Translating doxastic logics to epistemic logics

Mike van Niehoff

July 7, 2016

Bachelor Thesis Mathematics

Supervisor: Dr Nick Bezhanishvili

Korteweg-de Vries Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Universiteit van Amsterdam

Abstract

This thesis focuses on extending a translation recently proposed by researchers at theILLC between the doxastic logic KD45 and the epistemic logic S4.2. In this paper weconcentrate on the technical aspects of this translation and investigate its implicationsfor epistemic and doxastic logics that extend the basic logics of knowledge and belief,S4.2 and KD45.

We define the general notion of companionship that expresses the ‘link’ introduced bythe translation and, as main technical result, prove that S4.2 and S4.4 are the respec-tive least and greatest epistemic companions of KD45, which confirms a conjecture ofStalnaker [14]. We then lift our notion of companionship to the doxastic extensions ofKD45 and determine the individual least and greatest epistemic companions. In the endof the thesis we visualize the results in a diagram of the different companions.

Title: Translating doxastic logics to epistemic logicsAuthors: Mike van Niehoff, mikevanniehoff@me.com, 10593969Supervisor: Dr Nick BezhanishviliSecond grader: Prof.dr Sonja SmetsDate: July 7, 2016

Korteweg-de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math

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Contents

1 Introduction 4

2 Basics of Modal Logic 6

3 Doxastic and epistemic logics 93.1 KD45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 S4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Full and faithful translation 13

5 Epistemic and doxastic companions 175.1 S4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Epistemic logics in between S4.2 & S4.4 . . . . . . . . . . . . . . . . . . . 205.3 Extensions of KD45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Visualizing companionship . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Conclusion 26

7 Popular summary 27

Bibliography 29

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1 Introduction

Epistemic modal logic makes use of modal logic tools to give a formal account of theinformational attitude that agents may have. It is concerned with notions such as knowl-edge, belief, uncertainty, and hence incomplete or partial information. Although epis-temic modal logic in modern terms often covers both knowledge and belief, in this thesiswe will distinguish between epistemic and doxastic languages which make use of knowl-edge or belief respectively. Hintikka is broadly acknowledged as the founder of epistemicmodal logic because of his account of knowledge and belief based on Kripke models in[7] and [11]. Hintikka’s book [11] contains the modal logic analyses of “knowing p” and“believing p” and their duals, “for all that one knows p is possible” and “it is compatiblewith believing p”.

As the concepts of knowing and believing certain information often seems very natural,several intuitions occur within epistemic modal logic. An easy example for this is themodal formula �p → p, which forces a model to be reflexive, i.e. all states within themodel will have to be in a binary relation with itself. In models that describe theknowledge of agents we would certainly want this to be true for, if not, it would bepossible for an agent to be unaware of her own reality, which already intuitively leads toa contradiction. It would then for instance be possible for an agent to stand outside andknow for sure that it is not raining whilst it actually is raining. However, in a modelthat is concerned with belief, the formula need not necessarily be true. If an agent wereinside a room with no windows, it might be perfectly possible for her to believe that itis not raining whilst it actually is. So for belief and knowledge, we would want differentmodal formulas to be true.

Such intuitions can however often be more complicated to formalize than first appearsand can lead to unwanted results. For this reason discussions on the appropriate modallogics of knowledge and belief are still relevant and although the logic KD45 is generallyaccepted to be the logic of belief, there are to this day disputes on the true modal logicof knowledge. As it seems of course philosophically relevant for knowledge and belief tobe related, attempts have been made to state “knowing p” in terms of “believing p”,which were not all successful as Gettier’s case, [10], against justified true belief as knowl-edge showed. In [14] Stalnaker argues that the correct logic of knowledge is the modallogic S4.2 which would allow belief to be defined in terms of epistemic logic by letting“believing p” be equivalent to “not knowing that you don’t know p” without having touse the rather strong logic S5. Stalnaker describes this as “strong belief” (referred to as“full belief” in [1]), which introduces a strong connection between knowledge and belief.

The recent ILLC preprint, [1], generalizes Stalnaker’s formalization and provides atranslation from the language of belief to the language of knowledge which introducesa strong connection between epistemic and doxastic logics. The associated logic of

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knowledge in this setting is the system S4.2 and the logic of belief is the standardsystem KD45. The translation creates a link between the logic of knowledge and thelogic of belief. For more details and recent developments we refer to [1], [2], [3], [5], [7],[8], [10], [11], [12], [13], [14]

In this paper we will concentrate on the technical aspects of the translation introducedby [1] and [14] and investigate its implications for epistemic and doxastic logics thatextend the basic logics of knowledge and belief, S4.2 and KD45 respectively. We will callthe introduced relation between the epistemic logic S4.2 and the doxastic logic KD45a companionship and will further investigate how it extends to other logics. Stalnaker,[14], reasons S4.4 to be the greatest of logics from which its derived logic of belief wouldbe KD45, where S4.2 is the least one. This gives rise to the question of whether wecan find S4.2 and S4.4 to respectively be the least and greatest bound for the epistemiccompanions of KD45. Thus we will also look into different bounds of the companions ofthe extended logics and finally try to provide a full picture of the logics that extend thebasic companionship between S4.2 and KD45 to provide a better understanding of theaforementioned connections.

The aim of this thesis is therefore to investigate the relationship between S4.2 andKD45 that is introduced by the translation proposed in [1] and to look into how itextends to other epistemic and doxastic logics. These results will be visualized in adiagram which provides a better understanding of the aforementioned connections.

The thesis is organized as follows. In Chapter 2 we provide basic definitions of modallogic that are used throughout the thesis. Chapter 3 introduces the doxastic and epis-temic normal modal logics KD45 and S4.2 and describes the classes of finite and rootedframes they are sound and complete for. In Chapter 4 we focus on the translation in-troduced in [1] and prove the full and faithful link between KD45 and S4.2 that followsfrom it. Chapter 5 lifts the results of Chapter 4 to extensions of the basic case betweenKD45 and S4.2, for which we introduce the notion of companionship. We introduce theepistemic logic S4.4 and prove, as main technical result of the thesis, that S4.2 and S4.4are the respective least and greatest epistemic companion of KD45, which confirms aconjecture of Stalnaker [14]. We then extend our view of companionship to all doxasticextensions of KD45 and show that these have a determinable least and greatest epistemiccompanion as well. We complete Chapter 5 with a figure depicting all results of the the-sis which provides a better understanding of the aforementioned connections. Finallywe conclude the thesis with Chapter 6 by giving a brief summary and by providing somediscussion questions for possible future research.

