## 1. Introduction



Aristotelian texts set the groundwork for discussions of the logic of
knowledge and belief, particularly *De Sophisiticis Elenchis*
as well as the *Prior* and *Posterior Analytics*. While
Aristotle addressed the four alethic modes of possibility, necessity,
impossibility, and contingency, Buridan, Pseudo Scotus, Ockham, and
Ralph Strode, helped to extend Aristotle's insights to epistemic
themes and problems (Boh 1993; Knuuttila 1993). During this period,
Pseudo Scotus and William of Ockham supplemented Aristotle's
study of mental acts of cognition and volition (see Boh 1993: 130).
Ivan Boh's studies of the history of fourteenth and fifteenth
century investigations into epistemic logic provide an excellent
coverage of the topic, especially his *Epistemic Logic in the Later
Middle Ages* (1993).



According to Boh, the English philosopher Ralph Strode formulated a
fully general system of propositional epistemic rules in his
influential 1387 book *Consequences* (Boh 1993: 135).
Strode's presentation built on the earlier logical treatises of
Ockham and Burley. Problems of epistemic logic were also discussed
between the 1330s and 1360s by the so-called Oxford Calculators, most
prominently by William Heytesbury and Richard Kilvington. By the
fifteenth century, Paul of Venice and other Italian philosophers also
engaged in sophisticated reflection on the relationship between
knowledge, truth, and ontology.



Discussions of epistemic logic during the medieval period share a
similar set of foundational assumptions with contemporary discussions.
Most importantly, medieval philosophers explored the connection
between knowledge and veracity: If I know *p*, then *p* is
true. Furthermore, many medieval discussions begin with an assumption
similar to G.E. Moore's observation that an epistemic agent
cannot coherently assert "*p* but I do not believe (know)
*p*". Sentences of this form are generally referred to as
*Moore sentences*.



Modern treatments of the logic of knowledge and belief grew out of the
work of philosophers and logicians writing from 1948 through the
1950s. Rudolf Carnap, Jerzy Los, Arthur Prior, Nicholas
Rescher, G.H. von Wright, and others recognized that our discourse
concerning knowledge and belief admits of an axiomatic-deductive
treatment. Among the many important papers that appeared in the 1950s,
von Wright's seminal work (1951) is widely acknowledged as
having initiated the formal study of epistemic logic as we know it
today. Von Wright's insights were extended by Jaakko Hintikka in
his book *Knowledge and Belief: An Introduction to the Logic of the
Two Notions* (1962). Hintikka provided a way of interpreting
epistemic concepts in terms of possible world semantics and as such it
has served as the foundational text for the study of epistemic logic
ever since.



In the 1980s and 1990s, epistemic logicians focused on the logical
properties of systems containing groups of knowers and later still on
the epistemic features of so-called "multi-modal"
contexts. Since the 1990s work in
 dynamic epistemic logic
 has extended traditional epistemic logic by modeling the dynamic
process of knowledge acquisition and belief revision. In the past two
decades, epistemic logic has come to comprise a broad set of formal
approaches to the interdisciplinary study of knowledge and belief.



Interest in epistemic logic extends well beyond philosophers. Recent
decades have seen a great deal of interdisciplinary attention to
epistemic logic with economists and computer scientists actively
developing the field together with logicians and philosophers. In 1995
two important books signaled the fertile interplay between computer
science and epistemic logic: Fagin, Halpern, Moses, and Vardi (1995)
and Meyer and van der Hoek (1995). Work by computer scientists has
become increasingly central to epistemic logic in the intervening
years.



Among philosophers, there is increased attention to the interplay
between these formal approaches and traditional epistemological
problems (see for example, van Benthem 2006; Hendricks & Symons
2006; Stalnaker 2006; Artemov 2008; Holliday 2018; Baltag,
Bezhanishvili, et al. 2019).



Several introductory texts on epistemic logic exist, e.g., Fagin et
al. (1995), van Benthem (2011); van Ditmarsch, van der Hoek, and Kooi
(2007); van Ditmarsch, Halpern, et al. (2015); Gochet and Gribomont
(2006); and Meyer (2001) with Lenzen (1980) providing an overview of
early developments.


## 2. The Modal Approach to Knowledge



Until relatively recently, epistemic logic focused almost exclusively
on propositional knowledge. In cases of propositional knowledge, an
agent or a group of agents bears the propositional attitude of knowing
towards some proposition. For example, when one says: "Zoe knows
that there is a hen in the yard", one asserts that Zoe is the
agent who bears the propositional attitude *knowing* towards
the proposition expressed by the English sentence "there is a
hen in the yard". Now imagine that Zoe does not know whether
there is a hen in the yard. For example, it might be the case that she
has no access to information about whether there is or is not a hen in
the yard. In this case her lack of information means that she will
consider two scenarios as being possible, one in which there is a hen
in the yard and one in which there is not.



Perhaps she has some practical decision that involves not only hens
but also the presence of frightening dogs in the yard. She might wish
to feed the hens but will only do so if there is no dog in the yard.
If she were ignorant of whether there is a dog in the yard, the number
of scenarios she must consider in her deliberations grows to four.
Clearly, one needs to consider epistemic alternatives when one does
not have complete information concerning the situations that are
relevant to one's decisions. As we shall see below, possible
worlds semantics has provided a useful framework for understanding the
manner in which agents can reason about epistemic alternatives.



While epistemic logicians had traditionally focused on *knowing
that,* one finds a range of other uses of knowledge in natural
language. As Y. Wang (2018b) points out, the expressions *knowing
how*, *knowing what*, *knowing why* are very common,
appearing almost just as frequently (sometimes more frequently) in
spoken and written language as *knowing that*. Recently
non-standard epistemic logics of such know-wh expressions have been
developed, to which we will come back in
 Section 4.
 In the remainder of this section, we will introduce the basics of the
epistemic logic of knowing that.


### 2.1 The Formal Language of Epistemic Logic



Recent work in epistemic logic relies on a modal conception of
knowledge. In order to be clear about the role of modality in
epistemic logic it is helpful to introduce the basic elements of the
modern formalism. For the sake of simplicity we begin with the case of
knowledge and belief for a single agent, postponing consideration of
multiple agents to
 Section 3,



A prototypical epistemic logic language is given by first fixing a set
of *propositional variables* \(p\_{1}\), \(p\_{2}\),.... In
applications of epistemic logic, propositional variables are given
specific interpretations: For example, \(p\_{1}\) could be taken to
represent the proposition "there is a hen in the yard" and
\(p\_{2}\) the proposition "there is a dog in the yard",
etc. The propositional variables represent propositions which are
represented in no finer detail in the formal language. As such, they
are therefore often referred to as *atomic propositions* or
simply *atoms*. Let *Atom* denote the set of atomic
propositions.



Apart from the atomic propositions, epistemic logic supplements the
language of propositional logic with a modal operator, \(K\_{a}\), for
knowledge and \(B\_{a}\), for belief.




\(K\_{a}\varphi\) reads "Agent *a* knows that
\(\varphi\)"




and similarly




\(B\_{a}\varphi\) reads "Agent *a* believes that
\(\varphi\)".




In many recent publications on epistemic logic, the full set of
formulas in the language is given using a so-called *Backus-Naur
Form* (BNF). This is simply a notational technique derived from
computer science that provides a recursive definition of the formulas
deemed grammatically "correct", i.e., the set of
*well-formed formulas*:

 
\[\varphi::=p \mid \neg\varphi\mid (\varphi\wedge\varphi)\mid K\_{a}\varphi\mid B\_{a}\varphi,\text{ for } p\in\Atom.\]


It should be noted that the Greek letter \(\varphi\) stands for the
*syntactic category* of formula. So this definition says: an
atom *p* is a formula; \(\neg\varphi\) is a formula if
\(\varphi\) is a formula (read \(\neg\) as 'it is not the case
that'); \((\varphi\wedge\varphi)\) is a formula whenever any two
formulas are connected by the \(\wedge\) symbol (read \(\wedge\) as
'and'); and \(K\_{a}\varphi\) and \(B\_{a}\varphi\) are
formulas whenever \(\varphi\) is a formula (the readings were
indicated above). Note that in a non-BNF recursive specification of
the language, the Greek variable \(\varphi\) would be used a
*metavariable* ranging over formulas, and one would normally
state the clause for conjunctions as: \((\varphi \wedge \psi)\) is a
formula whenever \(\varphi\) and \(\psi\) are formulas. But the BNF
let's us get away with just using \(\varphi\) to get the same
effect.



We will call this basic language that includes both a *K*nowledge
and a *B*elief operator, \(\mathcal{L}\_{KB}\). As in
propositional logic, additional connectives are defined from \(\neg\)
and \(\wedge\): Typical notation is '\(\vee\)' for
'or', '\(\rightarrow\)' for 'if...,
then ...' and '\(\leftrightarrow\)' for
'... if, and only if, ...'. Also typically
\(\top\) ('top') and \(\bot\) ('bottom') is
used to denote the constantly true proposition and the constantly
false proposition, respectively.



As we shall see below, \(K\_{a}\varphi\) is read as stating that
\(\varphi\) holds in *all* of the worlds accessible to
*a*. In this sense, *K* can be regarded as behaving
similarly to the 'box' operator, \(\square\), often used
to denote necessity. In evaluating \(K\_{a}\varphi\) at a possible
world *w*, one is in effect evaluating a *universal
quantification* over all the worlds accessible from *w*. The
universal quantifier \(\forall\) in first-order logic has the
existential quantifier \(\exists\) as its *dual*: This means
that the quantifiers are mutually definable by taking either
\(\forall\) as primitive and defining \(\exists x\varphi\) as short
for \(\neg\forall x\neg\varphi\) or by taking \(\exists\) as primitive
and defining \(\forall x\varphi\) as \(\neg\exists x\neg\varphi\). In
the case of \(K\_{a}\), it may be seen that the formula \(\neg
K\_{a}\neg\varphi\) makes an *existential quantification*: It
says that there *exists* an accessible world that satisfies
\(\varphi\). In the literature, a dual operator for \(K\_{a}\) is often
introduced. The typical notation for \(\neg K\_{a}\neg\) includes
\(\langle K\_{a}\rangle\) and \(\widehat{K}\_{a}\). This notation mimics
the diamond-shape \(\lozenge\), which is the standard dual operator to
the box \(\square\), which in turn is standard notation for the
universally quantifying modal operator (see the entry on
 modal logic).



