EPISTEMIC MODAL LOGIC is a subfield of modal logic that is concerned
with reasoning about knowledge . While epistemology has a long
philosophical tradition dating back to
Ancient Greece
CONTENTS * 1 Historical development * 2 Standard possible worlds model * 2.1 Syntax * 2.2 Semantics * 3 The properties of knowledge * 3.1 The distribution axiom * 3.2 The knowledge generalization rule * 3.3 The knowledge or truth axiom * 3.4 The positive introspection axiom * 3.5 The negative introspection axiom * 3.6 Axiom systems * 4 Problems with the possible world model and modal model of knowledge * 5 See also * 6 Notes * 7 References * 8 External links HISTORICAL DEVELOPMENT Many papers were written in the 1950s that spoke of a logic of
knowledge in passing, but it was Finnish philosopher von Wright 's
paper An Essay in Modal Logic from 1951 that is seen as a founding
document. It was not until 1962 that another Finn, Hintikka , would
write
Knowledge
STANDARD POSSIBLE WORLDS MODEL Most attempts at modeling knowledge have been based on the possible worlds 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 . * 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 . 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 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 ( K 1 {displaystyle {mathit {K}}_{1}} , K 2 {displaystyle {mathit {K}}_{2}} , etc.) to indicate which agent one is talking about. So K a {displaystyle {mathit {K}}_{a}varphi } can be read as "Agent a {displaystyle a} knows that {displaystyle varphi } ." Thus, epistemic logic can be an example of multimodal logic applied for knowledge representation . The dual of K, which would be in the same relationship to K as {displaystyle Diamond } is to {displaystyle Box } , has no specific symbol, but can be represented by K a {displaystyle neg K_{a}neg varphi } , which can be read as " a {displaystyle a} does not know that not {displaystyle varphi } " or "It is consistent with a {displaystyle a} 's knowledge that {displaystyle varphi } is possible". The statement " a {displaystyle a} does not know whether or not {displaystyle varphi } " can be expressed as K a K a {displaystyle neg K_{a}varphi land neg K_{a}neg varphi } . In order to accommodate notions of common knowledge and distributed knowledge , three other modal operators can be added to the language. These are E G {displaystyle {mathit {E}}_{mathit {G}}} , which reads "every agent in group G knows;" C G {displaystyle {mathit {C}}_{mathit {G}}} , which reads "it is common knowledge to every agent in G;" and D G {displaystyle {mathit {D}}_{mathit {G}}} , which reads "it is distributed knowledge to every agent in G." If {displaystyle varphi } is a formula of our language, then so are E G {displaystyle {mathit {E}}_{G}varphi } , C G {displaystyle {mathit {C}}_{G}varphi } , and D G {displaystyle {mathit {D}}_{G}varphi } . Just as the subscript after K {displaystyle {mathit {K}}} can be omitted when there is only one agent, the subscript after the modal operators E {displaystyle {mathit {E}}} , C {displaystyle {mathit {C}}} , and D {displaystyle {mathit {D}}} 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 {displaystyle Phi } is a (n+2)-tuple ( S , , K 1 , . . . , K n ) {displaystyle (S,pi ,{mathcal {K}}_{1},...,{mathcal {K}}_{n})} , where S is a nonempty set of states or possible worlds, {displaystyle pi } is an interpretation, which associates with each state in S a truth assignment to the primitive propositions in {displaystyle Phi } , and K 1 , . . . , K n {displaystyle {mathcal {K}}_{1},...,{mathcal {K}}_{n}} are binary relations on S for n numbers of agents. It is important here not to confuse K i {displaystyle K_{i}} , our modal operator, and K i {displaystyle {mathcal {K}}_{i}} , our accessibility relation. The truth assignment tells us whether or not a proposition p is true or false in a certain state. So ( s ) ( p ) {displaystyle pi (s)(p)} tells us whether p is true in state s in model M {displaystyle {mathcal {M}}} . 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 {displaystyle varphi } is true at a certain world, one writes ( M , s ) {displaystyle (M,s)models varphi } , normally read as " {displaystyle varphi } is true at (M,s)," or "(M,s) satisfies {displaystyle varphi } ". It is useful to think of our binary relation K i {displaystyle {mathcal {K}}_{i}} as a possibility relation, because it is meant to capture what worlds or states agent i considers to be possible. In idealized accounts of knowledge (e.g., describing the epistemic status of perfect reasoners with infinite memory capacity), it makes sense for K i {displaystyle {mathcal {K}}_{i}} to be an equivalence relation , since this is the strongest form and is the most appropriate for the greatest number of applications. An equivalence relation is a binary relation that is reflexive , symmetric , and transitive . The accessibility relation does not have to have these qualities; there are certainly other choices possible, such as those used when modeling belief rather