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2 Basics of Modal Logic

In this chapter we recall the basic definitions that will be used throughout the thesis.We will closely follow the presentation of modal logic provided by [6].

Basic modal language

In this thesis we will only consider basic modal languages which are defined using a setof propositional letters Φ whose elements are usually denoted by p, q, r, etc., and a unarymodal operator ♦ (‘diamond’). The well-formed formulas ϕ of the basic modal languageare given by the rule

ϕ := p | ⊥ | ¬ϕ | ϕ ∨ ψ | ♦ϕ,

where p ranges over elements of Φ. For the modal operator we have a dual operator� (‘box’) which is defined by �ϕ := ¬♦¬ϕ. Note that we can also make use of thefollowing classical abbreviations:

> := ¬⊥ϕ ∧ ψ := ¬(¬ϕ ∨ ¬ψ)

ϕ→ ψ := ¬ϕ ∨ ψ)

ϕ↔ ψ := (ϕ→ ψ) ∧ (ψ → ϕ)

Frames and Models

Definition 1. (Frames)We call a frame a structure F = (W,R), where W is a non-empty set and R a binary

relation on W . We will write Rwv or say that “w sees v” if (w, v) ∈ R.

Definition 2. (Models)We say that M is a model if M = (F, V ), where F is a frame and V a valuation, i.e.

a function from Φ to P (W ), where Φ is the set of propositional letters and P (W ) thepowerset of W .

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Definition 3. (Satisfaction)Let M = (F, V ), where F = (W,R), and let w ∈W . We inductively define the notion

of a formula ϕ being satisfied (or true) in M at w (notation: M,w |= ϕ) as follows:

M,w |= p iff w ∈ V (p), where p ∈ Φ,

M,w |= ⊥ never,

M,w |= ¬ϕ iff not M,w |= ϕ (notation: M,w 6|= ϕ),

M,w |= ϕ ∨ ψ iff M,w |= ϕ or M,w |= ψ,

M,w |= ♦ϕ iff for some v ∈W with Rwv we have that M,v |= ϕ.

Note that from this definition it follows that M,w |= �ϕ iff for all v ∈ W such thatRwv we have that M,v |= ϕ. We further write M |= ϕ if M,w |= ϕ for all w ∈ W andF |= ϕ if for any V we have (F, V ) |= ϕ. Let C be a class of frames then C |= ϕ if F |= ϕfor all F ∈ C.

Definition 4. (Rooted models)Let M = (W,R, V ) be a model. If there is an r ∈W such that R∗(r) = W , where R∗

is the reflexive and transitive closure of R, we say that M is a rooted model with root r.

Definition 5. (Generated Submodels)Let M = (W,R, V ) and M ′ = (W ′, R′, V ′) be models and let M ′ be a submodel of M

(i.e. W ′ ⊆W , R′ is the restriction of R to W ′ and V ′ is the restriction of V to M ′). Wesay that M ′ is a generated submodel of M if M ′ is a submodel of M and the followingcondition holds:

if w is in M ′ and Rwv, then v is in M ′.

If M ′ is the smallest generated submodel of M that contains the set X ⊆ W thenM ′ is generated by X. If M ′ is generated by a singleton set, we say that M ′ is a point-generated submodel. Note that the definition can also be used to describe a (point-)generated subframe by deleting the clause concerning valuations.

Definition 6. (Bounded Morphisms)Let M = (W,R, V ) and M ′ = (W ′, R′, V ′) be models. A mapping f : M → M ′ is a

bounded morphism if it satisfies the following conditions:

(i) w and f(w) satisfy the same proposition letters.

(ii) If Rwv then R′f(w)f(v).

(iii) If R′f(w)v′ then there exists v such that Rwv and f(v) = v′.

Note that the definition can also be used to describe a bounded morphism betweenframes by deleting the clause concerning valuations.

Definition 7. (Clusters)Let M = (W,R, V ) be a model. We say that C ⊆W is a cluster on M if the restriction

of R to C is an equivalence relation and if this is not the case for any D ⊆W such thatC ( D.

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Definition 8. (Quasi-maximal (-minimal))Let M = (W,R, V ) be a model. We say that a point w ∈ W is quasi-maximal (or

-minimal) if ∀v ∈WRwv ⇒ Rvw (Rvw ⇒ Rwv)

If every point in a submodel M ′ of M is quasi-maximal, we say that M ′ is quasi-maximal (-minimal).

Normal Modal Logic

Using the basic modal language, a normal modal logic Λ is a set of formulas that containsall instances of propositional tautologies, �(p→ q)→ (�p→ �q), and ♦p↔ ¬�p, andthat is closed under modus ponens, uniform substitution and generalization. The min-imal normal modal logic which contains only all instances of propositional tautologies,�(p → q) → (�p → �q), and ♦p ↔ ¬�p will be called K (for Kripke). As K is theminimal normal modal logic, for any normal logic L we can write L = K + Σ, where Σis a set of formulas such that L is closed under modus ponens, uniform substitution andgeneralization.

Definition 9. (L-frames)Let L be a normal modal logic and F = (W,R) a frame. We say that F is an L-frame

if F |= ϕ for all ϕ ∈ L. We define a formula ϕ to be a theorem of L if ϕ ∈ L.

Definition 10. (Log(C) and FMP)Let C be a class of frames, we then define Log(C) to be the set of formulas ϕ such that

C |= ϕ. We will say that a normal modal logic L has the finite model property (FMP) ifthere exists a (not necessarily finite) class C of finite frames such that L = Log(C).

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3 Doxastic and epistemic logics

In this chapter we will introduce the doxastic and epistemic normal modal logics KD45and S4.2 as the logic of belief and the logic of knowledge, respectively.

3.1 KD45

For the doxastic language of Belief, LB, formulas will be defined as follows

ϕ := p | ⊥ | ¬ϕ | ϕ ∧ ψ | Bϕ,

where p ∈ Prop (the countable set of Propositional letters) and B is the ‘box’ operatorwith its dual 〈B〉 := ¬B¬.