More expressive languages in epistemic logic involve the addition of
operators for various notions of group knowledge (see
 Section 3).
 For example, as we discuss below, the *common knowledge*
operator and so-called *dynamic* operators are important
additions to the language of epistemic logic. Dynamic operators can
indicate for example the *truthful public announcement* of
\(\varphi\): \([\varphi!]\). A formula \([\varphi!]\psi\) is read
"if \(\varphi\) is truthfully announced to everybody, then after
the announcement, \(\psi\) is the case". The question of what
kinds of expressive power is added with the addition of operators is a
research topic that is actively being investigated in
 dynamic epistemic logic.
 So, for example, adding \([\varphi!]\) by itself to
\(\mathcal{L}\_{KB}\) does *not* add expressive power, but in a
language that also includes common knowledge, it does (e.g., van
Ditmarsch, van der Hoek, & Kooi 2007).


### 2.2 Higher-Order Attitudes



Notice that, for example, \(K\_{a}K\_{a}p\) is a formula in the language
we introduced above. It states that agent *a* knows that agent
*a* knows that *p* is the case. Formula with *nested*
epistemic operators of this kind express a *higher-order*
attitude: an attitude concerning the the attitude of some agent.



Higher-order attitudes is a recurring theme in epistemic logic. The
aforementioned Moore sentences, e.g., \(B\_{a}(p\wedge B\_{a}\neg p)\)
express a higher-order attitude. So do many of the epistemic
principles discussed in the literature and below. Consider the
following prominent epistemic principle involving higher-order
knowledge: \(K\_{a}\varphi\rightarrow K\_{a}K\_{a}\varphi\). Is it
reasonable to require that knowledge satisfies this scheme, i.e., that
if somebody knows \(\varphi\), then they know that they know
\(\varphi\)? In part, we might hesitate before accepting this
principle in virtue of the higher-order attitude involved. This is a
matter of ongoing discussion in epistemic logic and epistemology.


### 2.3 The Partition Principle and Modal Semantics



The semantics of the formal language introduced above is generally
presented in terms of so-called possible worlds. In epistemic logic
possible worlds are interpreted as epistemic alternatives. Hintikka
was the first to explicitly articulate such an approach (1962). This
is another central feature of his approach to epistemology which
continues to inform developments today. It may be stated,
 simplified,[1]
 as follows:




**Partition Principle:** Any propositional attitude
partitions the set of possible worlds into those that are in
accordance with the attitude and those that are not.




The partition principle may be used to provide a semantics for the
knowledge operator. Informally,




\(K\_{a}\varphi\) is true in world *w* if, and only if,
\(\varphi\) is true in every world \(w'\) compatible with the
information *a* has at *w*.




Here, agent *a* knows that \(\varphi\) just in case the agent has
information that rules out every possibility of error and rules out
every case where \(\neg\varphi\).


### 2.4 Kripke Models and The Indistinguishability Interpretation of Knowledge



Since the 1960s *Kripke models*, defined below, have served as
the basis of the most widely used semantics for all varieties of modal
logic. The use of Kripke models in the representation of epistemic
concepts involves taking a philosophical stance with respect to those
concepts. One widespread interpretation, especially in theoretical
economics and theoretical computer science, understands knowledge in
terms of informational indistinguishability between possible worlds.
What we will refer to here as the *indistinguishability
interpretation* goes back to Lehmann (1984), and Aumann (1976) has
an equivalent partition-based framework.



As the indistinguishability interpretation concerns knowledge, but not
belief, we will be working with a language without belief operators.
Therefore, let the language \(\mathcal{L}\_{K}\) be given by the
Backus-Naur form

 
\[\varphi::=p\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid K\_{a}\varphi\text{ for } p\in \Atom.\]


As we shall see, the indistinguishability interpretation involves very
stringent requirements in order for something to qualify as knowledge.
We introduce it here for pedagogical purposes, putting the formal
details of the interpretation in place so as to introduce and explain
relatively less extreme positions thereafter.



Consider again the case of Zoe, the hen, and the dog. The example
involves two propositions, which we will identify with the formal
atoms:




*p* read as "there is a hen in the yard".



and



*q* read as "there is a dog in the yard".




It is worth emphasizing that for the purposes of our formalization of
this scenario, these two are the *only* propositions of
interest. We are restricting our attention to \(\Atom=\{p,q\}\). In
early presentations of epistemic logic and in much of standard
epistemic logic at present, *all* the atoms of interest are
included from the outset. Obviously, this is an idealized scenario. It
is important to notice what this approach leaves out. Considerations
that are not captured in this way include the appearance of novel
atoms; the idea that other atomic propositions might be introduced at
some future state via some process of learning for example, or the
question of an agent's awareness of propositions; the scenario
in which an agent might be temporarily *unaware* of some atom
due to some psychological or other factor (see
 Section 5
 for references to so-called *awareness logic*). For now, the
main point is that standard epistemic logic begins with the assumption
that the set *Atom* exhausts the space of propositions for the
agent.



With two atoms, there are four different ways a world could
consistently be. We can depict each by a box:



![Basic Four Worlds: four boxes in a row with some space between them. The first labeled w1 and contains the pair: p, q. The second labeled w2 with the pair: p not q. The third, w3, with the pair: not p, q. The fourth, w4, with the pair: not p, not q. Almost all the subsequent images contain the same with some slight modifications.](fig1.svg)


The four boxes may be formally represented by a set
\(W=\{w\_{1},w\_{2},w\_{3},w\_{4}\}\), typically called a set of
***possible worlds***. Each world is further
labeled with the atoms true at that world. They are labeled by a
function *V*, the ***valuation***. The
valuation specifies which atoms are true at each world in the
following way: Given an atom *p*, \(V(p)\) is the subset of
worlds at which *p* is
 true.[2]
 That \(w\_{1}\) is labeled with *p* and *q* thus means that
\(w\_{1}\in V(p)\) and \(w\_{1}\in V(q)\). In the illustration,
\(V(p)=\{w\_{1},w\_{2}\}\) and \(V(q)=\{w\_{1},w\_{3}\}\).



For presentational purposes, assume that there really is a hen in the
yard, but no dog. Then \(w\_{2}\) would represent the
***actual world*** of the model. In illustrations,
the actual world is commonly highlighted:



![Basic Four Worlds except w2 is highlighted with a double line instead of a single line for the box.](fig2.svg)


Now, assume that the hen is always clucking, but that the dog never
barks, and that although Zoe has acute hearing, she cannot see the
yard. Then there are certain possible worlds that Zoe cannot
*distinguish*: possible ways things may be which she cannot
tell apart. For example, being in the world with only a hen \((p,\neg
q)\), Zoe cannot tell if she is in the world with both hen and dog
\((p,q)\): her situation is such that Zoe is aware of two ways things
could be but her information does not allow her to eliminate
either.



To illustrate that one possible world cannot be distinguished from
another, an arrow is typically drawn from the former to the
latter:



![Basic Four Worlds except w2 is highlighted and an arrow points from w2 to w1.](fig3.svg)


Here, arrows represent a *binary relation* on possible worlds.
In modal logic in general, it is referred to as the
***accessibility relation***. Under the
indistinguishability interpretation of epistemic logic, it is
sometimes called the ***indistinguishability
relation***. Formally, denote the relation \(R\_{a}\), with
the subscript showing the relation belongs to agent *a*. The
relation is a subset of the set of *ordered pairs* of possible
worlds, \(\{(w,w')\colon w,w'\in W\}\). One world *w*
"points" to another \(w'\) if \((w,w')\in R\_{a}\). In this
case, \(w'\) is said to be *accessible*
(*indistinguishable*) from *w*. In the literature, this is
often written \(wR\_{a}w'\) or \(R\_{a}ww'\). The notation
'\(w'\in R\_{a}(w)\)' is also common: the set \(R\_{a}(w)\)
is then the worlds accessible from *w*, i.e.,

 
\[R\_{a}(w):=\{w'\in W:(w,w')\in R\_{a}\}.\]


A final note: the set \(\{(w,w')\colon w,w'\in W\}\) is often written
\(W\times W\), the *Cartesian product* of *W* with
itself.



For \(R\_{a}\) to faithfully represent a relation of
indistinguishability, what worlds should it relate? If Zoe was plunged
in \(w\_{1}\) for example, could she tell that she is not in \(w\_{2}\)?
No: the relation of indistinguishability is
*symmetric*--if one cannot tell *a* from *b*,
neither can one tell *b* from *a*. That a relation is
symmetric is typically drawn by omitting arrow-heads altogether or by
putting them in both directions:



![Basic Four Worlds except w2 is highlighted and a double headed arrow connects w2 and w1.](fig4.svg)


Which of the remaining worlds are indistinguishable? Given that the
hen is always clucking, Zoe has information that allows her to
distinguish \(w\_{1}\) and \(w\_{2}\) from \(w\_{3}\) and \(w\_{4}\) and
*vice versa*, cf. symmetry. Hence, no arrows between these. The
worlds \(w\_{3}\) and \(w\_{4}\) are indistinguishable. This brings us
to the following representation:



![Basic Four Worlds except w2 is highlighted and a double headed arrow connects w2 and w1 and another double headed arrow connects w3 and w4.](fig5.svg)


Since no information will ever allow Zoe to distinguish something from
itself, any possible world is thus related to itself and the
indistinguishability relation is *reflexive*:



![Basic Four Worlds except w2 is highlighted and a double headed arrow connects w2 and w1 and another double headed arrow connects w3 and w4. Each world also has an arrow that loops back to the same world.](fig6.svg)


The standard interpretation of the Zoe example in terms of a possible
worlds model is now complete. Before turning to a general presentation
of the indistinguishability interpretation, let us look at what Zoe
knows.



