Translating doxastic logics to epistemic logics

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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

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Abstract

This thesis focuses on extending a translation recently proposed by researchers at the ILLC between the doxastic logic KD45 and the epistemic logic S4.2. In this paper we concentrate on the technical aspects of this translation and investigate its implications for 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 by the 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 of Stalnaker [14]. We then lift our notion of companionship to the doxastic extensions of KD45 and determine the individual least and greatest epistemic companions. In the end of the thesis we visualize the results in a diagram of the different companions.

Title: Translating doxastic logics to epistemic logics

Authors: Mike van Niehoff, mikevanniehoff@me.com, 10593969 Supervisor: Dr Nick Bezhanishvili

Second grader: Prof.dr Sonja Smets Date: July 7, 2016

Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam

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1 Introduction

Epistemic modal logic makes use of modal logic tools to give a formal account of the informational 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 thesis we will distinguish between epistemic and doxastic languages which make use of knowl-edge or belief respectively. Hintikka is broadly acknowlknowl-edged as the founder of epistemic modal 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 compatible with 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 the modal formula p → p, which forces a model to be reflexive, i.e. all states within the model will have to be in a binary relation with itself. In models that describe the knowledge of agents we would certainly want this to be true for, if not, it would be possible for an agent to be unaware of her own reality, which already intuitively leads to a contradiction. It would then for instance be possible for an agent to stand outside and know for sure that it is not raining whilst it actually is raining. However, in a model that is concerned with belief, the formula need not necessarily be true. If an agent were inside a room with no windows, it might be perfectly possible for her to believe that it is not raining whilst it actually is. So for belief and knowledge, we would want different modal formulas to be true.

Such intuitions can however often be more complicated to formalize than first appears and can lead to unwanted results. For this reason discussions on the appropriate modal logics of knowledge and belief are still relevant and although the logic KD45 is generally accepted to be the logic of belief, there are to this day disputes on the true modal logic of knowledge. As it seems of course philosophically relevant for knowledge and belief to be 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 knowlknowl-edge is the modal logic 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 to use 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 a

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knowledge in this setting is the system S4.2 and the logic of belief is the standard system KD45. The translation creates a link between the logic of knowledge and the logic 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 introduced by [1] and [14] and investigate its implications for epistemic and doxastic logics that extend the basic logics of knowledge and belief, S4.2 and KD45 respectively. We will call the introduced relation between the epistemic logic S4.2 and the doxastic logic KD45 a 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 would be KD45, where S4.2 is the least one. This gives rise to the question of whether we can find S4.2 and S4.4 to respectively be the least and greatest bound for the epistemic companions of KD45. Thus we will also look into different bounds of the companions of the extended logics and finally try to provide a full picture of the logics that extend the basic companionship between S4.2 and KD45 to provide a better understanding of the aforementioned connections.

The aim of this thesis is therefore to investigate the relationship between S4.2 and KD45 that is introduced by the translation proposed in [1] and to look into how it extends to other epistemic and doxastic logics. These results will be visualized in a diagram which provides a better understanding of the aforementioned connections.

The thesis is organized as follows. In Chapter 2 we provide basic definitions of modal logic 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 rooted frames 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 follows from it. Chapter 5 lifts the results of Chapter 4 to extensions of the basic case between KD45 and S4.2, for which we introduce the notion of companionship. We introduce the epistemic logic S4.4 and prove, as main technical result of the thesis, that S4.2 and S4.4 are the respective least and greatest epistemic companion of KD45, which confirms a conjecture of Stalnaker [14]. We then extend our view of companionship to all doxastic extensions of KD45 and show that these have a determinable least and greatest epistemic companion 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. Finally we conclude the thesis with Chapter 6 by giving a brief summary and by providing some discussion 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 set of propositional letters Φ whose elements are usually denoted by p, q, r, etc., and a unary modal operator ♦ (‘diamond’). The well-formed formulas ϕ of the basic modal language are 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 the following 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 ) the powerset 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,

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 M0 = (W0, R0, V0) be models and let M0 be a submodel of M (i.e. W0 ⊆ W , R0 is the restriction of R to W0 and V0 is the restriction of V to M0). We say that M0 is a generated submodel of M if M0 is a submodel of M and the following condition holds:

if w is in M0 and Rwv, then v is in M0.