Definition 11. KD45Define the following Axioms:

D : Bp→ 〈B〉p

4 : Bp→ BBp

5 : 〈B〉p→ B〈B〉p

The Logic of KD45 is then given by

K +D + 4 + 5

As all axioms of KD45 are Simple Sahlqvist Formulas, the Sahlqvist Algorithm can beused to find the first order order correspondents of the axioms ([6]). This gives that Dis the axiom of seriality, which corresponds to the first order sentence ∀x∃y(Rxy) andthat the axioms 4 and 5 correspond to transitivity (∀x∀y∀z((Rxy ∧Ryz)→ Rxz)) andEuclidianness (∀x∀y∀z((Rxy ∧ Rxz) → Rzy)). By Sahlqvist’s Theorem it then followsthat the logic KD45 is sound and complete with respect to the class of serial, transitiveand Euclidean frames.

It is a well known result that KD45 also has the finite model property, i.e. it is completewith respect to the class of finite, serial, transitive and Euclidean frames frames. This caneasily be shown by using the least filtration ([6]) and the Filtration Theorem (Theorem2.39 [6]). Also, by Proposition 2.6 from [6] about generated submodels, we know thatvalidity is preserved when taking point-generated subframes. Thus, if C is the class offinite, rooted, serial, transitive and Euclidean frames, the next corollary follows directly.

Corollary 12. KD45 = Log(C)

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KD45-frames

By Corollary 12 we know that it will be enough to consider finite and rooted KD45-frames, so only the class of frames C, thus the finite, rooted, serial, transitive andEuclidean frames. We will prove that any frame in C is of one of the following twoshapes, depending on whether the root of the frame is reflexive or not.

Cluster

Figure 3.1

Cluster

Figure 3.2

Note that in Figure 3.2, the relation between the root not in the cluster, say r, andthe cluster itself, indicates that r sees every point within the cluster.

Lemma 13. Let F be a frame in C, then F is of the same shape as in either Figure 3.1or Figure 3.2.

Proof. Let F = (W,R) ∈ C and suppose F has a reflexive root r. This root is connectedto all other points in W , by the definition of a root and transitivity of F . For anyw, v ∈ W , Rrv and Rrw, so Rwv as F is euclidian, but also Rrw and Rrv, so Rvw.Furthermore Rrw and Rrw, so Rww for any w ∈W . We thus observe that R restrictedto W \{r} gives an equivalence relation. However, for any w ∈W , Rrr and Rrw impliesRwr as F is Euclidean. So R is also an equivalence relation on W and, as F is finite,F only consists of one finite cluster of all points in W , thus is of the shape as in Figure3.1.

Now suppose F ∈ C has a non-reflexive root r. This root is connected to all otherpoints in W , by the definition of a root and transitivity of F . So again, as F is euclidian,Rwv and Rvw for any w, v ∈ W . Furthermore Rrw and Rrw, so Rww for any w ∈ W .We thus observe that R restricted to W \ {r} gives an equivalence relation. As r isnon-reflexive, W \ r is a maximal equivalence class under R and thus a cluster. Thisshows that F is of the shape as in Figure 3.2.

It will however, by Theorem 3.14 from [6], be enough to only consider the secondshape, as the first is a generated subframe of the second.

3.2 S4.2

For the epistemic language of Knowledge, LK , formulas will be defined as follows

ϕ := p | ⊥ | ¬ϕ | ϕ ∧ ψ | Kϕ,

where p ∈ Prop and K is the ‘box’ operator with its dual 〈K〉 := ¬K¬.

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Definition 14. S4.2Define the following Axioms:

T : Kp→ p

4 : Kp→ KKp

.2 : 〈K〉Kp→ K〈K〉p

The Logic of S4.2 is then given by

K + T + 4 + .2

As in 3.1, we can use Sahlqvist’s algorithm to find the first order correspondents ofthe axioms. This gives that T is the axiom of reflexivity (Section 3.2) and that theaxioms 4 and .2 correspond to transitivity and directedness (∀x∀y∀z((Rxy ∧ Rxz) →∃w(Ryw ∧Rzw))) . By Sahlqvist’s Theorem it then follows that the logic S4.2 is soundand complete with respect to the class of reflexive, transitive and directed frames.

It is a well known result that S4.2 has the finite model property as well. This can easilybe shown by using the transitive filtration ([6]) and the Filtration Theorem (Theorem2.39 [6]). Again, by Proposition 2.6 from [6] about generated submodels, we also knowthat validity is preserved when taking a point-generated subframe. Thus, if C is theclass of finite, rooted, reflexive, transitive and directed frames, the next corollary followsdirectly.

Corollary 15. S4.2 = Log(C)

S4.2-frames

By Corollary 15 we know that it will be enough to consider finite and rooted S4.2-frames,so only the class of frames C, , thus the finite, rooted, reflexive, transitive and directedframes. We will prove that any frame in C is of the following shape.

Q-min cl

Q-max cl

Figure 3.3

In this figure Q-min cl and Q-max cl denote a quasi-minimal cluster and a quasi-maximalcluster respectively. There can however be points that are not in either of the clusters,which is indicated by the points between the two clusters. As frames in C are rooted andtransitive, we have that the quasi-minimal cluster sees every other point in the frame(i.e. every point within the quasi-minimal cluster sees every other point). Note that

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instead of a single root we have a cluster as root as frames in C are reflexive. Because ofthe directedness of the frames in C we also have that all points are related to some pointwithin the quasi-maximal cluster, which exists because of the finiteness of the frames.

Lemma 16. Let G be a frame in C, then F is of the same shape as in Figure 3.3.

Proof. Let G = (W,R) ∈ C and let r be a root of G. This root is connected to all otherpoints in W , by the definition of a root and transitivity of G. As G is reflexive, we havethat {r} ∪ {v ∈ W | Rvr} forms a cluster (As it is a maximal equivalence class underR), say Q-min cl, and by transitivity and r being the root of G, all points in Q-min clsee all other points in G. Thus it also follows that no point outside Q-min cl sees anypoint in Q-min cl, as a cluster is a maximal equivalence class under R. So Q-min cl isquasi-minimal.G has a quasi-maximal cluster as well. As G is finite, we know that there is a quasi-

maximal element, but suppose that it does not form a quasi-maximal cluster. This mustmean that there are x, y ∈W quasi-maximal which do not see each other. Then as Rrxand Rry, by directedness of G, there is a point w ∈W not equal to x or y (as x and y donot see each other) such that Rxw and Ryw. But then, as y is quasi-maximal, we havethat Rwy and, as G is transitive, that Rxy which gives a contradiction. Thus there is aquasi-maximal cluster, say Q-max cl, in G.