Recall the informal modal semantics of the knowledge operator from
above:




\(K\_{a}\varphi\) is true in world *w* if, and only if,
\(\varphi\) is true in every world \(w'\) compatible with the
information *a* has at *w*.




To approach a formal definition, take '\(w\vDash\varphi\)'
to mean that \(\varphi\) is true in world *w*. Thus we can,
define truth of \(K\_{a}\varphi\) in *w* by




\(w\vDash K\_{a}\varphi\) iff \(w'\vDash\varphi\) for all \(w'\) such
that \(wR\_{a}w'\).




This definition states that *a* knows \(\varphi\) in world
*w* if, and only if, \(\varphi\) is the case in all the worlds
\(w'\) which *a* cannot distinguish from *w*.



So, where does that leave Zoe? First off, the definition allows us to
evaluate her knowledge in each of the worlds, but seeing as \(w\_{2}\)
is the actual world, it is the world of interest. Let *a* denote
Zoe, here are some examples of what we can say about Zoe's
knowledge in \(w\_{2}\):


1. \(w\_{2}\vDash K\_{a}p\). Zoe knows that the hen is in the yard as
all the worlds indistinguishable from \(w\_{2}\) that would be
\(w\_{1}\) and \(w\_{2}\) make *p* true.
2. \(w\_{2}\vDash\neg K\_{a}q\). Zoe does not know the dog is in the
yard, as one of the indistinguishable worlds in fact \(w\_{2}\) itself
makes *q* false.
3. \(w\_{2}\vDash K\_{a}K\_{a}p\). Zoe knows that she knows *p*
because
	1. \(w\_{2}\vDash K\_{a}p\) (cf. 1.) and
	2. \(w\_{1}\vDash K\_{a}p\).
4. \(w\_{2}\vDash K\_{a}\neg K\_{a}q\). Zoe knows that she does not know
*q* because
	1. \(w\_{2}\vDash\neg K\_{a}q\) (cf. 2.) and
	2. \(w\_{1}\vDash\neg K\_{a}q\).



We could say a lot more about Zoe's knowledge: every formula of
the epistemic language without belief operators may be evaluated in
the model. It thus represents all Zoe's higher-order information
about her own knowledge of which points 3. and 4. are the first
examples.



One last ingredient is required before we can state the
indistinguishability interpretation in its full generality. In the
example above, it was shown that the indistinguishability relation was
both *symmetric* and *reflexive*. Formally, these
properties may be defined as follows:




**Definition:** A binary relation \(R\subseteq W\times
W\) is


1. ***reflexive*** iff for all \(w\in
W,wRw\),
2. ***symmetric*** iff for all \(w,w'\in W,\) if
\(wRw'\), then \(w'Rw\).




The missing ingredient is then the relational property of
*transitivity*. 'Shorter than' is an example of a
transitive relation: Let *x* be shorter than *y*, and let
*y* be shorter than *z*. Then *x* must be shorter than
*z*. So, given \(w\_{1},w\_{2}\) and \(w\_{3}\), if the relation
*R* holds between \(w\_{1}\) and \(w\_{2}\) and between \(w\_{2}\)
and \(w\_{3}\), then the arrow between \(w\_{1}\) and \(w\_{3}\) is the
consequence of requiring the relation to be transitive:



![A diagram of three nodes: w1, w2, and w3. An arrow, labeled 'assumed' goes from w1 to w2 and another arrow with the same label goes from w2 to w3. A third arrow, labeled 'implied' goes from w1 to w3.](fig8.svg)


Formally, transitivity is defined as follows:




**Definition:** A binary relation \(R\subseteq W\times
W\) is ***transitive*** iff for all \(w,w',w''\in
W,\) if \(wRw'\) and \(w'Rw''\), then \(wRw''\)




A relation that is both reflexive, symmetric and transitive is called
an ***equivalence relation***.



With all the components in place, let us now define the Kripke
model:




**Definition:** A ***Kripke model***
for \(\mathcal{L}\_{K}\) is a tuple \(M=(W,R,V)\) where


* *W* is a non-empty set of possible worlds,
* *R* is a binary relation on *W*, and
* \(V\colon \Atom \longrightarrow\mathcal{P}(W)\) is a
valuation.




In the definition, '\(\mathcal{P}(W)\)' denotes the
*powerset* of *W*: It consists of all the subsets of
*W*. Hence \(V(p)\), the valuation of atom *p* in the model
*M*, is some subset of the possible worlds: Those where *p*
is true. In this general definition, *R* can be any relation on
*W*.



To specify which world is actual, one last parameter is added to the
model. When the actual world is specified a Kripke model is commonly
called *pointed*:




**Definition:** A ***pointed Kripke
model*** for \(\mathcal{L}\_{K}\) is a pair \((M,w)\)
where


* \(M=(W,R,V)\) is a Kripke model, and
* \(w\in W\).




Finally, we may formally define the semantics that was somewhat
loosely expressed above. This is done by defining a relation between
pointed Kripke models and the formulas of the formal language. The
relation is denoted '\(\vDash\)' and is often called the
***satisfaction relation***.



The definition then goes as follows:




**Definition:** Let \(M=(W,R\_{a},V)\) be a Kripke model
for \(\mathcal{L}\_{K}\) and let \((M,w)\) be a pointed Kripke model.
Then for all \(p\in \Atom\) and all
\(\varphi,\psi\in\mathcal{L}\_{K}\)


 
\[ \begin{align}
(M,w)&\vDash p & \textrm{ iff }& w\in V(p)\\
 (M,w)&\vDash\neg\varphi& \textrm{ iff }& \textrm{not } (M,w)\vDash\varphi\\
 (M,w)&\vDash(\varphi\wedge\psi)& \textrm{ iff }& (M,w)\vDash\varphi \textrm{ and }(M,w)\vDash\psi\\
 (M,w)&\vDash K\_{a}\varphi &\textrm{ iff }& (M,w')\vDash\varphi \textrm{ for all } w'\in W \textrm{ such that } wR\_{a}w'. 
\end{align} \]



The formula \(\varphi\) is ***satisfied*** in the
pointed model \((M,w)\) iff \((M,w)\vDash\varphi\).




In full generality, the indistinguishability interpretation holds that
for \(K\_{a}\) to capture knowledge, the relation \(R\_{a}\) must be an
equivalence relation. A pointed Kripke model for which this is
satisfied is often referred to as an ***epistemic
state***. In epistemic states, the relation is denoted by a
tilde with a subscript: \(\sim\_{a}\).



Given pointed Kripke models and the indistinguishability
interpretation, we have a semantic specification of one concept of
knowledge. With this approach, we can build models of situations
involving knowledge as we did with the toy example of Zoe and the
hens. We can use these models to determine what the agent does or does
not know. We also have the formal foundations in place to begin asking
questions concerning how the agent's knowledge or uncertainty
develops when it receives *new information*, a topic studied in
 dynamic epistemic logic.



We may also ask more general questions concerning the concept of
knowledge modeled using pointed Kripke models with
indistinguishability relations: Instead of looking at a particular
model at the time and asking which formulas the model makes true, we
can ask which general principles all such models agree on.


### 2.5 Epistemological Principles in Epistemic Logic



Settling on the correct formal representation of knowledge involves
reflecting carefully on the epistemological principles to which one is
committed. An uncontroversial example of such a principle which most
philosophers will accept is veridicality:




*If a proposition is known, then it is true*.

 
\[K\_{a}\varphi\rightarrow\varphi.\]



In a formal context this principle can be understood to say that if
\(\varphi\) is known then it should always be satisfied in one's
models. If it turns out that some of one's chosen models falsify
the veridicality principle, then most philosophers would simply deem
those models unacceptable.



Returning to pointed Kripke models, we can now ask which principles
these models commit one to. In order to begin answering this question,
we need to understand the most general features of our formalism. The
strategy in modal logic in general (see Blackburn, de Rijke, &
Venema 2001) is to abstract away from any given model's
*contingent* features. Contingent features would include, for
example, the specific number of worlds under consideration, the
specific valuation of the atoms, and the choice of an actual world. In
this case, the only features that are not contingent are those
required by the general definition of a pointed Kripke model.



To abstract suitably, take a pointed Kripke model \((M,w)=(W,R,V,w)\).
To determine whether the relation of this model is an equivalence
relation we only need to consider the worlds and the relation. The
pair of these elements constitute the fundamental level of the model
and is called the *frame* of the model:




**Definition:** Let \((M,w)=(W,R,V,w)\) be a pointed
Kripke model. Then the pair \((W,R)\) is called the
***frame*** of \((M,w)\). Any model \((M',w')\)
which shares the frame \((W,R)\) is said to be ***built
on*** \((W,R)\).




Consider again the epistemic state for Zoe from above:



![Basic Four Worlds except w2 is highlighted and a double headed arrow connects w2 and w1 and another double headed arrow connects w3 and w4. Each world also has an arrow that loops back to the same world.](fig9.svg)


Several other models may be built on the same frame. The following are
two examples:



![Basic Four Worlds except w3 (instead of w2) is highlighted and a double headed arrow connects w2 and w1 and another double headed arrow connects w3 and w4. Each world also has an arrow that loops back to the same world. In addition w2 has the pair: p, q instead of p, not q.](fig10.svg)
![Basic Four Worlds except w4 (instead of w2 or w3) is highlighted and a double headed arrow connects w2 and w1 and another double headed arrow connects w3 and w4. Each world also has an arrow that loops back to the same world. In addition w1 has the pair: not p, not q; w2, w3, and w4 each has the pair: p, q](fig11.svg)


With the notion of a frame, we may define the notion of validity of
interest. It is the second term defined in the following:




**Definition:** A formula \(\varphi\) is said to be
***valid in the frame*** \(F=(W,R)\) iff every
pointed Kripke model build on *F* satisfies \(\varphi\), i.e.,
iff for every \((M,w)=(F,V,w)=(W,R,V,w)\), \((M,w)\vDash\varphi\). A
formula \(\varphi\) is ***valid on the class of
frames*** \(\mathsf{F}\) (written
\(\mathsf{F}\vDash\varphi\)) iff \(\varphi\) is valid in every frame
*F* in \(\mathsf{F}\).