If M0 is the smallest generated submodel of M that contains the set X ⊆ W then M0 is generated by X. If M0 is generated by a singleton set, we say that M0 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 M0 = (W0, R0, V0) be models. A mapping f : M → M0 is a bounded morphism if it satisfies the following conditions:

(i) w and f (w) satisfy the same proposition letters. (ii) If Rwv then R0f (w)f (v).

(iii) If R0f (w)v0 then there exists v such that Rwv and f (v) = v0.

Note that the definition can also be used to describe a bounded morphism between frames 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 that C ( 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 ∈ W

Rwv ⇒ Rvw (Rvw ⇒ Rwv)

If every point in a submodel M0 of M is quasi-maximal, we say that M0 is quasi-maximal (-minimal).

Normal Modal Logic

Using the basic modal language, a normal modal logic Λ is a set of formulas that contains all instances of propositional tautologies, (p → q) → (p → q), and ♦p ↔ ¬p, and that 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 the minimal normal modal logic, for any normal logic L we can write L = K + Σ, where Σ is a set of formulas such that L is closed under modus ponens, uniform substitution and generalization.

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) if there 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 KD45 and 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’ operator with its dual hBi := ¬B¬.

Definition 11. KD45

Define the following Axioms: D : Bp → hBip

4 : Bp → BBp 5 : hBip → BhBip

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 be used to find the first order order correspondents of the axioms ([6]). This gives that D is the axiom of seriality, which corresponds to the first order sentence ∀x∃y(Rxy) and that the axioms 4 and 5 correspond to transitivity (∀x∀y∀z((Rxy ∧ Ryz) → Rxz)) and Euclidianness (∀x∀y∀z((Rxy ∧ Rxz) → Rzy)). By Sahlqvist’s Theorem it then follows that the logic KD45 is sound and complete with respect to the class of serial, transitive and Euclidean frames.

It is a well known result that KD45 also has the finite model property, i.e. it is complete with respect to the class of finite, serial, transitive and Euclidean frames frames. This can easily be shown by using the least filtration ([6]) and the Filtration Theorem (Theorem 2.39 [6]). Also, by Proposition 2.6 from [6] about generated submodels, we know that validity is preserved when taking point-generated subframes. Thus, if C is the class of finite, 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 and Euclidean frames. We will prove that any frame in C is of one of the following two shapes, 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, and the 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 3.1 or Figure 3.2.

Proof. Let F = (W, R) ∈ C and suppose F has a reflexive root r. This root is connected to all other points in W , by the definition of a root and transitivity of F . For any w, 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 restricted to W \ {r} gives an equivalence relation. However, for any w ∈ W , Rrr and Rrw implies Rwr 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 Figure 3.1.

Now suppose F ∈ C has a non-reflexive root r. This root is connected to all other points 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 is non-reflexive, W \ r is a maximal equivalence class under R and thus a cluster. This shows 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 second shape, 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,

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Definition 14. S4.2

Define the following Axioms: T : Kp → p

4 : Kp → KKp .2 : hKiKp → KhKip

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 of the axioms. This gives that T is the axiom of reflexivity (Section 3.2) and that the axioms 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 sound and 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 easily be shown by using the transitive filtration ([6]) and the Filtration Theorem (Theorem 2.39 [6]). Again, by Proposition 2.6 from [6] about generated submodels, we also know that validity is preserved when taking a point-generated subframe. Thus, if ˜C is the class of finite, rooted, reflexive, transitive and directed frames, the next corollary follows directly.

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 directed frames. 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-maximal cluster 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 and transitive, 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 of the directedness of the frames in ˜C we also have that all points are related to some point within 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 other points in W , by the definition of a root and transitivity of G. As G is reflexive, we have that {r} ∪ {v ∈ W | Rvr} forms a cluster (As it is a maximal equivalence class under R), say Q-min cl, and by transitivity and r being the root of G, all points in Q-min cl see all other points in G. Thus it also follows that no point outside Q-min cl sees any point in Q-min cl, as a cluster is a maximal equivalence class under R. So Q-min cl is quasi-minimal.

