Epistemic modal logic is a subfield of
modal logic that is concerned with reasoning about
knowledge
Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is disti ...
. While
epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
has a long philosophical tradition dating back to
Ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
, epistemic logic is a much more recent development with applications in many fields, including
philosophy,
theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumsc ...
,
artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machine
A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, moveme ...
,
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...
and
linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Lingu ...
. While philosophers since
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
have discussed modal logic, and
Medieval philosophers such as
Avicenna
Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islam ...
,
Ockham, and
Duns Scotus
John Duns Scotus ( – 8 November 1308), commonly called Duns Scotus ( ; ; "Duns the Scot"), was a Scottish Catholic priest and Franciscan friar, university professor, philosopher, and theologian. He is one of the four most important ...
developed many of their observations, it was
C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of
Kripke.
Historical development
Many papers were written in the 1950s that spoke of a logic of knowledge in passing, but the Finnish philosopher
G. H. von Wright's 1951 paper titled ''An Essay in Modal Logic'' is seen as a founding document. It was not until 1962 that another Finn,
Hintikka, would write ''Knowledge and Belief'', the first book-length work to suggest using modalities to capture the semantics of knowledge rather than the
alethic statements typically discussed in modal logic. This work laid much of the groundwork for the subject, but a great deal of research has taken place since that time. For example, epistemic logic has been combined recently with some ideas from
dynamic logic to create
dynamic epistemic logic Dynamic epistemic logic (DEL) is a logical framework dealing with knowledge and information change. Typically, DEL focuses on situations involving multiple Intelligent agent, agents and studies how their knowledge changes when Event (philosophy), ev ...
, which can be used to specify and reason about information change and exchange of information in
multi-agent systems
A multi-agent system (MAS or "self-organized system") is a computerized system composed of multiple interacting intelligent agents.Hu, J.; Bhowmick, P.; Jang, I.; Arvin, F.; Lanzon, A.,A Decentralized Cluster Formation Containment Framework fo ...
. The seminal works in this field are by Plaza,
Van Benthem Van Benthem is a Dutch surname meaning "from Bentheim". It can refer to:
*Evert van Benthem (b. 1958), Dutch speedskater, twice winner of the Elfstedentocht
* Jean Van Benthem (fl. 1908), Belgian cyclist
* Johan van Benthem (b. 1949), Dutch pro ...
, and Baltag, Moss, and Solecki.
Standard possible worlds model
Most attempts at modeling knowledge have been based on the
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their ...
s model. In order to do this, we must divide the set of possible worlds between those that are compatible with an agent's knowledge, and those that are not. This generally conforms with common usage. If I know that it is either Friday or Saturday, then I know for sure that it is not Thursday. There is no possible world compatible with my knowledge where it is Thursday, since in all these worlds it is either Friday or Saturday. While we will primarily be discussing the logic-based approach to accomplishing this task, it is worthwhile to mention here the other primary method in use, the
event-based approach. In this particular usage, events are sets of possible worlds, and knowledge is an operator on events. Though the strategies are closely related, there are two important distinctions to be made between them:
* The underlying mathematical model of the logic-based approach are
Kripke semantics, while the event-based approach employs the related
Aumann structures based on
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
.
* In the event-based approach logical formulas are done away with completely, while the logic-based approach uses the system of modal logic.
Typically, the logic-based approach has been used in fields such as philosophy, logic and AI, while the event-based approach is more often used in fields such as
game theory and
mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...
. In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe.
Syntax
The basic
modal operator
A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non- truth-functional in the following se ...
of epistemic logic, usually written ''K'', can be read as "it is known that," "it is epistemically necessary that," or "it is inconsistent with what is known that not." If there is more than one agent whose knowledge is to be represented, subscripts can be attached to the operator (
,
, etc.) to indicate which agent one is talking about. So
can be read as "Agent
knows that
." Thus, epistemic logic can be an example of
multimodal logic A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.
Overview
A modal logic with ''n'' primitive unary modal operators \Box_i, i\in \ is called an ...
applied for
knowledge representation
Knowledge representation and reasoning (KRR, KR&R, KR²) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks such as diagnosing a medic ...
. The dual of ''K'', which would be in the same relationship to ''K'' as
is to
, has no specific symbol, but can be represented by
, which can be read as "
does not know that not
" or "It is consistent with
's knowledge that
is possible". The statement "
does not know whether or not
" can be expressed as
.
In order to accommodate notions of
common knowledge and
distributed knowledge, three other modal operators can be added to the language. These are
, which reads "every agent in group G knows" (
mutual knowledge);
, which reads "it is common knowledge to every agent in G"; and
, which reads "it is distributed knowledge to the whole group G." If
is a formula of our language, then so are
,
, and
. Just as the subscript after
can be omitted when there is only one agent, the subscript after the modal operators
,
, and
can be omitted when the group is the set of all agents.
Semantics
As we mentioned above, the logic-based approach is built upon the possible worlds model, the semantics of which are often given definite form in Kripke structures, also known as Kripke models. A Kripke structure ''M'' for ''n'' agents over
is an (''n'' + 2)-tuple
, where S is a nonempty set of ''states'' or ''possible worlds'',
is an ''interpretation'', which associates with each state in S a truth assignment to the primitive propositions in
(the set of all primitive propositions), and
are
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
s on S for ''n'' numbers of agents. It is important here not to confuse
, our modal operator, and
, our accessibility relation.
The truth assignment tells us whether or not a proposition ''p'' is true or false in a certain state. So
tells us whether ''p'' is true in state ''s'' in model
. Truth depends not only on the structure, but on the current world as well. Just because something is true in one world does not mean it is true in another. To state that a formula
is true at a certain world, one writes
, normally read as "
is true at (M,s)," or "(M,s) satisfies
".
It is useful to think of our binary relation
as a ''possibility'' relation, because it is meant to capture what worlds or states agent ''i'' considers to be possible; In other words,
if and only if