Epistemic logic

Examples

# Epistemic logic

## Idea

Epistemic logic is the branch of modal logic which is concerned with epistemology, hence with notions of knowledge and sometimes also notions of belief, although these may be treated separately by doxastic logic. In its applied form it has found considerable use in computer science and Artificial Intelligence.

The key modal operators for epistemic logic are forms of necessity and possibility corresponding to “It is known to be the case that” and “It may be the case that (so far as is known)”. As a form of modal logic, the semantics typically used is possible worlds semantics where the worlds correspond to maximally specific ways the world could have been. Philosophers disagree as to whether to distinguish the space of metaphysically possible worlds from the space of epistemically possible worlds (Kment 21).

## Introduction

Epistemic modalities are captured in epistemic modal logic by interpreting “necessarily $\phi$” as saying “It is known that proposition $\phi$ is true”. It is common also to indicate the epistemic agent, so that corresponding to agent $x$ we have “$x$ knows that proposition $\phi$ is true”.

In the closely related ‘provability logic’, the basic modal operator interprets as “it is provable that $\phi$.

Notice that the notions of possibility and necessity have different senses in ordinary language. For example, if we say ‘$P$ is possible’, we may mean that $P$ is: epistemically possible, not ruled out by anything I know; physically possible, not ruled out by the laws of physics; logically possible, not ruled out by the laws of logic. Some suggest that there is a further type of possibility, metaphysical possibility intermediate between logical and physical possibility. Metaphysical possibility would allow that different laws of physics might apply.

## Epistemic formulae

These are variants of the formulae of the basic modal language. The basic modal operators are, here, labelled $K_i$ since they relate to ‘knowledge’ and indicate the epistemic agent, the knower, $i$. These correspond to the $\Box$ operators in the standard form, and are used in preference to the dual $\lozenge$ forms since they are more immediately relevant to applications.

More formally, we have $P$ or $Prop$, is a set of countably (finite or infinite) many atomic formulae. there is also a set $A$, often called the set of ‘agents’ and taken to be $A = \{1,\ldots,m\}$. The set of epistemic formulae (= basic $m$-agent epistemic language) will be denoted $\mathcal{L}^m_K(P)$ is given by the rules

$\phi ::= p \mid \bot \mid \neg \phi \mid \phi_1 \wedge \phi_2 \mid K_i\phi for i\in A.$

We read $K_i \phi$ as ‘’agent $i$ knows that $\phi$’’.

The converse or dual operators, denoted $M_i$ (so that $M_i \phi = \neg K_i\neg \phi$) reads as ‘’agent $i$ considers $\phi$ is possible’’.

###### Note

The ‘agent’ terminology is extremely useful, but in pure modal logic texts is not used so much. It does provide an ‘intuition’ and an interpretation however.

## Axioms for epistemic logic

The question then arises as to which axioms of modal logic are appropriate to the epistemic case. Answers will depend on what interpretation is given to ‘know’ in specifying how the operator $K$ behaves. E.g., there is a difference between the knowledge of a realistic, cognitively-limited agent and that of some idealized agent with boundless resources.

It is generally admitted that axiom (T) should hold, here

$\mathbf{T}: K \phi \to \phi,$

which states that if $\phi$ is known then $\phi$ is true. Truth is generally taken to be a precondition of knowledge.

A much more contentious issue in the field is whether to admit an epistemic version of axiom (4). Known as the KK principle or KK thesis, this corresponds to

$\mathbf{4}: K \phi \to K K \phi,$

which states that when $\phi$ is known to be true then it is also known that it is known. T

It is even possible to challenge the admission of two of the most fundamental modal axioms, axioms (N) and (K). These correspond to

$\mathbf{N} \phantom{,}\colon\phantom{,} \text{If}\phantom{,}P\phantom{,}\text{is a theorem, then so is}\phantom{,}K P$

and

$\mathbf{K}: K(P \to Q) \to (K P \to K Q).$

An understanding of N that it states that all theorems are known is evidently problematic. Similarly for K, it is not clear that we know the deductive consequences of the collection of propositions that we know, such as the application of modus ponens to every instance of known $P \to Q$ and $P$.

On the other hand, we might consider epistemic logic to represent the reasoning of an ideal, logically omniscient, agent, and so admit N and K.

