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 Avicenna, 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 Saul Kripke.
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, Jaakko 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.
The seminal works in this field are by Plaza, Van Benthem, and Baltag, Moss, and Solecki.
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.
Though the strategies are closely related, there are two important distinctions to be made between them: 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.
If there is more than one agent whose knowledge is to be represented, subscripts can be attached to the operator (
In order to accommodate notions of common knowledge (e.g. in the Muddy Children Puzzle) and distributed knowledge, three other modal operators can be added to the language.
As 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.
In idealized accounts of knowledge (e.g., describing the epistemic status of perfect reasoners with infinite memory capacity), it makes sense for
to be an equivalence relation, since this is the strongest form and is the most appropriate for the greatest number of applications.
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.
is an equivalence relation, and that the agents are perfect reasoners, a few properties of knowledge can be derived.
This axiom is traditionally known as K. In epistemic terms, it states that if an agent knows
This axiom logically establishes modus ponens as a rule of inference for every epistemically possible world.
This property and the next state that an agent has introspection about its own knowledge, and are traditionally known as 4 and 5, respectively.
This axiom may seem less obvious than the ones listed previously, and Timothy Williamson has argued against its inclusion forcefully in his book, Knowledge and Its Limits.
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.
KT45, the modal logic that results from the combining of K, T, 4, 5, and the Knowledge Generalization Rule, is primarily known as S5.
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: which states that the agent does not believe a contradiction, or that which is false.
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.
[3] 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 Generalization Rule, respectively), which are axioms that are minimally true of all normal modal logics, we can prove that we know all the logical consequences of our beliefs.
The fallacy is "epistemic" because it posits an immediate identity between a subject's knowledge of an object with the object itself, failing to recognize that Leibniz's Law is not capable of accounting for intensional contexts.
The name of the fallacy comes from the example: The premises may be true and the conclusion false if Bob is the masked man and the speaker does not know that.
Another example: Expressed in doxastic logic, the above syllogism is: The above reasoning is invalid (not truth-preserving).