The set of formulas valid on a class of frames \(\mathsf{F}\) is
called the ***logic*** of \(\mathsf{F}\). Denote
this logic that is, the set
\(\{\varphi\in\mathcal{L}\_{K}\colon\mathsf{F}\vDash\varphi\}\) by
\(\Lambda\_{\mathsf{F}}\). This is a *semantic* approach to
defining logics, each just a set of formulas. One may also define
logics *proof-theoretically* by defining a logic as the set of
formulas provable in some system. With logics as just sets of
formulas, *soundness* and *completeness* results may
then be expressed using set inclusion. To exemplify, let
\(\mathsf{A}\) be a set of axioms and write
\(\mathsf{A}\vdash\varphi\) when \(\varphi\) is provable from
\(\mathsf{A}\) using some given set of deduction rules. Let the
resulting logic the set of theorems be denoted
\(\Lambda\_{\mathsf{A}}\). It is the set of formulas from
\(\mathcal{L}\_{K}\) provable from \(\mathsf{A}\), i.e., the set
\(\{\varphi\in\mathcal{L}\_{K}\colon\mathsf{A}\vdash\varphi\}\). The
logic \(\Lambda\_{\mathsf{A}}\) is sound with respect to \(\mathsf{F}\)
iff \(\Lambda\_{\mathsf{A}}\subseteq\Lambda\_{\mathsf{F}}\) and complete
with respect to \(\mathsf{F}\) iff

\(\Lambda\_{\mathsf{F}}\subseteq\Lambda\_{\mathsf{A}}\).[3]



Returning to the indistinguishability interpretation of knowledge, we
may then seek to find the epistemological principles which the
interpretation is committed to. There is a trivial answer of little
direct interest: Let \(\mathsf{EQ}\) be the class of frames with
equivalence relations. Then the logic of the indistinguishability
interpretation is the set of formulas of \(\mathcal{L}\_{K}\) which are
valid over \(\mathsf{EQ}\), i.e., the set
\(\Lambda\_{\mathsf{EQ}}:=\{\varphi\in\mathcal{L}\_{K}\colon\mathsf{EQ}\vDash\varphi\}\).
Not very informative.



Taking an *axiomatic* approach to specifying the logic,
however, yields a presentation in terms of easy to grasp principles.
To start with the simplest, then the principle T states that knowledge
is *factual*: If the agent knows \(\varphi\), then \(\varphi\)
must be true. The more cumbersome K states that if the agent knows an
implication, then if the agent knows the antecedent, it also knows the
consequent. I.e., if we include the derivation rule *modus
ponens* (from \(\varphi\rightarrow\psi\) and \(\varphi\), conclude
\(\psi\)) as rule of our logic of knowledge, K states that knowledge
is *closed under implication*. The principle B states that if
\(\varphi\) is true, then the agent knows that it considers
\(\varphi\) possible. Finally, 4 states that if the agent knows
\(\varphi\), then it knows that it knows \(\varphi\). T, B and 4 in
the table below (the names are historical and not all meaningful).



\[ \begin{align}
\textrm{K} & & K\_{a}(\varphi\rightarrow\psi) & \rightarrow(K\_{a}\varphi\rightarrow K\_{a}\psi)\\
 \textrm{T} & & K\_{a}\varphi & \rightarrow\varphi\\
 \textrm{B} & & \varphi & \rightarrow K\_{a}\widehat{K}\_{a}\varphi\\
 \textrm{4} & & K\_{a}\varphi & \rightarrow K\_{a}K\_{a}\varphi\\
 
\end{align} \]


In lieu of epistemological intuitions, we could discuss a concept of
knowledge by discussing these and other principles. Should we accept T
as a principle that knowledge follows? What about the others? Before
we proceed, let us first make clear how the four above principles
relate to the indistinguishability interpretation. To do so, we need
the notion of a *normal modal logic*. In the below definition,
as in the above principles, we are technically using *formula
schemas*. For example, in \(K\_{a}\varphi\rightarrow\varphi\), the
\(\varphi\) is a variable ranging over formulas in
\(\mathcal{L}\_{K}\). Thus, strictly speaking,
\(K\_{a}\varphi\rightarrow\varphi\) is not a formula, but a
*scheme* for obtaining a formula. A *modal instance* of
\(K\_{a}\varphi\rightarrow\varphi\) is then the formula obtained by
letting \(\varphi\) be some concrete formula from \(\mathcal{L}\_{K}\).
For example, \(K\_{a}p\rightarrow p\) and \(K\_{a}(p\wedge
K\_{a}q)\rightarrow(p\wedge K\_{a}q)\) are both modal instances of
T.




**Definition:** Let \(\Lambda\subseteq\mathcal{L}\_{K}\)
be a set of modal formulas. Then \(\Lambda\) is a ***normal
modal logic*** iff \(\Lambda\) satisfies all of the
following:


1. \(\Lambda\) contains all modal instances of the classical
propositional tautologies.
2. \(\Lambda\) contains all modal instances of K.
3. \(\Lambda\) is closed under *modus ponens*: If
\(\varphi\in\Lambda\) and \(\varphi\rightarrow\psi\in\Lambda\), then
\(\psi\in\Lambda\).
4. \(\Lambda\) is closed under *generalization* (a.k.a.
*necessitation*): If \(\varphi\in\Lambda\), then
\(K\_{a}\varphi\in\Lambda\).




There is a unique *smallest* normal modal logic (given the set
*Atom*)that which contains exactly what is required by the
definition and *nothing more*. It is often called the
***minimal normal modal logic*** and is denoted by
the boldface **K** (not to be confused with the
non-boldface K denoting the schema).



The logic **K** is just a set of formulas from
\(\mathcal{L}\_{K}\). I.e.,
**K** \(\subseteq\mathcal{L}\_{K}\). Points 1.4.
gives a perspective on this set: They provide an
*axiomatization*. Often, as below, the schema K is referred to
as an axiom, though really the instantiations of K are axioms.



To **K**, we can add additional principles as axioms
(axiom schemes) to obtain stronger logics (logics that have additional
theorems: Logics \(\Lambda\) for which
**K** \(\subseteq\Lambda\)). Of immediate interest
is the logic called **S5**:




**Definition:** The logic **S5** is the
smallest normal modal logic containing all modal instances of T, B,
and 4.




Here, then, is the relationship between the above four principles and
the indistinguishability interpretation:




**Theorem 1:** The logic **S5** is the logic
of the class of pointed Kripke models build on frames with equivalence
relations. I.e., \(\textbf{S5} =\Lambda\_{\mathsf{EQ}}\).




What does this theorem tell us with respect to the principles of
knowledge, then? In one direction it tells us that if one accepts the
indistinguishability interpretation, then one has implicitly accepted
the principles K, T, B and 4 as reasonable for knowledge. In the other
direction, it tells us that if one finds that **S5** is
the appropriate logic of knowledge *and* one finds that pointed
Kripke models are the right way to semantically represent knowledge,
then one must use an equivalence relation. Whether one should
interpret this relation in terms of indistinguishability, though, is a
matter on which logic is silent.



In discussing principles for knowledge, it may be that some of the
four above seem acceptable, while others do not: One may disagree with
the acceptability of B and 4, say, while accepting K and T. In
understanding the relationship between **S5** and
equivalence relations, a more fine-grained perspective is beneficial:
Theorem 1 may be chopped into smaller pieces reflecting the
contribution of the individual principles K, T, 4 and B to the
equivalence requirementi.e., that the relation should be at the same
time reflexive, symmetric and transitive.




**Theorem 2:** Let \(F=(W,R)\) be a frame. Then:


* All modal instances of K are valid in *F*.
* All modal instances of T are valid in *F* iff *R* is
reflexive.
* All modal instances of B are valid in *F* iff *R* is
symmetric.
* All modal instances of 4 are valid in *F* iff *R* is
transitive.




There are a number of insights to gain from Theorem 2. First, if one
wants to use *any* type of Kripke model to capture knowledge,
then one must accept K. Skipping some details, one must in fact accept
the full logic **K** as this is the logic of the class of
*all* Kripke models (see, e.g., Blackburn, de Rijke, &
Venema 2001).



Second, the theorem shows that there is an intimate relationship
between the individual epistemic principles and the properties on the
relation. This, in turn, means that one, in general, may approach the
"logic" in epistemic logic from two sides from intuitions
about the accessibility relation or from intuitions about epistemic
principles.



Several normal modal logical systems weaker than **S5**
have been suggested in the literature. Here, we specify the logics by
the set of their modal axioms. For example, the logic
**K** is given by \(\{\text{K}\}\), while
**S5** is given by
\(\{\text{K},\text{T},\text{B},\text{4}\}\). To establish
nomenclature, the following table contains a selection of principles
from the literature with the frame properties they characterize, cf.
Aucher (2014) and Blackburn, de Rijke, & Venema (2001), on the
line below them. The frame conditions are not all straightforward.