G has a 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-quasi-maximal cluster. This must mean that there are x, y ∈ W quasi-maximal which do not see each other. Then as Rrx and Rry, by directedness of G, there is a point w ∈ W not equal to x or y (as x and y do not see each other) such that Rxw and Ryw. But then, as y is quasi-maximal, we have that Rwy and, as G is transitive, that Rxy which gives a contradiction. Thus there is a quasi-maximal cluster, say Q-max cl, in G.

Between these two clusters there could be other reflexive points. These all see some point in the quasi-maximal cluster, because they are seen themselves by the root and G is directed. For let z ∈ W be some arbitrary point that is not in Q-min cl and not in 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, y must 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 the classes 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 prove the full and faithful link between KD45 and S4.2 that follows from it. This relation between two logics will be known as companionship and this notion can be extended to other doxastic and epistemic logics, which will be done in the following chapters. Definition 17. Translation (.) : LB → LK

For 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)∗= hKiKϕ∗

5. (hBiϕ)∗ = KhKiϕ∗ (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 from 3.1, that KD45 = Log(C), where C is the class of finite, rooted, serial, transitive and Euclidean 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 3.1 that F is of the shape

CL

or

CL

But as the first shape is a generated subframe of the second, by Theorem 3.14 [6], we only 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+ ∈ ˜C and 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 ∈ L} B,

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 both M 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 with}

complexity 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 = hBiχ.

Let M, w |= ψ, so M, w |= hBiχ. 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 of the shape of F+, for all points in M+ it holds that hKiχ∗ is true and we have that M+, w |= KhKiχ∗. So M+, w |= ψ∗ as required.

Now let M+, w |= ψ∗, so M+, w |= KhKiχ∗. As hKiχ∗ has to hold for all successors of w, 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 in

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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. But then by Claim 19 it follows that M+, w 6|= ϕ∗, where M+= (F+, V ). Thus F+6|= ϕ∗ and we have proven that S4.2 6` ϕ∗. As we considered arbitrary ϕ ∈ LB, by contrapositive it

holds 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 from 3.2 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 to CL. Note that G−∈ C_KD45 (Which is the class of finite KD45-frames. For this direction we 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 with complexity 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, so we omit the proof. Now consider ψ = hBiχ.

Let M, w |= ψ∗, so M, w |= KhKiχ∗. As hKiχ∗ 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∗. By induction hypothesis M−, u |= χ and as u ∈ CL and G− is of the shape as described in the beginning of the prove, we have that M−, w |= hBiχ. So M−, w |= ψ as required.

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Now let M−, w |= ψ, so M−, w |= hBiχ. Thus there is v ∈ W such that Rwv and M−, v |= χ. However, because of the shape of G− mentioned in the beginning, 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 of the shape of G as described in the beginning, for all points in M it holds that hKiχ∗ is true and we have that M, w |= KhKiχ∗. So M, 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∗. But then by Claim 20 it follows that M−, w 6|= ϕ, where M−= (G−, V ). Thus G−6|= ϕ∗ _and

we have proven that KD45 6` ϕ∗. As we considered arbitrary ϕ ∈ LB, by contrapositive

it holds for all ϕ ∈ LBthat KD45 ` ϕ ⇒ S4.2 ` ϕ∗. This concludes the proof of Theorem

18.

In this chapter we have proven the translation, [1], to be full and faithful in Theorem 18. However, as the translation is not limited to KD45 and S4.2 but is expressed for all doxastic and epistemic logics in the languages of belief and knowledge, we can ask ourselves 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 we can impose bounds on this special link between doxastic and epistemic logics. It is also interesting to look at extensions of KD45, all of which we will do in the following chapters.

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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 between KD45 and S4.2. We will thus introduce the more general notion of companionship in the beginning of this chapter.