Between these two clusters there could be other reflexive points. These all see somepoint in the quasi-maximal cluster, because they are seen themselves by the root andG is directed. For let z ∈ W be some arbitrary point that is not in Q-min cl and notin Q-max cl. Then Rrz, but as also Rrx for some x in Q-max cl, by directedness of G,we know that there is some y such that Rzy and Rxy. However, as x is in Q-max cl, ymust be in Q-max cl as well.

Thus G is of the shape as in Figure 3.3.

In this chapter we introduced the normal modal logics KD45 and S4.2 and described theclasses of finite and rooted frames they are sound and complete for.

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4 Full and faithful translation

In this chapter we will give the definition of the translation introduced in [1] and provethe full and faithful link between KD45 and S4.2 that follows from it. This relationbetween two logics will be known as companionship and this notion can be extended toother doxastic and epistemic logics, which will be done in the following chapters.

Definition 17. Translation (.) : LB → LKFor any ϕ ∈ LB, where LB is as in 3.1, the translation (ϕ)∗ of ϕ into LK , where LK

is as in 3.2, is defined recursively, as in [1], by:

1. (p)∗ = p where p ∈ Prop (the set of all Propositional letters)

2. (¬ϕ)∗ = ¬ϕ∗

3. (ϕ ∧ ψ)∗ = ϕ∗ ∧ ψ∗

4. (Bϕ)∗ = 〈K〉Kϕ∗

5. (〈B〉ϕ)∗ = K〈K〉ϕ∗ (This follows from 4.)

Theorem 18. For each formula ϕ ∈ LB:

KD45 ` ϕ ⇐⇒ S4.2. ` ϕ∗

Proof. We will do both parts of the proof by contrapositive. Note that we know from3.1, that KD45 = Log(C), where C is the class of finite, rooted, serial, transitive andEuclidean frames. Further, as seen in 3.2, S4.2 = Log(C), where C is the class of finite,rooted, reflexive, transitive and directed frames. Let ϕ ∈ LB be arbitrary.

“⇐”

Let KD45 6` ϕ, then there is some F = (W,R) ∈ C such that F 6|= ϕ. We know from 3.1that F is of the shape

CLor

CL

But as the first shape is a generated subframe of the second, by Theorem 3.14 [6], weonly need to consider the second case. So let F be of that form.

Now consider F+ = (W,R+) where we make the root of F reflexive . So now F+ ∈ Cand is of the shape

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CL

Claim 19. Let M = (F, V ) and M+ = (F+, V ). For all w ∈M and all ϕ ∈ LB,

M,w |= ϕ ⇐⇒ M+, w |= ϕ∗.

Proof. We will proof this by induction on ϕ.

Basis case

Consider first ϕ = p. Then obviously M,w |= p⇔M+, w |= p∗, as the valuation of bothM and M+ is V and p∗ = p.

Induction case

Our induction hypothesis (IH) is that M,w |= ψ ⇔ M+, w |= ψ∗ holds for all ψ withcomplexity smaller than the complexity of ϕ.

Let ψ = ¬χ. Then

M,w |= ψ ⇔M,w |= ¬χ⇔M,w 6|= χ

(IH)⇔ M+, w 6|= χ∗

⇔M+, w |= ¬χ∗

⇔M+, w |= ψ∗

Let ψ = θ ∧ χ. Then

M,w |= ψ ⇔M,w |= θ and M,w |= χ

(IH)⇔ M+, w |= θ∗ and M+, w |= χ∗

⇔M+, w |= ψ∗

Now consider ψ = 〈B〉χ.Let M,w |= ψ, so M,w |= 〈B〉χ. Thus there is v ∈ W such that Rwv and M,v |= χ.

However, because of the shape of F , we know that v must be in the cluster CL. Further,by induction hypothesis, we have that M+, v |= χ∗. Thus, as v ∈ CL, because ofthe shape of F+, for all points in M+ it holds that 〈K〉χ∗ is true and we have thatM+, w |= K〈K〉χ∗. So M+, w |= ψ∗ as required.

Now let M+, w |= ψ∗, so M+, w |= K〈K〉χ∗. As 〈K〉χ∗ has to hold for all successors ofw, it also has to hold for v ∈ CL. Thus there is u ∈ CL such that R+vu and M+, u |= χ∗.By induction hypothesis M,u |= χ and as u ∈ CL and F is of the shape as described inthe beginning of the proof, we have that M,w |= 〈B〉χ. So M,w |= ψ as required.

We thus proved Claim 19 by induction on ϕ.

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As F 6|= ϕ, we know there is some M = (F, V ) and w ∈ W such that M,w 6|= ϕ. Butthen by Claim 19 it follows that M+, w 6|= ϕ∗, where M+ = (F+, V ). Thus F+ 6|= ϕ∗ andwe have proven that S4.2 6` ϕ∗. As we considered arbitrary ϕ ∈ LB, by contrapositive itholds for all ϕ ∈ LB that S4.2 ` ϕ∗ ⇒ KD45 ` ϕ.

“⇒”

Let S4.2 6` ϕ∗, then there is some G = (W,R) ∈ C such that G 6|= ϕ∗. We know from3.2 that G is of the shape

CL

x

Now consider the following G− = (W,R−) where we leave the quasi-maximal cluster(CL) of G intact, all the other points of W\CL we pull apart, only with a relation toCL. Note that G− ∈ CKD45 (Which is the class of finite KD45-frames. For this directionwe do not require G− to be rooted.) and of the form

CL

x

Claim 20. Let M = (G,V ) and M− = (G−, V ). For all w ∈M and all ϕ ∈ LB,

M,w |= ϕ ⇐⇒ M−, w |= ϕ∗.

Proof. We will prove this by induction on ϕ.

Basis case

Obviously for ϕ = p the claim holds.

Induction case

Our induction hypothesis (IH) is that M,w |= ψ∗ ⇔ M−, w |= ψ holds for all ψ withcomplexity smaller than the complexity of ϕ.

The cases ψ = ¬χ, ψ = δ ∧χ can easily be verified in the same way as in Claim 19, sowe omit the proof. Now consider ψ = 〈B〉χ.

