of: HIRTLE, STEPHEN C., & FRANK, ANDREW U.(eds), Spatial Information Theory. Lecture Notes in Computer Science, vol.1329 Springer, for COSIT 97.

VAN BENTHEM, JOHAN. 2006. ”One is a Lonely Number”: Logic and Commu-nication. In: CHATZIDAKIS, Z., KOEPKE, P., & POHLERS, W. (eds), LogicColloquium ’02. Lecture Notes in Logic, vol. 27. Association for SymbolicLogic.

VAN BENTHEM, JOHAN. 2007. Rational Dynamics and Epistemic Logic in Games.International Game Theory Review, 9(1), 13–45.

VAN BENTHEM, JOHAN, & BEZHANISHVILI, GURAM. 2007. Modal Logics of Space.In: AIELLO, MARCO, PRATT-HARTMAN, IAN E., & VAN BENTHEM, JOHAN

(eds), Handbook of Spatial Logics. Springer.

VAN BENTHEM, JOHAN, & GHEERBRANT, AMELIE. 2010. Game Solution, Epis-temic Dynamics and Fixed-Point Logics. Fundamenta Infomaticae, 100(1-4), 19–41.

VAN BENTHEM, JOHAN, & SARENAC, DARKO. 2004. The Geometry of Knowledge.Pages 1–31 of: Aspects of Universal Logic. Travaux Logic, vol. 17.

VAN BENTHEM, JOHAN, VAN EIJCK, JAN, & KOOI, BARTELD. 2005. Logics ofCommunication and Change. Tech. rept. Institute for Logic, Language andComputation.

VAN BENTHEM, JOHAN, BEZHANISHVILI, GURAM, CATE, BALDER TEN, &SARENAC, DARKO. 2006. Modal Logics for Product Topologies. Studia Log-ica, 84(3), 375–99.

VAN DITMARSCH, HANS, VAN DER HOEK, WIEBE, & KOOI, BARTELD. 2007. Dy-namic Epistemic Logic. Springer.

YANOFSKY, NOSON S. 2003. A Universal Approach to Self-Referential Paradoxes,Incompleteness and Fixed Points. The Bulletin of Symbolic Logic, 9(3), 362–386.

ZVESPER, JONATHAN, & PACUIT, ERIC. 2010. A Note on Assumption-Completeness in Modal Logic. Pages 190–206 of: BONANNO, GIACOMO,LOWE, BENEDIKT, & VAN DER HOEK, WIEBE (eds), Proceedings of the 8thinternational conference on Logic and the foundations of game and decisiontheory. LOFT ’08.

51