In Table 1, the subscript on \(R\_{a}\) is omitted to ease readability,
and so is the domain of quantification *W* over which the worlds
variables \(x,y,z\) range.






|  |  |
| --- | --- |
| K | \(K\_{a}(\varphi\rightarrow\psi)\rightarrow(K\_{a}\varphi\rightarrow
K\_{a}\psi)\)

None: *Not applicable* |
| D | \(K\_{a}\varphi\rightarrow\widehat{K}\_{a}\varphi\)

Serial: \(\forall x\exists y,xRy\). |
| T | \(K\_{a}\varphi\rightarrow\varphi\)

Reflexive: \(\forall x,xRx\). |
| 4 | \(K\_{a}\varphi\rightarrow K\_{a}K\_{a}\varphi\)

Transitive: \(\forall x,y,z,\text{if }xRy\text{ and }yRz\text{, then
}xRz\). |
| B | \(\varphi\rightarrow K\_{a}\widehat{K}\_{a}\varphi\)

Symmetric: \(\forall x,y,\text{if }xRy\text{, then }yRx\). |
| 5 | \(\neg K\_{a}\varphi\rightarrow K\_{a}\neg K\_{a}\varphi\)

Euclidean: \(\forall x,y,z,\text{if }xR\_{a}y\text{ and }xR\_{a}z\text{,
then }yRz\). |
| .2 | \(\widehat{K}\_{a}K\_{a}\varphi\rightarrow
K\_{a}\widehat{K}\_{a}\varphi\)

Confluent: \(\forall x,y,\text{if }xRy\text{ and }xRy',\text{ then
}\exists z,yRz\text{ and }y'Rz\). |
| .3 | \((\widehat{K}\_{a}\varphi\wedge\widehat{K}\_{a}\psi)\rightarrow(\widehat{K}\_{a}(\varphi\wedge\widehat{K}\_{a}\psi)\vee\widehat{K}\_{a}(\varphi\wedge\psi)\vee\widehat{K}\_{a}(\psi\wedge\widehat{K}\_{a}\varphi))\)

No branching to the right: \(\forall x,y,z,\text{if }xRy\text{ and
}xRz,\text{then }yRz\text{ or }y=z\text{ or }zRy\) |
| .3.2 | \((\widehat{K}\_{a}\varphi\wedge\widehat{K}\_{a}K\_{a}\psi)\rightarrow
K\_{a}(\widehat{K}\_{a}\varphi\vee\psi)\)

Semi-Euclidean: \(\forall x,y,z,\) if \(xRy\) and \(xRz\), then
\(zRx\) or \(yRz\). |
| .4 | \((\varphi\wedge\widehat{K}\_{a}K\_{a}\varphi)\rightarrow
K\_{a}\varphi\)

Name unknown to authors: \(\forall x,y,\) if \(xRy\), then \(\forall
z\), if \(xRz\), then \(x = z\) or \(yRz\). |



Table 1. Epistemic Principles and their
frame conditions.




Adding epistemic principles as axioms to the basic minimal normal
modal logic **K** yields new, normal modal logics. A
selection is:





|  |  |
| --- | --- |
| **K** | \(\{\text{K}\}\) |
| **T** | \(\{\text{K},\text{T}\}\) |
| **D** | \(\{\text{K},\text{D}\}\) |
| **KD4** | \(\{\text{K},\text{D},\text{4}\}\) |
| **KD45** | \(\{\text{K},\text{D},\text{4},\text{5}\}\) |
| **S4** | \(\{\text{K},\text{T},\text{4}\}\) |
| **S4.2** | \(\{\text{K},\text{T},\text{4},\text{.2}\}\) |
| **S4.3** | \(\{\text{K},\text{T},\text{4},\text{.3}\}\) |
| **S4.4** | \(\{\text{K},\text{T},\text{4},\text{.4}\}\) |
| **S5** | \(\{\text{K},\text{T},\text{5}\}\) |



Table 2. Logic names and axioms




Different axiomatic specifications may produce the same logic. Notice,
e.g., that the table's axiomatic specification
\(\{\text{K},\text{T},\text{5}\}\) of **S5** does not
match that given in the definition preceding Theorem 1,
\(\{\text{K},\text{T},\text{B},\text{4}\}\). Note also, there is more
than one axiomatization of **S5**: the axioms
\(\{\text{K},\text{T},\text{5}\}\),
\(\{\text{K},\text{T},\text{B},\text{4}\}\),
\(\{\text{K},\text{D},\text{B},\text{4}\}\) and
\(\{\text{K},\text{D},\text{B},\text{5}\}\) all give the
**S5** logic (e.g., Chellas 1980). An often seen variant
is \(\{\text{K},\text{T},\text{4},\text{5}\}\). However, it is
redundant to add it as all its instances can be proven from K, T and
5. But as both 4 and 5 capture important epistemic principles (see
 Section 2.6),
 4 is often sometimes included for the sake of philosophical
transparency. For more equivalences between modal logics, see, e.g.,
the entry on
 modal logic
 or Chellas (1980) or Blackburn, de Rijke, and Venema (2001).



Logics may be stronger or weaker than each other, and knowing the
frame properties of their axioms may help us to understand their
relationship. For example, as 4 is derivable from
\(\{\text{K},\text{T},\text{5}\}\), all the theorems of
**S4** are derivable in **S5**.
**S5** is thus *at least as strong* as
**S4**. In fact, **S5** is also *strictly
stronger*: It can prove things which **S4**
cannot.



That **S5** may be axiomatized both by
\(\{\text{K},\text{T},\text{B},\text{4}\}\) and
\(\{\text{K},\text{T},\text{5}\}\) may be seen through the frame
properties of the axioms: every reflexive and euclidean relation (T
and 5) is an equivalence relation (T, B, and 4). This also shows the
redundancy of 4: If one has assumed a relation reflexive and
euclidean, then it adds nothing new to additionally assume it to be
transitive. In general, having an understanding of the interplay
between relational properties is of great aid in seeing relationships
between modal logics. For example, noticing that every reflexive
relation is also serial means that all formulas valid on the class of
serial models are also valid on the class of reflexive models. Hence,
every theorem of **D** is thus a theorem of
**T**. Hence **T** is at least as strong as
**D** (i.e., \(\textbf{D}\subseteq\textbf{T}\)). That
**T** is also strictly stronger (not
\(\textbf{T}\subseteq\textbf{D}\)) can be shown by finding a serial,
non-reflexive model which does not satisfy some theorem of
**T** (for example \(K\_{a}p\rightarrow p\)).


### 2.6 Principles of Knowledge and Belief



With the formal background of epistemic logic in place, it is
straightforward to slightly vary the framework in order to accommodate
the concept of belief. Return to the language \(\mathcal{L}\_{KB}\) of
both knowledge and belief:

 
\[\varphi::=p\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid K\_{a}\psi\mid B\_{a}\psi,\text{ for } p\in \Atom.\]


To interpret knowledge and belief formulas together in pointed Kripke
models, all that is needed is an additional relation between possible
worlds:




**Definition:** A ***pointed Kripke
model*** for \(\mathcal{L}\_{KB}\) is a tuple
\((M,w)=(W,R\_{K},R\_{B},V,w)\) where


* *W* is a non-empty set of possible worlds,
* \(R\_{K}\) and \(R\_{B}\) are a binary relations on *W*,
* \(V\colon \Atom\longrightarrow\mathcal{P}(W)\) is a valuation,
and
* \(w\in W\).




\(R\_{K}\) is the relation for the knowledge operator and \(R\_{B}\) the
relation for the belief operator. The definition makes no further
assumptions about their properties. In the figure below we provide an
illustration, where the arrows are labeled in accordance with the
relation they correspond to. The reflexive loop at \(w\_{3}\) is a
label indicating that it belongs to both relations, i.e.,
\((w\_{3},w\_{3})\in R\_{K}\) and \((w\_{3},w\_{3})\in R\_{B}\).



![Four boxes labeled w1 (containing 'p'), w2 (containing 'not p'), w3 (containing 'p') and w4 (containing 'not p'). w1 is highlighted and an arrow, labeled 'K', goes from it to w2. w2 has arrows, each labeled 'B', pointing to w3 and w4. w3 has an arrow, labeled 'K,B', looping back to it.](fig12.svg)


The satisfaction relation is defined as above, but with the obvious
changes for knowledge and belief:




\((M,w)\vDash K\_{a}\varphi\) iff \((M,w')\vDash\varphi\) for all
\(w'\in W\) such that \(wR\_{K}w'\).



\((M,w)\vDash B\_{a}\varphi\) iff \((M,w')\vDash\varphi\) for all
\(w'\in W\) such that \(wR\_{B}w'\).




The indistinguishability interpretation puts very strong requirements
on the accessibility relation for knowledge. These have now been
stripped away and so has any commitment to the principles T, B, D, 4
and 5. Taking Kripke models as basic semantics, we are still committed
to K, though this principle is not unproblematic as we shall see below
in our discussion of the problem of logical omniscience.



Of the principles from Table 1, T, D, B, 4 and 5 have been discussed
most extensively in the literature on epistemic logic, both as
principles for knowledge and as principles for belief. The principle T
for knowledge

 
\[ K\_{a}\varphi\rightarrow\varphi \]


is broadly accepted. Knowledge is commonly taken to be
*veridical*--only true propositions can be known. For,
e.g., Hintikka (1962) and Fagin et al. (1995), the failure of T for
belief is the defining difference between the two notions.



Though belief is not commonly taken to be veridical, beliefs are
typically taken to be *consistent*. I.e., agents are taken to
*never* believe the contradiction that is, any formula
equivalent with \((p\wedge\neg p)\) or \(\bot\), for short. That
beliefs should be consistent is then captured by the principle



\[\neg B\_{a}\bot.\]


The principle \(\neg B\_{a}\bot\) is, on Kripke models, equivalent with
the principle D, \(B\_{a}\varphi\rightarrow\widehat{B}\_{a}\varphi\).
Hence the validity of \(\neg B\_{a}\bot\) requires serial frames.
Witness, e.g., its failure in \(w\_{1}\) above: As there are no worlds
accessible through \(R\_{B}\), *all* accessible worlds satisfy
\(\bot\). Hence \(w\_{1}\) satisfies \(B\_{a}\bot\), violating
consistency. Notice also that \(\neg B\_{a}\bot\) may be re-written to
\(\widehat{B}\_{a}\top\), which is true at a world just in case some
world is accessible through \(R\_{B}\). Its validity thus ensures
seriality.