Let M,w |= ψ∗, so M,w |= K〈K〉χ∗. As 〈K〉χ∗ has to hold for all successors of w,it has to hold for v ∈ CL. Thus there is u ∈ CL such that Rvu and M,u |= χ∗. Byinduction hypothesis M−, u |= χ and as u ∈ CL and G− is of the shape as described inthe beginning of the prove, we have that M−, w |= 〈B〉χ. So M−, w |= ψ as required.

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Now let M−, w |= ψ, so M−, w |= 〈B〉χ. Thus there is v ∈ W such that Rwv andM−, v |= χ. However, because of the shape of G− mentioned in the beginning, weknow that v must be in the cluster CL. Further, by induction hypothesis, we have thatM,v |= χ∗. Thus, as v ∈ CL, because of the shape of G as described in the beginning,for all points in M it holds that 〈K〉χ∗ is true and we have that M,w |= K〈K〉χ∗. SoM,w |= ψ∗ as required.

We thus proved Claim 20 by induction on ϕ.

As G 6|= ϕ∗, we know there is some M = (G,V ) and w ∈W such that M,w 6|= ϕ∗. Butthen by Claim 20 it follows that M−, w 6|= ϕ, where M− = (G−, V ). Thus G− 6|= ϕ∗ andwe have proven that KD45 6` ϕ∗. As we considered arbitrary ϕ ∈ LB, by contrapositiveit holds for all ϕ ∈ LB that KD45 ` ϕ⇒ S4.2 ` ϕ∗. This concludes the proof of Theorem18.

In this chapter we have proven the translation, [1], to be full and faithful in Theorem18. However, as the translation is not limited to KD45 and S4.2 but is expressed forall doxastic and epistemic logics in the languages of belief and knowledge, we can askourselves what implications the translation can have for logics extending KD45 and S4.2.First of all, we should explore whether Theorem 18 holds only for S4.2 and whether wecan impose bounds on this special link between doxastic and epistemic logics. It isalso interesting to look at extensions of KD45, all of which we will do in the followingchapters.

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5 Epistemic and doxastic companions

In this chapter we generalize the previous results to other doxastic and epistemic logics.In order to do this we will have to extend our understanding of the translation betweenKD45 and S4.2. We will thus introduce the more general notion of companionship in thebeginning of this chapter.

Firstly, we will introduce the epistemic logic S4.4, which Stalnaker [14] reasons to bethe greatest of logics from which its derived logic of belief would be KD45. It is a strongerlogic than S4.2, but weaker than S5. We furthermore prove that S4.4 is a companion ofKD45 as well, showing that a logic may have more than one companion.

Next, we will extend the result of Chapter 4 to other doxastic logics above S4.2 andprove that it is actually possible to provide a bound for the companions of KD45, whereS4.4 is the greatest companion of KD45.

Finally, we will look at all extensions of the doxastic logic KD45 by using the work of[4] and investigate their individual epistemic companions.

Definition 21. (Companions)Let L be a doxastic logic and L′ an epistemic logic. If or each formula ϕ ∈ LB

L ` ϕ ⇐⇒ L′ ` ϕ∗,

then we that L′ is an epistemic companion of L and that L is a doxastic companion of L′.

5.1 S4.4

In this section we introduce the epistemic normal modal logics S4.4.

Definition 22. S4.4Define the following Axioms:

T : Kp→ p

4 : Kp→ KKp

.4 : p ∧ 〈K〉Kq → K(p ∨ q)

The Logic of S4.2 is then given by

K + T + 4 + .4

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As all axioms of S4.4 are Simple Sahlqvist Formulas, the Sahlqvist Algorithm can beused to find the first order order correspondents of the axioms ([6]). This gives that Tis the axiom of reflexivity, which corresponds to the first order sentence ∀x(Rxx) andthat the axioms 4 and .4 correspond to transitivity and .4 (∀x∀y∀z((Rxy∧Rxz)→ (z =x ∨ Ryz))). By Sahlqvist’s Theorem it then follows that the logic S4.4 is sound andcomplete with respect to the class of reflexive, transitive and .4 frames.

As S4 ⊆ S4.4 and S4.4 is of finite height, we know by Theorem 12.21 from [8] that S4.4is locally tabular ([8]), which implies that S4.4 has the finite model property. Also, byProposition 2.6 from [6] about generated submodels, we know that validity is preservedwhen taking a point-generated subframe. Thus, if C is the class of finite, rooted, reflexive,transitive and .4 frames, the next corollary follows directly.

Corollary 23. S4.4 = Log(C)

S4.4-frames

By Corollary 23 we know that it will be enough to consider finite and rooted S4.4-frames,so only the class of frames C, thus the finite, rooted, reflexive, transitive and .4 frames.We will prove that any frame in C is of one of the following two shapes.

Cluster

Figure 5.1

Cluster

Figure 5.2

Note that in Figure 5.2, the relation between the root not in the cluster, say r, andthe cluster itself, indicates that r sees every point within the cluster.

Lemma 24. Let F be a frame in C, then F is of the same shape as in either Figure 5.1or Figure 5.2.

Proof. Let F = (W,R) ∈ C and let r be a root in F . This root is connected to all otherpoints in W , by the definition of a root and transitivity of F . For any w, v ∈ W , suchthat r is distinct from w and v, we have that Rrv and Rrw implies Rvw as F is .4, butalso Rrw and Rrv, so Rwv. Furthermore Rrw and Rrw, so Rww for any w ∈W distinctfrom r. We thus observe that R restricted to W \ {r} gives an equivalence relation.

Suppose that there is some y ∈ W distinct from r such that Ryr. But as for anyw ∈W distinct from r we already had Rwy, we know by transitivity of F that Rwr forall w ∈ W . So in this case R is also an equivalence relation on W and, as F is finite,F only consists of one finite cluster of all points in W , thus is of the shape as in Figure5.1.

Now suppose that there is no z ∈ W distinct from r such that Rzr. Then obviously,by the results in the beginning of the proof, W \{r} gives a cluster to which r is related.So F is of the shape as in Figure 3.2.

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It will however, by Theorem 3.14 from [6], be enough to only consider the secondshape, as the first is a generated subframe of the second.

Companionship

We will show that S4.4 is an epistemic companion of KD45 by proving the following:

Theorem 25. For each formula ϕ ∈ LB.

KD45 ` ϕ ⇐⇒ S4.4 ` ϕ∗.

Note that the proof very closely resembles the proof of Theorem 18 (the companionshipof KD45 and S4.2) in Section 4. We will therefore only give a sketch of the proof thatS4.4 is a companion of KD45.