Firstly, we will introduce the epistemic logic S4.4, which Stalnaker [14] reasons to be the greatest of logics from which its derived logic of belief would be KD45. It is a stronger logic than S4.2, but weaker than S5. We furthermore prove that S4.4 is a companion of KD45 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 and prove that it is actually possible to provide a bound for the companions of KD45, where S4.4 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 L0 an epistemic logic. If or each formula ϕ ∈ LB

L ` ϕ ⇐⇒ L0 ` ϕ∗,

then we that L0 is an epistemic companion of L and that L is a doxastic companion of L0.

5.1 S4.4

In this section we introduce the epistemic normal modal logics S4.4. Definition 22. S4.4

Define the following Axioms: T : Kp → p

4 : Kp → KKp

.4 : p ∧ hKiKq → K(p ∨ q)

The Logic of S4.2 is then given by

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As all axioms of S4.4 are Simple Sahlqvist Formulas, the Sahlqvist Algorithm can be used to find the first order order correspondents of the axioms ([6]). This gives that T is the axiom of reflexivity, which corresponds to the first order sentence ∀x(Rxx) and that 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 and complete 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 S4.4 is locally tabular ([8]), which implies that S4.4 has the finite model property. Also, by Proposition 2.6 from [6] about generated submodels, we know that validity is preserved when taking a point-generated subframe. Thus, if ˆC is the class of finite, rooted, reflexive, transitive and .4 frames, the next corollary follows directly.

Corollary 23. S4.4 = Log( ˆC)

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 ˆC, thus the finite, rooted, reflexive, transitive and .4 frames. We will prove that any frame in ˆC 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, and the cluster itself, indicates that r sees every point within the cluster.

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

Proof. Let F = (W, R) ∈ ˆC and let r be a root in F . This root is connected to all other points in W , by the definition of a root and transitivity of F . For any w, v ∈ W , such that r is distinct from w and v, we have that Rrv and Rrw implies Rvw as F is .4, but also Rrw and Rrv, so Rwv. Furthermore Rrw and Rrw, so Rww for any w ∈ W distinct from 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 any w ∈ W distinct from r we already had Rwy, we know by transitivity of F that Rwr for all 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 Figure 5.1.

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It will however, by Theorem 3.14 from [6], be enough to only consider the second shape, 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 companionship of KD45 and S4.2) in Section 4. We will therefore only give a sketch of the proof that S4.4 is a companion of KD45.

Proof. We will again do both parts of the proof by contrapositive. Note that we know from 3.1, KD45 = Log(C), where C is the class of finite, rooted, serial, transitive and Euclidean frames and from before that, S4.4 = Log( ˆC), where ˆC 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) ∈ ˆC such that F 6|= ϕ. Let F+ = (W, R+) where we add a reflexive relation to the root of F . So now F+∈ ˆC and F and F0 are of the 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. But then by Claim 19, which we proved in Section 4, it follows that M+, w 6|= ϕ∗, where M+= (F+, V ). Thus F+ 6|= ϕ∗ _{and we have proven that S4.4 6` ϕ}∗_{. As we considered}

arbitrary 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) ∈ ˆC such that G 6|= ϕ∗. We know from before that G is of the shape

CL

or

CL

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

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

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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 simpler than that of Claim 20 as G is of the same shape as in Claim 20 but G− here is a rooted version 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∗. But then by Claim 26 it follows that M−, w 6|= ϕ, where M−= (G−, V ). Thus G−6|= ϕ∗ and we have proven that KD45 6` ϕ∗. As we considered arbitrary ϕ ∈ LB, by contrapositive

it holds for all ϕ ∈ LBthat KD45 ` ϕ ⇒ S4.2 ` ϕ∗. This concludes the proof of Theorem

25.

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 epistemic companion of KD45. However, in Theorem 25 we saw that S4.4 is an epistemic companion of 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 these companions. As it can be shown easily that no companion of KD45 can be greater or equal to the logic S5 we already know that an upper bound, at least of some kind, can be 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 suffices/

to show that S5 is not an epistemic companion of KD45. Let ϕ := Bp → p. Then per definition of S5, ϕ∗ := hKiKp → p ∈ S5. As ϕ /∈ KD45, it follows that S5 is not an epistemic 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, which leads 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.