Notice that the veridicality of knowledge ensures its consistency: Any
reflexive frame is automatically serial. Hence accepting
\(K\_{a}\varphi\rightarrow\varphi\) implies accepting \(\neg
K\_{a}\bot\).



Of the principles D, 4 and 5, the two latter have received so far the
most attention, both for knowledge and for belief. They are commonly
interpreted as governing of *principled access* to own mental
states. The 4 principles

 
\[ \begin{align}
K\_{a}\varphi & \rightarrow K\_{a}K\_{a}\varphi\\
 B\_{a}\varphi & \rightarrow B\_{a}B\_{a}\varphi\\
 
\end{align} \]


are often referred to as *principles of positive
introspection*, or for knowledge the *'KK'
principle*. Both principles are deemed acceptable by, e.g.,
Hintikka (1962) on grounds *different* from introspection. He
argues based on an autoepistemic analysis of knowledge, using a
non-Kripkean possible worlds semantics called *model systems*.
Hintikka holds that when an agent commits to knowing \(\varphi\), the
agent commits to holding the same attitude no matter what new
information the agent will encounter in the future. This entails that
in all the agent's epistemic alternatives--for Hintikka,
all the model sets (partial descriptions of possible worlds) where the
agent knows at least as much they now do&\madash the agent still
knows \(\varphi\). As \(K\_{a}\varphi\) thus holds in all the
agent's epistemic alternatives, Hintikka concludes that
\(K\_{a}K\_{a}\varphi\). Likewise, Hintikka endorses 4 for belief, but
Lenzen raises objections (Lenzen 1978: ch. 4).



Williamson argues against the general acceptability of the principle
(Williamson 2000: ch. 5) for a concept of knowledge based on slightly
inexact observations, a so-called *margin of error principle*
(see, e.g., Aucher 2014 for a short summary).



The 5 principles

 
\[ \begin{align}
\neg K\_{a}\varphi &\rightarrow K\_{a}\neg K\_{a}\varphi\\
 \neg B\_{a}\varphi &\rightarrow B\_{a}\neg B\_{a}\varphi\\
 
\end{align} \]


are often referred to as *principles of negative
introspection*. Negative introspection is quite controversial as
it poses very high demands on knowledge and belief. The schema 5 may
be seen as a *closed world assumption* (Hendricks 2005): The
agent has a complete overview of all the possible worlds and own
information. If \(\neg\psi\) is considered possible
(\(\widehat{K}\_{a}\neg\psi\), i.e., \(\neg K\_{a}\psi\)), then the
agent knows it is considered possible (\(K\_{a}\neg K\_{a}\psi\)). Such
a closed world assumption is natural when constructing hyper-rational
agents in, e.g., computer science or game theory, where the agents are
assumed to reason as hard as logically possible about their own
information when making decisions.



Arguing against 5 is Hintikka (1962), using his conception of
epistemic alternatives. Having accepted T and 4 for knowledge, 5
stands or falls with the assumption of a symmetric accessibility
relation. But, Hintikka argues, the accessibility relation is not
symmetric: If the agent possesses some amount of information at model
set \(s\_{1}\), then the model set \(s\_{2}\) where the agent has
learned something more will be an epistemic alternative to \(s\_{1}\).
But \(s\_{1}\) will not be an epistemic alternative to \(s\_{2}\),
because in \(s\_{1}\), the agent does by hypothesis not know as much as
it does in \(s\_{2}\). Hence the relation is not symmetric, so 5 is not
a principle of knowledge, on Hintikka's account.



Given Hintikka's non-standard semantics, it is a bit difficult
to pin down whether he would accept a normal modal logic as the logic
of knowledge and belief, but if so, then **S4** and
**KD4** would be the closest candidates (see Hendricks
& Rendsvig 2018 for this point). By contrast, for knowledge von
Kutschera argued for **S4.4** (1976), Lenzen suggested
**S4.2** (1978), van der Hoek argued for
**S4.3** (1993), and Fagin, Halpern, Moses, and Vardi
(1995) and many others use **S5** for knowledge and
**KD45** for belief.



Beyond principles governing knowledge and principles governing belief,
one may also consider principles governing the interplay between
knowledge and belief. Three principles of interest are



\[\begin{align}
\tag\*{KB1} K\_{a}\varphi & \rightarrow B\_{a}\varphi\\
 \tag\*{KB2} B\_{a}\varphi & \rightarrow K\_{a}B\_{a}\varphi\\
 \tag\*{KB3} B\_{a}\varphi & \rightarrow B\_{a}K\_{a}\varphi\\
 
\end{align} \]


The principles KB1 and KB2 were introduced by Hintikka (1962), who
endorses only the former (see SS3.7), noting that Plato is also
committed to KB1 in *Theatetus*. The first principle, KB1,
captures the intuition that knowledge is a stronger notion than
belief. The second--like 4 and 5--captures the idea that one
has privileged access to one's own beliefs. The third, stemming
from Lenzen (1978), captures the notion that beliefs are held with
some kind of conviction: if something is believed, it is believed to
be known.



Though the interaction principles KB1--KB3 may look innocent on
their own, they may lead to counterintuitive conclusions when combined
with specific logics of knowledge and belief. First, Voorbraak (1993)
shows that combining 5 for knowledge and D for belief with KB1,
implies that

 
\[B\_{a}K\_{a}\varphi\rightarrow K\_{a}\varphi\]


is a theorem of the resulting logic. Assuming that knowledge is
truthful, this theorem entails that agents cannot believe to know
something which happens to be false.



If additionally KB3 is added, the notions of knowledge and belief
*collapse.* I.e., it may be proven that
\(B\_{a}\varphi\rightarrow K\_{a}\varphi\), which, in combination with
KB1 entails that

 
\[B\_{a}\varphi\leftrightarrow K\_{a}\varphi.\]


Hence, the two notions have collapsed to one. This was stated in 1986,
by Kraus and Lehmann.



If one is not interested in knowledge and belief collapsing, one must
thus give something up: One cannot have both 5 for knowledge, D for
belief and KB1 and KB3 governing their interaction. Again, results
concerning correspondence between principles and relational properties
may assist: In 1993, van der Hoek showed, based on a semantic analysis
that where the four principles are jointly sufficient for collapse,
*no subset of them is*. Giving up any one principle will thus
eliminate the collapse. Weakening KB1 to hold only for non-modal
formulas is also sufficient to avoid collapse (cf. Halpern 1996).



For more on epistemic interaction principles, the principles .2, .3,
.3.2. and .4, and relations to so-called *conditional beliefs*,
see Aucher (2014). For an introduction to conditional beliefs and
relations to several other types of knowledge from the philosophical
literature, see Baltag and Smets (2008). The latter also includes
discussion concerning the interdefinability of various notions, as
does Halpern, Samet, and Segev (2009) for knowledge and
(non-conditional) belief.



Note that although we mainly focus on the Kripke semantics in this
entry, there are other epistemic models for potentially weaker logic,
such as neighborhood models (e.g., van Benthem et al. 2014) and
topological models (e.g., Baltag, Bezhanishvili, et al. 2019). 


## 3. Knowledge in Groups



We human beings are preoccupied with the epistemic states of other
agents. In ordinary life, we reason with varying degrees of success
about what others know. We are especially concerned with what others
know about us, and often specifically about what they know about what
we know.




Does she know that I know where she buried the treasure?



Does she know that I know that she knows?



And so on.




Epistemic logic can reveal interesting epistemic features of systems
involving groups of agents. In some cases, for example, emergent
social phenomena depend on agents reasoning in particular ways about
the knowledge and beliefs of other agents. As we have seen,
traditional systems of epistemic logic applied only to single-agent
cases. However, they can be extended to groups or multi-agent systems
in a relatively straightforward manner.



As David Lewis noted in his book *Convention* (1969) many
prominent features of social life depend on agents assuming that the
rules of some practice are matters of *common knowledge*. For
example, drivers know that a red traffic light indicates that they
should stop at an intersection. However, for the convention of traffic
lights to be in place at all, it is first necessary that drivers must
also know that other drivers know that *red* means
*stop*. In addition, drivers must also know that everyone knows
that everyone knows that .... The conventional role of traffic
lights relies on all drivers knowing that all drivers know the rule,
that the rule is a piece of
 common knowledge.



A variety of norms, social and linguistic practices, agent
interactions and games presuppose common knowledge, first formalized
by Aumann (1976) and with earliest epistemic logical treatments by
Lehmann (1984) and by Halpern and Moses (1984). In order to see how
epistemic logic sheds light on these phenomena, it is necessary to
introduce a little more formalism. Following the standard treatment
(see, e.g., Fagin et al. 1995), we can syntactically augment the
language of propositional logic with *n* knowledge operators, one
for each agent involved in the group of agents under consideration.
The primary difference between the semantics given for a mono-agent
and a multi-agent semantics is roughly that *n* accessibility
relations are introduced. A modal system for *n* agents is
obtained by joining together *n* modal logics where for
simplicity it may be assumed that the agents are homogenous in the
sense that they may all be described by the same logical system. The
epistemic logical system for *n* agents consists of *n*
copies of a certain modal system. In such an extended epistemic logic
it is possible to express that some agent in the group knows a certain
fact that an agent knows that another agent knows a fact etc. It is
possible to develop the logic even further: Not only may an agent know
that another agent knows a fact, but they may all know this fact
simultaneously.


### 3.1 Multi-Agent Languages and Models



To represent knowledge for a set \(\mathcal{A}\) of *n* agents,
first let's stipulate a language. Let \(\mathcal{L}\_{Kn}\) be
given by the *Backus-Naur form*

 
\[\varphi::=p\mid \neg\varphi\mid (\varphi\wedge\varphi)\mid K\_{i}\varphi\,\text{ for } p\in\Atom,i\in\mathcal{A}.\]


To represent knowledge for all *n* agents jointly in pointed
Kripke models, all that is needed is to add suitably many
relations:




**Definition:** A ***pointed Kripke
model*** for \(\mathcal{L}\_{Kn}\) is a tuple
\((M,w)=(W,\{R\_{i}\}\_{i\in\mathcal{A}},V,w)\) where


* *W* is a non-empty set of possible worlds,
* For every \(i\in\mathcal{A}\), \(R\_{i}\) is a binary relation on
*W*,
* \(V\colon \Atom\longrightarrow\mathcal{P}(W)\) is a valuation,
and
* \(w\in W\).