Proof. We will again do both parts of the proof by contrapositive. Note that we knowfrom 3.1, KD45 = Log(C), where C is the class of finite, rooted, serial, transitive andEuclidean frames and from before that, S4.4 = Log(C), where C is the class of finite,rooted, reflexive, transitive and .4 frames. Let ϕ ∈ LB be arbitrary.

“⇐”

Let KD45 6` ϕ, then there is some F = (W,R) ∈ C such that F 6|= ϕ. Let F+ = (W,R+)where we add a reflexive relation to the root of F . So now F+ ∈ C and F and F ′ are ofthe same shape as in the first part of the proof of Theorem 18.

As F 6|= ϕ, we know there is some M = (F, V ) and w ∈ W such that M,w 6|= ϕ. Butthen by Claim 19, which we proved in Section 4, it follows that M+, w 6|= ϕ∗, whereM+ = (F+, V ). Thus F+ 6|= ϕ∗ and we have proven that S4.4 6` ϕ∗. As we consideredarbitrary ϕ ∈ LB, by contrapositive it holds for all ϕ ∈ LB that KD45 ` ϕ⇐ S4.4 ` ϕ∗.

“⇒”

Let S4.4 6` ϕ∗, then there is some G = (W,R) ∈ C such that G 6|= ϕ∗. We know frombefore that G is of the shape

CLor

CL

But as the first shape is a generated subframe of the second, by Theorem 3.14 [6], weonly need to consider the second case. So let G be of that form.

But then consider the following G− = (W,R−) ∈ C

CL

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Claim 26. Let M = (G,V ) and M− = (G−, V ). For all w ∈M and all ϕ ∈ LB,

M,w |= ϕ ⇐⇒ M−, w |= ϕ∗

Proof. This can again be proved by induction on ϕ. In fact, this proof is actually simplerthan that of Claim 20 as G is of the same shape as in Claim 20 but G− here is a rootedversion of the G− in Claim 20. For this reason we will omit the proof.

As G 6|= ϕ∗, we know there is some M = (G,V ) and w ∈W such that M,w 6|= ϕ∗. Butthen by Claim 26 it follows that M−, w 6|= ϕ, where M− = (G−, V ). Thus G− 6|= ϕ∗ andwe have proven that KD45 6` ϕ∗. As we considered arbitrary ϕ ∈ LB, by contrapositiveit holds for all ϕ ∈ LB that KD45 ` ϕ⇒ S4.2 ` ϕ∗. This concludes the proof of Theorem25.

5.2 Epistemic logics in between S4.2 & S4.4

From our basic case, Theorem 18 in Section 4, we already know that S4.2 is an epistemiccompanion of KD45. However, in Theorem 25 we saw that S4.4 is an epistemic companionof KD45 as well. This means that a logic can have more than one epistemic companion,which gives rise to the question whether we can find a least and greatest bound for thesecompanions. As it can be shown easily that no companion of KD45 can be greater orequal to the logic S5 we already know that an upper bound, at least of some kind, canbe determined.

Lemma 27. If S5 ⊆ L, then L is not an epistemic companion of KD45

Proof. As S5 ⊆ L, for all ϕ∗ ∈ LK , if ϕ∗ /∈ L we have that ϕ∗ /∈ S5. It thus sufficesto show that S5 is not an epistemic companion of KD45. Let ϕ := Bp → p. Then perdefinition of S5, ϕ∗ := 〈K〉Kp → p ∈ S5. As ϕ /∈ KD45, it follows that S5 is not anepistemic companion of KD45.

This already provides us with an upper bound for any epistemic companion of KD45.But we will now go on to show that we can find an even stricter bound in the logic S4.4.In fact we will see that this is the smallest upper bound on companions of KD45, whichleads to the following theorem.

Theorem 28. Let S4.2 ⊆ L and let all L have the finite modal property. Then

L is an epistemic companion of KD45 iff L ⊆ S4.4

Proof.“⇒”

Suppose S4.2 ⊆ L ⊆ S4.4. We want to show that for each formula ϕ ∈ LB.

KD45 ` ϕ ⇐⇒ L ` ϕ∗.

This is easily shown by contrapositive, as ϕ /∈ KD45 ⇒(1)ϕ∗ /∈ S4.4 ⇒

(2)ϕ∗ /∈ L.

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(1) By Theorem 25 (S4.4 companion of KD45),

(2) L ⊆ S4.4.

Conversely, ϕ∗ /∈ L⇒(1)ϕ∗ /∈ S4.2 ⇒

(2)ϕ /∈ KD45

(3) S4.2 ⊆ L,

(4) By Theorem 18 (S4.2 companion of KD45).

“⇐”

Let S4.2 ⊆ L and let L be a companion of KD45. Note that, by Lemma 27, we know thatS5 6⊆ L. Suppose for the sake of contradiction that L 6⊆ S4.4. So there is some ψ ∈ Lsuch that ψ /∈ S4.4. This means that there is an F |= S4.4 such that F 6|= ψ, which, bySection 5.1 and Corollary 23, must be of the following shape

CL

where |CL| = k, for some k ∈ N.

So F 6|= L. We will now show that this leads to a contradiction. Let ϕ := ϕn ∧(p ∧ 〈B〉B¬p), where ϕn is a formula that makes sure that there are n points in thequasi-maximal cluster, ϕn := 〈B〉(p1 ∧ . . . ∧ pn−1 ∧ pn) ∧ 〈B〉(p1 ∧ . . . ∧ pn−1 ∧ ¬pn))and where we introduce p ∧ 〈B〉B¬p, which is the negation of the symmetric axiom, tomake sure that only KD45-frames that are not S5-frames validate ϕ. Observe that in aKD45-frame, of the shape as in Section 3.1, with n points in the quasi-maximal clusterand a seperate root, ϕ holds. Which means that KD45 6` ¬ϕ.

Thus, as we assumed L to be a companion of KD45, KD45 6` ¬ϕ implies L 6` (¬ϕ)∗.Thus there exists some G |= L such that G 6|= (¬ϕ)∗.

However (¬ϕ)∗ := ¬ϕ∗n ∨ ¬(p ∧ 〈B〉B¬p)∗, where

¬(p ∧ 〈B〉B¬p)∗ = ¬(p ∧K〈K〉〈K〉K¬p)

and

¬ϕ∗n = ¬(K〈K〉(p1 ∧ . . . ∧ pn−1 ∧ pn) ∧K〈K〉(p1 ∧ . . . ∧ pn−1 ∧ ¬pn)).