than knowledge. THE PROPERTIES OF KNOWLEDGE Assuming that K i {displaystyle {mathcal {K}}_{i}} is an equivalence relation, and that the agents are perfect reasoners, a few properties of knowledge can be derived. The properties listed here are often known as the "S5 Properties," for reasons described in the Axiom Systems section below. THE DISTRIBUTION AXIOM This axiom is traditionally known as K. In epistemic terms, it states that if an agent knows {displaystyle varphi } and knows that {displaystyle varphi implies psi } , then the agent must also know {displaystyle ,psi } . So, ( K i K i ( ) ) K i {displaystyle (K_{i}varphi land K_{i}(varphi implies psi ))implies K_{i}psi } THE KNOWLEDGE GENERALIZATION RULE Another property we can derive is that if {displaystyle phi } is valid, then K i {displaystyle K_{i}phi } . This does not mean that if {displaystyle phi } is true, then agent i knows {displaystyle phi } . What it means is that if {displaystyle phi } is true in every world that an agent considers to be a possible world, then the agent must know {displaystyle phi } at every possible world. This principle is traditionally called N. if M then M K i . {displaystyle {text{if }}Mmodels varphi {text{ then }}Mmodels K_{i}varphi .,} THE KNOWLEDGE OR TRUTH AXIOM This axiom is also known as T. It says that if an agent knows facts, the facts must be true. This has often been taken as the major distinguishing feature between knowledge and belief. We can believe a statement to be true when it is false, but it would be impossible to know a false statement. K i {displaystyle K_{i}varphi implies varphi } THE POSITIVE INTROSPECTION AXIOM This property and the next state that an agent has introspection
about its own knowledge, and are traditionally known as 4 and 5,
respectively. The Positive Introspection Axiom, also known as the KK
Axiom, says specifically that agents know that they know what they
know. This axiom may seem less obvious than the ones listed
previously, and
Timothy Williamson has argued against its inclusion
forcefully in his book,
Knowledge
THE NEGATIVE INTROSPECTION AXIOM The Negative Introspection Axiom says that agents know that they do not know what they do not know. K i K i K i {displaystyle neg K_{i}varphi implies K_{i}neg K_{i}varphi } AXIOM SYSTEMS Different modal logics can be derived from taking different subsets
of these axioms, and these logics are normally named after the
important axioms being employed. However, this is not always the case.
KT45, the modal logic that results from the combining of K, T, 4, 5,
and the
Knowledge
Epistemic logic also deals with belief, not just knowledge. The basic modal operator is usually written B instead of K. In this case though, the knowledge axiom no longer seems right—agents only sometimes believe the truth—so it is usually replaced with the Consistency Axiom, traditionally called D: B i {displaystyle neg B_{i}bot } which states that the agent does not believe a contradiction, or that which is false. When D replaces T in S5, the resulting system is known as KD45. This results in different properties for K i {displaystyle {mathcal {K}}_{i}} as well. For example, in a system where an agent "believes" something to be true, but it is not actually true, the accessibility relation would be non-reflexive. The logic of belief is called doxastic logic . PROBLEMS WITH THE POSSIBLE WORLD MODEL AND MODAL MODEL OF KNOWLEDGE The notion of knowledge discussed does not take into account computational constraints on inference. If we take the possible worlds approach to knowledge, it follows that our epistemic agent a knows all the logical consequences of his or her or its beliefs. If Q {displaystyle Q} is a logical consequence of P {displaystyle P} , then there is no possible world where P {displaystyle P} is true but Q {displaystyle Q} is not. So if a knows that P {displaystyle P} , it follows that all of the logical consequences of P {displaystyle P} are true of all of the possible worlds compatible with a 's beliefs. Therefore, a knows Q {displaystyle Q} . It is not epistemically possible for a that not- Q {displaystyle Q} given his knowledge that P {displaystyle P} . This consideration was a part of what led Robert Stalnaker to develop two dimensionalism , which can arguably explain how we might not know all the logical consequences of our beliefs even if there are no worlds where the propositions we know come out true but their consequences false. Even when we ignore possible world semantics and stick to axiomatic
systems, this peculiar feature holds. With K and N (the Distribution
Rule and the
Knowledge
SEE ALSO * Logic portal * Common knowledge
*
Epistemic closure
*
Epistemology
NOTES * ^ p. 257 in: Ferenczi, Miklós (2002). Matematikai logika (in
Hungarian). Budapest: Műszaki könyvkiadó. ISBN 963-16-2870-1 .
257
* ^ Stalnaker, Robert. "Propositions." Issues in the
Philosophy
REFERENCES * Anderson, A. and N. D. Belnap. Entailment: The Logic of Relevance and Necessity. Princeton: Princeton University Press |