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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 that S5 6⊆ L. Suppose for the sake of contradiction that L 6⊆ S4.4. So there is some ψ ∈ L such that ψ /∈ S4.4. This means that there is an F |= S4.4 such that F 6|= ψ, which, by Section 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 ∧ hBiB¬p), where ϕn is a formula that makes sure that there are n points in the

quasi-maximal cluster, ϕn := hBi(p1 ∧ . . . ∧ pn−1∧ pn) ∧ hBi(p1 ∧ . . . ∧ pn−1∧ ¬pn))

and where we introduce p ∧ hBiB¬p, which is the negation of the symmetric axiom, to make sure that only KD45-frames that are not S5-frames validate ϕ. Observe that in a KD45-frame, of the shape as in Section 3.1, with n points in the quasi-maximal cluster and 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 ∧ hBiB¬p)∗, where

¬(p ∧ hBiB¬p)∗= ¬(p ∧ KhKihKiK¬p) and

¬ϕ∗_n= ¬(KhKi(p1∧ . . . ∧ pn−1∧ pn) ∧ KhKi(p1∧ . . . ∧ pn−1∧ ¬pn)).

Note that (p ∧ hBiB¬p)∗ still implies symmetry, as can be easily checked, and that ϕ∗_n, because of the K, still makes sure that there must be n points in any quasi-maximal cluster. As G 6|= ¬ϕ∗_n∨ ¬(p ∧ hBiB¬p)∗, we know that there is some valuation V and point x in G for which G, V, x 6|= ¬ϕ∗_n∨ ¬(p ∧ hBiB¬p)∗, so G, V, x |= ϕ∗_n∧ (p ∧ hBiB¬p)∗. Thus, as S4.2 ⊆ L and we assumed L to have the FMP, we know that G is of the following form, where x is some point in G that is not in the quasi-maximal cluster (because of the 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 the cluster CL and every other point in G to the reflexive root of F . This is illustrated in the 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 and find their respective companions.

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

CL

or

CL

Let (N, ≤) denote the set of natural numbers with its usual ordering and consider the set 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, 1) (1, 1) (2, 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 N t N, 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 set

of all extensions of KD45. We can then visualize this result in the following way, where

an_n+k+1, n, k ∈ N, denotes ↓ (n, 1) ∪ ↓ (n + k + 1, 0) and we again closely follow the

results of [4]. (N t N) = KD45 (N × {0})∪ ↓ (2, 1) a2 3 ↓ (2, 1) (N × {0})∪ ↓ (1, 1) a1 3 a1 2 ↓ (1, 1) (N × {0})∪ ↓ (0, 1) a0 3 a0 2 a0 1 ↓ (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 will make it easier to find their companions.

Companions of the extensions of KD45

The proves concerning companionship will now be more combinatorial and similar to earlier proves, therefore we will merely sketch some of the proves so that we can provide a 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 in

the 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 that these finite frames correspond also to the doxastic logic S5n, where we only consider

frames with up to and including n points in our quasi-maximal cluster. Thus it is easy to prove, very much in the same way as in chapter 4, that S5nis a companion of ↓ (n, 0)

for all n ∈ N. This shows that in the logics L such that S5 ⊆ L the doxastic and epistemic language overlap.

Theorem 31. Let L be any down-set of N × {1}. Let S4.2n⊆ L0 ⊆ S4.4n. Then

L0 is a epistemic companion of L.

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

CL

or

CL

where |CL| ranges up to and including n. Note that actually ↓ N × {1} = KD45 and that the finite frames correspond also to the Logic KD45n, where we only consider frames

with up to and including n points in our quasi-maximal cluster. In exactly the same way as in chapter 4 it follows that S4.2n is a companion of ↓ (n, 1) for all n ∈ N and

in much the same way as in the proof Theorem 28 if follows that S4.2n and S4.4n are

respectively the least and greatest companion of ↓ (n, 1) for all n ∈ N.

Theorem 32. Let L be an_n+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).