## Models for epistemic logics

The geometric or combinatorial semantics of epistemic models follows the same techniques of Kripke frames as at geometric models for modal logics, whilst the algebraic models are BAOs that is Boolean algebras with operators. As usual the Kripke frames semantics is an example of coalgebraic semantics?.

### General

Early discussion of epistemic modal logic:

and introducing the formalization as K modal logic:

A fairly recent book on epistemic logics and their applications (which was used for some of the material above):

• J.- J. Ch. Meyer and W. Van der Hoek, Epistemic logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995,

General books on modal logic with discussion of epistemic logic:

• Patrick Blackburn, M. de Rijke and Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

• Marcus Kracht, Tools and Techniques in Modal Logic, Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999,

• Johan van Benthem, Dynamic logic for belief revision (pdf)

• Boris Kment, Varieties of Modality, SEP

### S5 modal logic as epistemic logic

Discussion of S5 modal logic as epistemic logic:

• Robert J. Aumann, Interactive epistemology I: Knowledge, International Journal of Game Theory 28 (1999) 263–300 $[$doi:10.1007/s001820050111$]$

$[$Ditmarsch, Hoek & Kooi (2008):$]$ “in Aumann’s survey paper on interactive epistemology the reader will immediately recognise the system S5.”

p. 35: “in a precise sense the S5 properties completely characterize our definition of knowledge”

pp. 121: “What standard logics of knowledge capture is not actual knowledge, but potential knowledge — what one is entitled to know. The switch to potential knowledge means we drop all considerations of complexity $[...]$ It is easy to see that, under such an assumption, a knowledge modality should be a normal modal operator. But, what else should be required? $[...]$ All these together make a knowledge operator obey the S5 conditions.”

p. 198: “Modal epistemic logic, the logic of knowledge, provides a very natural interpretation to the accessibility relation in Kripke models. For an agent $i$, two worlds $w$ and $v$ are connected (written $R_i w v$), if the agent cannot (epistemically) distinguish them. In other words, we have $R_i v w$ if, according to $i$’s information at $w$, the world might as well be in state $v$, or that $v$ is compatible with i’s information at w. Using this interpretation of access, $R_i$ is obviously an equivalence relation. $[...]$ Thus, we are in the realm of the multi-modal logic $S5_m$.”

• Hans Ditmarsch, Wiebe Hoek, Barteld Kooi, Epistemic Logic, chapter 2 in: Dynamic Epistemic Logic, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) 337, Springer (2008) $[$doi:10.1007/978-1-4020-5839-4_2, pdf$]$

p. 11: “The logical system S5 is by far the most popular and accepted epistemic logic”

• Dov Samet, S5 knowledge without partitions, Synthese 172 (2010) 145–155 $[$doi:10.1007/s11229-009-9469-0$]$

• Meghyn Bienvenu, Hélène Fargier, Pierre Marquis, Knowledge Compilation in the Modal Logic S5, Proceedings of the AAAI Conference on Artificial Intelligence 24 1 (2010) $[$doi:10.1609/aaai.v24i1.7587, pdf$]$

p. 1: “Propositional epistemic logic S5 is a well-known modal logic which is suitable for representing and reasoning about the knowledge of a single agent”

• Rasmus Rendsvig, John Symons, Epistemic Logic, The Stanford Encyclopedia of Philosophy (2011) $[$web$]$

Fagin, Halpern, Moses, and Vardi (1995) and many others use S5 for knowledge”

• Rachel Boddy, Epistemic Issues and Group Knowledge, Amsterdam (2014) $[$pdf$]$

p. 13: “Formal approaches to epistemology – such as game theory and computer science – typically assume the S5 conditions for knowledge, which is (partly) explained by the convenient formal properties of the logic. Philosophers typically opt for a weaker notion. Hintikka (1962), for instance, argues that the proper logic for knowledge is the modal system S4”

• Yakoub Salhi and Michael Sioutis, A Resolution Method for Modal Logic S5, EPiC Series in Computer Science 36 (2015) 252–262 $[$pdf$]$

• Ronald de Haan, Iris van de Pol, On the Computational Complexity of Model Checking for Dynamic Epistemic Logic with S5 Models $[$arXiv:1805.09880$]$