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[edit]
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
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[edit] 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 φ displaystyle mathit K _ a varphi can be read as "Agent a displaystyle a knows that φ displaystyle varphi ." Thus, epistemic logic can be an example of multimodal logic applied for knowledge representation.[1] 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 ¬ φ displaystyle neg K_ a neg varphi , which can be read as " a displaystyle a does not know that not φ displaystyle varphi " or "It is consistent with a displaystyle a 's knowledge that φ displaystyle varphi is possible". The statement " a displaystyle a does not know whether or not φ displaystyle varphi " can be expressed as ¬ K a φ ∧ ¬ K a ¬ φ 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 φ displaystyle varphi is a formula of our language, then so are E G φ displaystyle mathit E _ G varphi , C G φ displaystyle mathit C _ G varphi , and D G φ 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[edit] 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 Φ displaystyle Phi is a (n+2)-tuple ( S , π , K 1 , . . . , K n ) displaystyle (S,pi , mathcal K _ 1 ,..., mathcal K _ n ) , where S is a nonempty set of states or possible worlds, π displaystyle pi is an interpretation, which associates with each state in S a truth assignment to the primitive propositions in Φ 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 π ( 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 φ displaystyle varphi is true at a certain world, one writes ( M , s ) ⊨ φ displaystyle (M,s)models varphi , normally read as " φ displaystyle varphi is true at (M,s)," or "(M,s) satisfies φ 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[edit] 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[edit] This axiom is traditionally known as K. In epistemic terms, it states that if an agent knows φ displaystyle varphi and knows that φ ⟹ ψ displaystyle varphi implies psi , then the agent must also know ψ displaystyle ,psi . So, ( K i φ ∧ K i ( φ ⟹ ψ ) ) ⟹ K i ψ displaystyle (K_ i varphi land K_ i (varphi implies psi ))implies K_ i psi The knowledge generalization rule[edit] Another property we can derive is that if ϕ displaystyle phi is valid, then K i ϕ displaystyle K_ i phi . This does not mean that if ϕ displaystyle phi is true, then agent i knows ϕ displaystyle phi . What it means is that if ϕ displaystyle phi is true in every world that an agent considers to be a possible world, then the agent must know ϕ displaystyle phi at every possible world. This principle is traditionally called N. if M ⊨ φ then M ⊨ K i φ . displaystyle text if Mmodels varphi text then Mmodels K_ i varphi ., The knowledge or truth axiom[edit] 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 φ ⟹ φ displaystyle K_ i varphi implies varphi The positive introspection axiom[edit]
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
K i φ ⟹ K i K i φ displaystyle K_ i varphi implies K_ i K_ i varphi The negative introspection axiom[edit] The Negative Introspection Axiom says that agents know that they do not know what they do not know. ¬ K i φ ⟹ K i ¬ K i φ displaystyle neg K_ i varphi implies K_ i neg K_ i varphi Axiom systems[edit]
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
¬ 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[edit] 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
Q displaystyle Q is a logical consequence of P displaystyle P , then we can derive K a ( P → Q ) displaystyle mathcal K _ a (Prightarrow Q) with N and the conditional proof and then K a P → K a Q displaystyle mathcal K _ a Prightarrow mathcal K _ a Q with K. When we translate this into epistemic terms, this says that if Q displaystyle Q is a logical consequence of P displaystyle P , then a knows that it is, and if a knows P displaystyle P , a knows Q displaystyle Q . That is to say, a knows all the logical consequences of every proposition. This is necessarily true of all classical modal logics. But then, for example, if a knows that prime numbers are divisible only by themselves and the number one, then a knows that 8683317618811886495518194401279999999 is prime (since this number is only divisible by itself and the number one). That is to say, under the modal interpretation of knowledge, when a knows the definition of a prime number, a knows that this number is prime. It should be clear at this point that a is not human. This shows that epistemic modal logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge (if anything).[3] See also[edit] Logic portal Common knowledge Epistemic closure Epistemology Logic in computer science Modal logic Philosophical Explanations Two-dimensionalism Notes[edit] ^ 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
References[edit] Anderson, A. and N. D. Belnap. Entailment: The Logic of Relevance and
Necessity. Princeton: Princeton University Press, 1975. ASIN
B001NNPJL8.
Brown, Benjamin, Thoughts and Ways of Thinking: Source Theory and Its
Applications. London: Ubiquity Press, 2017. [1].
van Ditmarsch Hans, Halpern Joseph Y., van der Hoek Wiebe and Kooi
Barteld (eds.), Handbook of Epistemic Logic, London: College
Publications, 2015.
Fagin, Ronald; Halpern, Joseph; Moses, Yoram; Vardi, Moshe (2003).
Reasoning about Knowledge. Cambridge: MIT Press.
ISBN 978-0-262-56200-3. . A classic reference.
Ronald Fagin, Joseph Halpern, Moshe Vardi. "A nonstandard approach to
the logical omniscience problem." Artificial Intelligence, Volume 79,
Number 2, 1995, p. 203-40.
Hendricks, V.F. Mainstream and Formal Epistemology. New York:
Cambridge University Press, 2007.
Hintikka, Jaakko (1962).
External links[edit] "Dynamic Epistemic Logic". Internet Encyclopedia of Philosophy.
Hendricks, Vincent; Symons, John. "Epistemic Logic". In Zalta, Edward
N. Stanford Encyclopedia of Philosophy.
Garson, James. "Modal logic". In Zalta, Edward N. Stanford
Encyclopedia of Philosophy.
Vanderschraaf, Peter. "Common Knowledge". In Zalta, Edward N. Stanford
Encyclopedia of Philosophy.
"Epistemic modal logic"—Ho Ngoc Duc. v t e Non-classical logic Modal Alethic Axiologic Deontic Doxastic Epistemic Temporal Intuitionistic Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory Fuzzy Degree of truth
Fuzzy rule
Fuzzy set
Fuzzy finite element
Substructural Structural rule Relevance logic Linear logic Paraconsistent Dialetheism Description Ontology
Many-valued Three-valued Four-v |

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