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5.4 Visualizing companionship

In this section we conclude the results of Chapter 5 and provide a full picture of the logics and their individual companions that extend the basic companionship between S4.2 and KD45. These results are visualized in the following figure in order to provide a better understanding of the aforementioned connections.

S4.2 KD45 S4.4 S4.2n S4.4n a2 3 a1 3 a1 2 a0 3 a02 a01 S5 S5

The figure shows as basis the logics KD45 and S4.2 and their companionship introduced by the translation, defined in Definition 17 and proven in Chapter 4. As we proved in Theorem 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 the diagram shows the doxastic extensions of KD45 and their companions that we found in Section 5.3. Observe that in the case of S5 and its extensions the translation provides an identity companionship. As described in the previous section, the companions of the logics an_n+k+1 can be found by taking the intersection between the corresponding companions of ↓ (n, 1) and ↓ (n + k + 1, 0), which can be found in the diagram. To preserve the clarity of the figure the companions of the logics an_n+k+1 are not depicted.

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6 Conclusion

In this paper we have concentrated on the technical aspects of the translation introduced by [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 to the epistemic companions of KD45. We then used Bezhanishvili’s paper, [4], to find all the extensions of KD45 in order to determine the corresponding companions. As was the aim of the thesis we also visualized the results in a diagram of all companionships in order provides a better understanding of the aforementioned connections.

An interesting question for further research from a philosophical perspective would be to 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 look at the implications of the translation within a more complex dynamic epistemic modal logic setting.

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7 Popular summary

Epistemic modal logic makes use of modal logic tools to give a formal account of the informational attitude that agents may have. It is thus concerned with what is known or believed by these agents in a modal system. Although these notions seem very intuitive to us, it can lead to complex and interesting situations, especially when using higher order 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 of dynamic epistemic modal logic. In the following picture epistemic modal reasoning and information updates are used to derive a new situation of knowledge.

With this use of dynamic epistemic modal logic it is possible for the third person to know that everyone wants beer. Namely, if the first person would not have wanted beer, she would have an-swered no, as not everyone would have wanted beer. However, the first person cannot yet say yes as she does not know if the other two also want beer. The same is true for the second per-son, who does not know whether the last one will want beer. So after the two statements, the third person now knows that the other two want beer and is able to answer yes.

To solve this example a modal logician would use models to describe what the agents know and change it whenever the common knowledge situation changes. However, until now it might still seem easy to keep track and solve the puzzle without the use of modal logic, but what if we have a situation in which an agent also knows that some other agent knows that she knows? This is called higher order knowledge and already appears more complex and hard to keep up with. Take for example a look at this scene from the film Pirates of the Caribbean where Jack Sparrow (Mr. Smith) arrives in Port Royal and 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 black guts out.

1

This 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 told it to you.

In this situation it already requires a bit more brainpower to keep track of what which agent knows or believes. Unknowingly, the three agents in this scene use higher order epistemic 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, it would be impossible to study the information attitude of all of them without help. Due to the use of models to visualize the problem and results within epistemic modal logic however, it will be possible to investigate such complex circumstances.

In this thesis we use epistemic modal logic to investigate a translation between the language of belief and the language of knowledge given in [1]. Just as with a normal everyday system of translation, it is important that the translation works well. One would not want to follow in the footsteps of Google Translate and make the same mistake as 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 and S4.2 respectively, our translation works well enough. However, more importantly, we also examine for which other logics this holds. Just as Google Translate might work better for a translation from Dutch to English than a translation from Dutch to Chinese, our translation between logics only works well within certain bounds. It is obviously relevant to know for which cases these good ‘links’ exist and this is why the main technical resuls of this thesis is the specification of the bounds to which KD45 can be translated well enough. Furthermore, it is possible to extend this basic case to extensions of KD45, just like extending our starting point from Dutch to other languages, to see to which logics these translate well enough. We can then even visualize these results in a diagram which show the ‘links’ between these epistemic modal logics. For more information and details we suggest reading the thesis and for a more detailed background on modal logic we refer to [6].

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

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Translating doxastic logics to epistemic logics