Note that (p ∧ 〈B〉B¬p)∗ still implies symmetry, as can be easily checked, and thatϕ∗n, because of the K, still makes sure that there must be n points in any quasi-maximalcluster. As G 6|= ¬ϕ∗n ∨ ¬(p ∧ 〈B〉B¬p)∗, we know that there is some valuation V andpoint x in G for which G,V, x 6|= ¬ϕ∗n∨¬(p∧〈B〉B¬p)∗, so G,V, x |= ϕ∗n∧(p∧〈B〉B¬p)∗.Thus, as S4.2 ⊆ L and we assumed L to have the FMP, we know that G is of the followingform, where x is some point in G that is not in the quasi-maximal cluster (because ofthe formula ϕ∗):

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CLn

x

where |CLn| = n.

As |CLn| = n for any n ∈ N, we can choose n ≥ k, but then, as also |W \ CL| 6= ∅,we can define a map from G to F in the following way: We map the cluster CLn to thecluster CL and every other point in G to the reflexive root of F . This is illustrated inthe following figure and it is easy to see that this map is a bounded morphism.

CLn

Bounded morphism

CL

But then, as by our assumption F 6|= ψ, we know by Theorem 3.14 [6] that G 6|= ψ.So it follows that ψ /∈ L, contradicting L 6⊆ S4.4. Thus it follows that S4.2 ⊆ L ⊆ S4.4.

5.3 Extensions of KD45

In this chapter we will closely follow the work of paper [4] about extensions of KD45.We will try to visualize all extensions of KD45 in order to better understand them andfind their respective companions.

Let the logic L be an extension of KD45, so KD45 ⊆ L. So, as any frame F of anextension of KD45 is a KD45 frame as well (i.e. F |= KD45), by 3.1, we know thepossible shape of F is given by

CLor

CL

Let (N,≤) denote the set of natural numbers with its usual ordering and consider theset N t N = (N× {0}) ∪ (N× {1}) of two disjoint copies of the set of natural numbers.Now observe the following lattice where we defined an order R on N t N by putting((n, i), (m, j)) ∈ R iff n ≤ m and i ≤ j for n,m ∈ N and i, j ∈ {0, 1}

(0, 0) (1, 0) (2, 0)N× {0}

(0, 1) (1, 1) (2, 1)N× {1}

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We assign frames to these points in the lattice in the following way:

.to (0,0) and

. .to (1,0) and

. . .to (2,0) etc.

.

to (0,1) and

. .

to (1,1) and

. . .

to (2,1) etc.

Note that then N t N = KD45.

Definition 29. (Down-sets)We say that a set A ⊆ N t N is a down-set, notation ↓ A, if for all (n, i) ∈ A

(m, j)R(n, i)⇒ (m, j) ∈ A.

Now observe the set of all down-sets of NtN, thus of the previously described lattice.By the result of [4] it follows that this set, say Down↓(N t N), is isomorphic to the setof all extensions of KD45. We can then visualize this result in the following way, whereann+k+1, n, k ∈ N, denotes ↓ (n, 1) ∪ ↓ (n + k + 1, 0) and we again closely follow theresults of [4].

(N t N) = KD45

(N× {0})∪ ↓ (2, 1)

a23

↓ (2, 1)

(N× {0})∪ ↓ (1, 1)

a13

a12

↓ (1, 1)

(N× {0})∪ ↓ (0, 1)

a03

a02

a01

↓ (0, 1)

N× {0}

↓ (3, 0)

↓ (2, 0)

↓ (1, 0)

{(0, 0)}

∅

We have thus found a way to visualize and specify all extensions of KD45, which willmake it easier to find their companions.

Companions of the extensions of KD45

The proves concerning companionship will now be more combinatorial and similar toearlier proves, therefore we will merely sketch some of the proves so that we can providea full picture.

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Theorem 30. Let L be any down-set of N× {0}. Then

S5n is an epistemic companion of L,

where S5n contain all and only those S5-frames with up to and including n points inthe quasi-maximal cluster.

Proof. Because of our way of assigning frames, any down-set (N × {0})n = ↓ (n, 0)consists of all frames of the form

CL

where |CL| ranges up to and including n. Note that actually ↓ N×{0} = S5 and thatthese finite frames correspond also to the doxastic logic S5n, where we only considerframes with up to and including n points in our quasi-maximal cluster. Thus it is easyto prove, very much in the same way as in chapter 4, that S5n is a companion of ↓ (n, 0)for all n ∈ N. This shows that in the logics L such that S5 ⊆ L the doxastic and epistemiclanguage overlap.

Theorem 31. Let L be any down-set of N× {1}. Let S4.2n ⊆ L′ ⊆ S4.4n. Then

L′ is a epistemic companion of L.

Proof. Observe that any down-set (N×{1})n = ↓ (n, 1) consists of all frames of the form

CLor

CL

where |CL| ranges up to and including n. Note that actually ↓ N × {1} = KD45 andthat the finite frames correspond also to the Logic KD45n, where we only consider frameswith up to and including n points in our quasi-maximal cluster. In exactly the sameway as in chapter 4 it follows that S4.2n is a companion of ↓ (n, 1) for all n ∈ N andin much the same way as in the proof Theorem 28 if follows that S4.2n and S4.4n arerespectively the least and greatest companion of ↓ (n, 1) for all n ∈ N.

Theorem 32. Let L be ann+k+1, for some n, k ∈ N. Then

L is a doxastic companion of Comp1 ∩ Comp2,

where Comp1 and Comp2 are the companions of ↓ (n, 1) and ↓ (n+ k + 1, 0).

Proof. Remembering ann+k+1, n, k ∈ N, to be defined as ↓ (n, 1) ∪ ↓ (n + k + 1, 0) it iseasy to find their companions. As we know the companions of ↓ (n, 1) and ↓ (n+k+1, 0),it directly follows that the companion of ann+k+1, is Comp = Comp1 ∩ Comp2.

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5.4 Visualizing companionship

In this section we conclude the results of Chapter 5 and provide a full picture of thelogics and their individual companions that extend the basic companionship betweenS4.2 and KD45. These results are visualized in the following figure in order to provide abetter understanding of the aforementioned connections.