To also incorporate beliefs, simply apply the same move as in the
single agent case: augment the language and let there be two relations
for each agent.



The definition uses a family of relations
\(\{R\_{i}\}\_{i\in\mathcal{A}}\). In the literature, the same is
denoted \((W,R\_{i},V,w)\_{i\in\mathcal{A}}\). Alternatively, *R*
is taken to be a function sending agents to relations, i.e.,
\(R:\mathcal{A\rightarrow}\mathcal{P}(W\times W)\). Then for each
\(i\in\mathcal{A}\), \(R(i)\) is a relation on *W*, often denoted
\(R\_{i}\). These are stylistic choices.



When considering only a single agent, it is typically not relevant to
include more worlds in *W* than there are possible valuations of
atoms. In multi-agent cases, this is not the case: to express the
different forms of available higher-order knowledge, many copies of
"the same" world are needed. Let us exemplify for
\(\mathcal{A}=\{a,b\}\), \(\Atom=\{p\}\) and each
\(R\_{i},i\in\mathcal{A},\) an equivalence relation. Let us represent
that both *a* and *b* know *p*, but *b* does not
know that *a* knows *p*, i.e., \(K\_{a}p\wedge
K\_{b}p\wedge\neg K\_{b}K\_{a}p\). Then we need three worlds:



![Three boxes labeled w1 (containing 'p'), w2 (containing 'p'), and w3 (containing 'not p'). Each box has an arrow labeled 'a,b' looping back to it. w1 is highlighted and is connected to w2 by a double headed arrow labeled 'b'. w2 is connected to w3 by a double headed arrow labeled 'a'.](fig13.svg)


If we delete \(w\_{1}\) and let \(w\_{2}\) be the actual world, then
*a* would lose knowledge in *p*: both *p* worlds are
needed. In general, if *W* is assumed to have any fixed, finite
size, there will be some higher-order information formula that cannot
be satisfied in it.


### 3.2 Notions of Group Knowledge



Multi-agent systems are interesting for other reasons than to
represent higher-order information. The individual agents'
information may also be pooled to capture what the agents know
jointly, as group knowledge (see Baltag, Boddy, & Smets 2018 for a
recent discussion). A standard notion is this style is *distributed
knowledge*: The knowledge the group *would have* if the
agents share all their individual knowledge. To represent it, augment
the language \(\mathcal{L}\_{Kn}\) with operators

 
\[D\_{G}\text{ for }G\subseteq\mathcal{A},\]


to make \(D\_{G}\varphi\) a well-formed formula. Where
\(G\subseteq\mathcal{A}\) is a group of agents, the formula
\(D\_{G}\varphi\) reads that it is *distributed knowledge in the
group *G* that \(\varphi\)*.



To evaluate \(D\_{G}\varphi\), we define a new relation from those
already present in the model. The idea behind the definition is that
if some one agent has eliminated a world as an epistemic alternative,
then so will the group. Define the relation as the intersection of the
individual agents' relations:

 
\[R\_{G}^{D}=\bigcap\_{i\in G}R\_{i}\]


In the three state model, \(R\_{G}^{D}\) contains only the three loops.
To evaluate a distributed knowledge formula, use the same form as for
other modal operators:

 
\[(M,w)\vDash D\_{G}\varphi\text{ iff }(M,w')\vDash\varphi\text{ for all }w'\in W\text{ such that }wR\_{G}^{D}w'.\]


It may be the case that some very knowing agent knows all that is
distributed knowledge in *G*, but it is not guaranteed. To
capture that all the agents know \(\varphi\), we could use the
conjunction of the formulas \(K\_{i}\varphi\) for \(\in\mathcal{A}\),
i.e., \(\bigwedge\_{i\in\mathcal{A}}K\_{i}\varphi\). This is a
well-defined formula if \(\mathcal{A}\) is finite (which it typically
is). If \(\mathcal{A}\) is not finite, then
\(\bigwedge\_{i\in\mathcal{A}}K\_{i}\varphi\) is not a formula in
\(\mathcal{L}\_{Kn}\), as it has only finite conjunctions. As a
shorthand for \(\bigwedge\_{i\in\mathcal{A}}K\_{i}\varphi\), it is
standard to introduce the *everybody knows* operator,
\(E\_{G}\):

 
\[E\_{G}\varphi:=\bigwedge\_{i\in\mathcal{A}}K\_{i}\varphi.\]


In the three world model, \(K\_{a}p\wedge K\_{b}p\), so
\(E\_{\{a,b\}}p\).



That everybody knows something does not mean that this knowledge is
shared between the members of the group. The three world model
exemplifies this: Though \(E\_{\{a,b\}}p\), it also the case that
\(\neg K\_{b}E\_{\{a,b\}}p\).



To capture that there is no uncertainty in the group about \(\varphi\)
nor *any higher-order* uncertainty about \(\varphi\) being
known by all agents, no formula in the language \(\mathcal{L}\_{Kn}\)
is enough. Consider the formula

 
\[E\_{G}^{k}\varphi\]


where \(E\_{G}^{k}\) is short for *k* iterations of the \(E\_{G}\)
operator. Then for no natural number *k* will the formula
\(E\_{G}^{k}\varphi\) be enough: it could be the case that *b*
doesn't know it! To rectify this situation, one could try



\[\bigwedge\_{k\in\mathbb{N}}E\_{G}^{k}\varphi\]


but this is not a formula as \(\mathcal{L}\_{Kn}\) only contains finite
conjunctions.



Hence, though the \(E\_{G}\) operator is definable in the language
\(\mathcal{L}\_{Kn}\), a suitable notion of *common knowledge*
is not. For that, we again need to define a new relation on our model.
This time, we are interested in capturing that nobody considers
\(\varphi\) epistemically possible *anywhere*. To build the
relation, we therefore first take the union the relations of all the
agents in *G*, but this is not quite enough: to use the standard
modal semantic clause, we must also be able to reach all of the worlds
in this relation *in a single step*. Hence, let



\[R\_{G}^{C}:=\left(\bigcup\_{i\in G}R\_{i}\right)^{\*}\]


where \((\cdotp)^{\*}\) is the operation of taking the *reflecxive
transitive closure.* If *R* is a relation, then \((R)^{\*}\)
is *R* plus the reflexive arrow and all the pairs missing to make
*R* a transitive relation. Consider the three world model: With
the relation \(\bigcup\_{i\in\{a,b\}}R\_{i}\), we can reach \(w\_{3}\)
from \(w\_{1}\) in two steps, stopping over at \(w\_{2}\). With
\((\bigcup\_{i\in\{a,b\}}R\_{i})^{\*}\), \(w\_{3}\) is reachable in one
step: By the newly added transitive link from \(w\_{1}\) to
\(w\_{3}\).



To represent common knowledge, augment the *Backus-Naur form*
of \(\mathcal{L}\_{Kn}\) with operators

 
\[C\_{G}\text{ for }G\subseteq\mathcal{A},\]


to make \(C\_{G}\varphi\) a well-formed formula. Evaluate such formulas
by the semantic clause

 
\[(M,w)\vDash C\_{G}\varphi\text{ iff }(M,w')\vDash\varphi\text{ for all }w'\in W\text{ such that }wR\_{G}^{C}w'.\]


Varying the properties of the accessibility relations
\(R\_{1},R\_{2},\ldots,R\_{n}\), as described above results in different
epistemic logics. For instance, system **K** with common
knowledge is determined by all frames, while system
**S4** with common knowledge is determined by all
reflexive and transitive frames. Similar results can be obtained for
the remaining epistemic logics (Fagin et al. 1995, van Ditmarsch et
al. 2007). For an informal discussion, consult
 the entry on common knowledge.


## 4. Beyond Knowing That



So far, we have focused on *know-that*, the standard notion
analyzed in epistemic logic. However, as mentioned earlier, in natural
language, we also express knowledge in terms of know-wh, i.e., the
verb "know" followed by an embedded question, such as:


* Alice *knows whether* the claim is true.
* Bob *knows what* the password is.
* Charlie *knows how* to prove the theorem.
* Dave *knows why* Charlie *knows how* to prove the
theorem.



Logical research on know-wh also dates back to Hintikka (1962), who
dedicated a chapter to the formal account of *know-who*, and
proposed to formalize other *know-wh* alike. According to
Hintikka, to know "who Mary is" is to know the answer to
the corresponding question "Who is Mary?". Essentially,
one must be able to identify the person denoted by the name
*Mary* (supposing it is a proper name). Therefore, it is a
piece of *de re* knowledge, i.e., the knowledge of an
*object*. To render it formally, Hintikka treats "Bob
knows who Mary is" as "there is a person such that Bob
*knows that* this person is Mary", which connects
know-that with know-wh by using quantification over objects. To
express Hintikka's formalization, we need to extend our basic
language of propositional modal logic to *first-order modal
logic*, the modal logic with quantifiers and predicates. As
another example, "Bob knows who murdered Dave" can be
formalized in first-order epistemic logic as \(\exists x K\_{\Bob}
\Murder(x, \Dave)\). Note that the order of the quantifier and the
know-that modality is crucial to capture *de re* knowledge. The
swapped *de dicto* version \(K\_{\Bob} \exists x\Murder(x,
\Dave)\) merely says "Bob knows that Dave is murdered".
For Hintikka, it is also important how the agent is identified, cf.
(Hintikka & Symons 2003).



