Epistemic logic

Epistemic logic

Epistemic 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, epistemic logic is a much more recent development with applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Ockham and Duns Scotus 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.

Many papers were written in the fifties that spoke of a logic of knowledge in passing, but it was 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 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 public announcement logic and product update logic, which attempt to model the epistemic subtleties of conversations. The seminal works in this field are by Plaza, van Benthem, and Baltag, Moss, and Solecki.

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. 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 structures, 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 (mathit{K}_1, mathit{K}_2, etc.) to indicate which agent one is talking about. So mathit{K}_avarphi can be read as "Agent a knows that 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 Diamond is to Box, has no specific symbol, but can be represented by neg K_a neg varphi, which can be read as "a does not know that not varphi" or "a holds varphi possible." "a does not know whether or not varphi" can be expressed as neg K_avarphi land neg K_anegvarphi.

In order to accommodate notions of common knowledge and distributed knowledge, three other modal operators can be added to the language. These are mathit{E}_mathit{G}, which reads "every agent in group G knows;" mathit{C}_mathit{G}, which reads "it is common knowledge to every agent in G;" and mathit{D}_mathit{G}, which reads "it is distributed knowledge to every agent in G." If varphi is a formula of our language, then so are mathit{E}_G varphi, mathit{C}_G varphi, and mathit{D}_G varphi. Just as the subscript after mathit{K} can be omitted when there is only one agent, the subscript after the modal operators mathit{E}, mathit{C}, and 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 Phi is a tuple (S, pi, mathcal{K}_1, ..., mathcal{K}_n), where S is a nonempty set of states or possible worlds, pi is an interpretation which associates with each state in S a truth assignment to the primitive propositions in Phi, and 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, our modal operator, and 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 pi (s)(p) tells us whether p is true in state s in model 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 show that a formula varphi is true at a certain world, one writes (M,s) models varphi, normally read as "varphi is true at (M,s)," or "(M,s) satisfies varphi".

It is useful to think of our binary relation mathcal{K}_i as a possibility relation, because it is meant to capture what worlds or states agent i considers to be possible. It also usually makes sense for 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 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 varphi and knows that varphi implies psi, then the agent must also know psi. So,

(K_ivarphi land K_i(varphi implies psi)) implies K_ipsi

The knowledge generalization rule

Another property we can derive is that if varphi is valid, then K_ivarphi. It does not mean that if varphi is true, that agent i knows varphi. What it means is that if varphi is true in every world that an agent considers to be a possible world, then the agent must know varphi at every possible world.

text{if }M models varphitext{ then }M models 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. While you can believe something that is false, you cannot know something that is false.

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 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 recent book, Knowledge and Its Limits.

K_i varphi implies K_i K_i varphi

The negative introspection axiom

The Negative Introspection Axiom says that agents know what they do not know.

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 Generalization Rule, is primarily known as S5. This is why the properties of knowledge described above are often called the S5 Properties.

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:

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 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.

Notes

References

  • Fagin, Ronald et al. Reasoning about Knowledge. Cambridge: MIT Press, 2003.
  • Meyer, J-J C., 2001, "Epistemic Logic," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
  • Anderson, A. and N. D. Belnap. Entailment: The Logic of Relevance and Necessity. Princeton: Princeton University Press, 1975.
  • Fagin et al. "A nonstandard approach to the logical omniscience problem." Artificial Intelligence, Volume 79, Number 2, 1995, p. 203-40.
  • Hintikka, J. Knowledge and Belief. Ithaca: Cornell University Press, 1962.
  • Montague, R. "Universal Grammar". Theoretica, Volume 36, 1970, p. 373-398.

See also

External links

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