S4.2 KD45

S4.4

S4.2n

S4.4n

a23

a13

a12a03

a02

a01

S5S5

The figure shows as basis the logics KD45 and S4.2 and their companionship introducedby the translation, defined in Definition 17 and proven in Chapter 4. As we proved inTheorem 28, it is possible to find an upper bound for the epistemic companions of KD45,namely the logic S4.4 which proves the conjecture of Stalnaker in [14]. The rest of thediagram shows the doxastic extensions of KD45 and their companions that we found inSection 5.3. Observe that in the case of S5 and its extensions the translation providesan identity companionship. As described in the previous section, the companions ofthe logics ann+k+1 can be found by taking the intersection between the correspondingcompanions of ↓ (n, 1) and ↓ (n + k + 1, 0), which can be found in the diagram. Topreserve the clarity of the figure the companions of the logics ann+k+1 are not depicted.

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6 Conclusion

In this paper we have concentrated on the technical aspects of the translation introducedby [1] between doxastic and epistemic logics and investigated its implications for a rela-tionship, which we called companionship, between KD45 and S4.2 and their extensions.After proving that the translation is full and faithfull, we have generalized the result and,following the reasoning of Stalnaker ([14]), shown that S4.4 provides an upperbound tothe epistemic companions of KD45. We then used Bezhanishvili’s paper, [4], to find allthe extensions of KD45 in order to determine the corresponding companions. As wasthe aim of the thesis we also visualized the results in a diagram of all companionshipsin order provides a better understanding of the aforementioned connections.

An interesting question for further research from a philosophical perspective would beto examine what the results of this paper mean for the general notions of “belief” and“knowledge”. Also for future research from a mathematical perspective, one could lookat the implications of the translation within a more complex dynamic epistemic modallogic setting.

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7 Popular summary

Epistemic modal logic makes use of modal logic tools to give a formal account of theinformational attitude that agents may have. It is thus concerned with what is known orbelieved by these agents in a modal system. Although these notions seem very intuitiveto us, it can lead to complex and interesting situations, especially when using higherorder knowledge and belief or epistemic updates which are relevant in real life.

Let us look at an example of a real life situation in which 3 agents make use ofdynamic epistemic modal logic. In the following picture epistemic modal reasoning andinformation updates are used to derive a new situation of knowledge.

With this use of dynamic epistemic modal logicit is possible for the third person to know thateveryone wants beer. Namely, if the first personwould not have wanted beer, she would have an-swered no, as not everyone would have wantedbeer. However, the first person cannot yet sayyes as she does not know if the other two alsowant beer. The same is true for the second per-son, who does not know whether the last one willwant beer. So after the two statements, the thirdperson now knows that the other two want beerand is able to answer yes.

To solve this example a modal logician would use models to describe what the agentsknow and change it whenever the common knowledge situation changes. However, untilnow it might still seem easy to keep track and solve the puzzle without the use of modallogic, but what if we have a situation in which an agent also knows that some otheragent knows that she knows? This is called higher order knowledge and already appearsmore complex and hard to keep up with. Take for example a look at this scene from thefilm Pirates of the Caribbean where Jack Sparrow (Mr. Smith) arrives in Port Royaland states his business to two guards. 1

Mullroy: What’s your purpose in Port Royal, Mr. Smith?Murtogg: Yeah, and no lies.Jack Sparrow: Well, then, I confess, it is my intention to commandeer one of these ships,pick up a crew in Tortuga, raid, pillage, plunder and otherwise pilfer my weasely blackguts out.

1This example was introduced by ILLC researcher Alexandru Baltag in his lectures on dynamic epis-temic logic at the University of Amsterdam.

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Murtogg: I said no lies.Mullroy: I think he’s telling the truth.Murtogg: Don’t be stupid: if he were telling the truth, he wouldn’t have told it to us.Jack Sparrow: Unless, of course, he knew you wouldn’t believe the truth even if he toldit to you.

In this situation it already requires a bit more brainpower to keep track of what whichagent knows or believes. Unknowingly, the three agents in this scene use higher orderepistemic modal logic and might be better off if they used the modal perspective as well.Now if we would add even more, say some natural number n, agents to this situation, itwould be impossible to study the information attitude of all of them without help. Dueto the use of models to visualize the problem and results within epistemic modal logichowever, it will be possible to investigate such complex circumstances.

In this thesis we use epistemic modal logic to investigate a translation between thelanguage of belief and the language of knowledge given in [1]. Just as with a normaleveryday system of translation, it is important that the translation works well. Onewould not want to follow in the footsteps of Google Translate and make the same mistakeas for example translating “Elke avond laat ik mijn hond uit in het park om de hoek” to“Every night I leave my dog in the park around the corner”, which might sound similar,but definitely has a different implication.

We thus prove in this thesis that for the basic logics of belief and knowledge, KD45 andS4.2 respectively, our translation works well enough. However, more importantly, we alsoexamine for which other logics this holds. Just as Google Translate might work betterfor a translation from Dutch to English than a translation from Dutch to Chinese, ourtranslation between logics only works well within certain bounds. It is obviously relevantto know for which cases these good ‘links’ exist and this is why the main technical resulsof this thesis is the specification of the bounds to which KD45 can be translated wellenough. Furthermore, it is possible to extend this basic case to extensions of KD45, justlike extending our starting point from Dutch to other languages, to see to which logicsthese translate well enough. We can then even visualize these results in a diagram whichshow the ‘links’ between these epistemic modal logics. For more information and detailswe suggest reading the thesis and for a more detailed background on modal logic werefer to [6].

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[3] van Benthem, J. “Logic and the Dynamics of Information”, Minds and Machines,2003, Vol.13(4), pp.503-519

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[8] Chagrov, A., Zakharyaschev M. “Modal Logic”, Clarendon Press, Oxford, June 1963.

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[10] Gettier, E. “Is Justified True Belief Knowledge”, Analysis, June 1963, Volume 23,No. 6, pp. 121-123

[11] Hintikka, J. “Knowledge and Belief. An Introduction to the Logic of the Two No-tions”, Cornell University Press, Ithaca, N.Y., 1962.

[12] Hintikka, J. “Epistemic logic and the methods of philosophical analysis”, Aus-tralasian Journal of Philosophy, Jan 1, 1968, Vol.46, p.37.

[13] Hintikka, J. “ ’Knowing that one knows’ reviewed”, Synthese, 1970, Vol.21(2),pp.141-162.

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