Such treatments of know-wh are also supported by a large body of
research on the semantics of (embedded) questions in linguistics
(e.g., Groenendijk & Stokhof 1982; Harrah 2002), and epistemology
of knowledge-wh (e.g., Stanley & Williamson 2001). Linguists have
discussed various readings of epistemic expressions with embedded
questions, in particular the so-called *mention-some* and
*mention-all* interpretations. The mention-some interpretation
requires that one knows at least one (correct) answer in order to
possess know-wh knowledge, while the mention-all interpretation
requires that one knows all the (correct) answers (cf. Groenendijk
& Stokhof 1982). For example, a (strong) exhaustive mention-all
reading for "Bob knows who came to the party" is
formalized by \(\forall x (K\_{\Bob} \Came(x) \lor K\_{\Bob}\neg \Came
(x))\) where \(\forall\) replaces \(\exists\) as the leading
quantifier, contra the previous examples. The interpretation of
know-wh is also context-dependent in general, which introduces further
complications (e.g., Aloni 2001, 2018). Coming back to logic, Hintikka
(1962) discussed the notion of consistency in such a logic of know-who
and various philosophical issues, such as trans-world identity. In his
latter work, Hintikka (2003) also made use of his *Independent
Friendly Logic* to go beyond the first-order quantification, in
order to capture wh-knowledge about higher-order entities, such as
"I know whom every young mother should trust" with the
intention of "trusting *her own mother*".



Although Hintikka gave a basic semantics of know-wh, the systematical
study of the epistemic logics of know-wh was not developed until
recently. This is partly due to the underdevelopment of first-order
modal logic, compared to its propositional brother (cf., Gochet &
Gribomont 2006 for a survey on first-order epistemic logic). Various
well-known systems of first-order modal logic do not enjoy the
technically desirable properties of propositional modal logic, but
suffers incompleteness, undecidability and the failure of
interpolation (cf., Brauner & Ghilardi 2007). In particular,
it is very hard to find decidable fragments of first-order modal logic
to make it computationally appealing (e.g., Hodkinson, Wolter, &
Zakharyaschev 2000).



Nevertheless, there are works on clarifying the *de re* and
*de dicto* distinctions in the setting of quantified epistemic
logic (e.g., Grove 1995; Corsi & Orlandelli 2013; Holliday &
Perry 2014; Occhipinti Liberman & Rendsvig 2022), and a number of
application-driven frameworks based on first-order epistemic logic
(e.g., Mika Cohen & Dam 2007; Kaneko & Nagashima 1996; Sturm,
Wolter, & Zakharyaschev 2000; Belardinelli & Lomuscio 2011),
although they are not directly about know-wh. As one of a few
exceptions directly related to know-wh based on quantified modal
logic, Rendsvig (2012) discusses Frege's Puzzle about Identity
in a framework where agents may have knowledge-who about the referents
of proper names.



Another line of research directly related to know-who is based on
*term-modal logic* proposed by (Fitting, Thalmann, &
Voronkov 2001), a variant of quantified modal logic where modalities
are subscripted by *terms* instead of indexes. This allows
formulas such as \(\exists y \exists x K\_{y} \Murder(x, \Dave)\)
("there is someone that knows who murdered Dave"), or
\(\exists x K\_{\Bob}( \Murder(x, \Dave) \wedge x = \Adam)\)
("Bob knows who murdered Dave and that is Adam"). Starting
from (Kooi 2007), various versions of (dynamic) term-modal logic were
proposed to discuss know-who, such as Occhipinti Liberman, Achen,
& Rendsvig 2020; and Y. Wang, Wei, & Seligman 2022. These are
closer to the original systems suggested by Hintikka (1962), where the
names of agents are also treated as terms. Term-modal logic
formalization of know-who in social networks, and the dynamics that
influence it, is a main theme in Occhipinti Liberman & Rendsvig
2022. A few decidable fragments of term-modal logics were discovered
in Orlandelli & Corsi 2017; Padmanabha & Ramanujam 2019a,
2019b; and Occhipinti Liberman, Achen, and Rendsvig 2020, where a
recent review of the term-modal logic literature can be found.



Concerning a different type of know-wh, some formal discussions on
know-how have appeared in the setting of *strategy logics*. It
is observed that in the setting of *Epistemic Alternating Temporal
Logic*, just combining "know" and "can"
does not capture know-how (cf., Jamroga & van der Hoek 2004;
Herzig 2015) but *de dicto* knowledge: I know there is some way
to guarantee \(\varphi\). Instead of explicitly using the quantifiers,
people introduced alternative semantics for the coalition operators in
epistemic variants of ATL (e.g., Jamroga & Agotnes 2007;
Maubert, Pinchinat, Schwarzentruber, & Stranieri 2020; Maubert,
Murano, Pinchinat, Schwarzentruber, & Stranieri 2020).



Recently, a *know-wh first* approach has emerged that treats
know-wh as an independent modality, just as the modality of know-that
(cf. the survey by Y. Wang 2018b). The general motivation of this line
of research is to focus on the logical behavior of each know-wh as a
*primitive concept*, rather than breaking it down in the syntax
into quantifiers, modalities, and predicates in the full language of
predicate modal logic. This approach is inspired by the early work of
Plaza 1989 and Ma & Guo 1983, where a \(Kv\) operator is
introduced to capture the notion of "knowing the value of a
constant". Semantically, \(Kv d\) can be viewed as a
"bundle" of \(\exists x K(d=x)\), which says the agent
*knows what* the value of \(d\) is. In this line of research,
various languages were proposed with both the know-that operator and
the know-wh operator of certain types, whose semantics is given by the
first-order modal interpretation, as we described above for the \(Kv\)
operator.



As an example of a *know-wh first* framework, consider the
following epistemic language of (goal-directed) know-how and know-that
discussed by Li & Wang (2021a), where \(K\!h\_i\varphi\) says that
agent \(i\) *knows how* to achieve the goal \(\varphi\).



\[\varphi::=p\mid\neg\varphi\mid (\varphi\land\varphi)\mid K\_i\varphi\mid 
 K\!h\_i\varphi\]


The semantics is given over a type of Kripke models with both
epistemic relations \(\sim\_i, i \in \mathcal{A}\) and labeled action
relations \(\stackrel{a}{\rightarrow}, a \in \Act\), where
\(\Act=\cup\_{i\in \mathcal{A}}\Act\_i\) is the union of the sets of
actions of each agent, which do not explicitly described in the formal
language. Such a model takes the form \(\mathcal{M}= (W,\{\sim\_i\}\_{i
\in \mathcal{A}}, \{\stackrel{a}{\rightarrow}\}\_{a \in \Act},V)\). The
satisfaction relation between a pointed Kripke model \(\mathcal{M},w\)
and a \(K\!h\_i\varphi\) formula is given by the following \(\exists x
K\)-style schema:




|  |  |  |
| --- | --- | --- |
| \(\mathcal{M},w\vDash K\!h\_i\varphi\) | \(\iff\) | *there is* a plan \(\pi\) such that *for all* \(w'
\sim\_i w\):


1.
 \(\pi\) is *strongly executable* by \(i\) on \(w'\);
2.
 \(\mathcal{M},v\vDash\varphi\) for each final state \(v\)
reachable by executing \(\pi\).
 |



A plan \(\pi\) can have different forms, such as a finite linear
sequence of actions, a conditional strategy given the knowledge of
agents, or even a program with loops (cf., Li & Wang 2021b). A
plan is *strongly executable* intuitively means the plan will
never get stuck and always terminates successfully.



To exemplify the framework, suppose that a patient is experiencing
some rare symptom \(p\). To know the cause (\(q\) or \(\neg q\)),
Doctor 1 suggests the patient first take a special test (\(a\)), which
is only available at Doctor 1's hospital, and send the results
to Doctor 2 in another city, who is more experienced in examining the
results. Doctor 2 will then know the cause, depending on which,
different medicines (\(b\) or \(c\)) should be subscribed to cure the
patient. The situation is depicted below as a model with both
epistemic relations (dotted lines labeled by \(1,2\); reflexive ones
omitted) and action relations (solid lines labeled by \(a,b,c\)). Note
that only Doctor 1 can execute action \(a\), and only Doctor 2 can
perform \(b\) or \(c\). After action \(a\), Doctor 2 knows whether she
is at \(w\_3\) or at \(w\_4\), but Doctor 1 is still uncertain about
it.



![In the model, the dotted lines represent the epistemic relations labeled by agents, and the solid lines represent the action transitions labeled by actions. E.g., on w3 performing action b will result in w5, while performing action c will result in w6. The self-loops are omitted for epistemic relations, e.g., on w3, Doctor 2 only consider w3 possible.](fig14.svg)


Intuitively, Doctor 1 knows how to let Doctor 2 know how to cure the
patient (\(\neg p\)), although neither Doctor 1 nor Doctor 2 knows how
to cure the patient alone. This can be expressed by \(\neg K\!h\_1 \neg
p\land \neg K\!h\_2 \neg p\land K\!h\_1((K\_2 q\lor K\_2\neg q)\land
K\!h\_2 \neg p)\), which is true at \(w\_1\) and \(w\_2\), given a proper
notion of plans.



Finally, logics of know-how may be obtained for this framework. The
axioms depend on the specific assumptions made of the Kripke models,
as in standard epistemic logic. For example, if we allow
*conditional knowledge-based plans* and consider agents with
*perfect recall* (as in Fervari, Herzig, Li, & Wang 2017),
then the following axioms and rules on top of the S5 proof system for
\(K\_i\) are complete:




|  |  |  |  |
| --- | --- | --- | --- |
| \(\mathtt{EMP}\) | \(K\_i p \to K\!h\_i p\) | \(\mathtt{KhK}\) | \(K\!h\_i p \to K\!h\_i K\_i p\) |
| \(\mathtt{KKh}\) | \(K\!h\_i p \to K\_iK\!h\_i p\) | \(\mathtt{KhKh}\) | \(K\!h\_i K\!h\_i p \to K\!h\_i p\) |
| \(\mathtt{Khbot}\) | \(\neg K\!h\_i \bot\) | \(\mathtt{MONOKh}\) | \(\dfrac{\vdash\varphi\to\psi}{K\!h\_i\varphi\to
K\!h\_i\psi}\) |