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Modern Origins of Modal Logic

The Negative Introspection Axiom says that agents know that they do not know what they do not know. This axiom is valid on any Euclidean frame. 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.

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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 :. 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.

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. 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.

Modal Logic |

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. 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 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.

Interpretations of Modal Logic

This shows that epistemic modal logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge if anything. From Wikipedia, the free encyclopedia. Philosophy portal. Matematikai logika in Hungarian. 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.

Lewis who created the first symbolic and systematic approach to the topic, in It continued to mature as a field, reaching its modern form in with the work of Kripke. Many papers were written in the fifties that spoke of a logic of knowledge in passing, but it was Finnish philosopher von Wright 's paper An Essay in Modal Logic from that is seen as a founding document.

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It was not until 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. 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. 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.

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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:. 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.

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. In order to accommodate notions of common knowledge and distributed knowledge , three other modal operators can be added to the language.

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  3. Modal Logic |

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. The truth assignment tells us whether or not a proposition p is true or false in a certain state. 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. In idealized accounts of knowledge e. 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 listed here are often known as the "S5 Properties," for reasons described in the Axiom Systems section below. When Amanda looked Junior's download an essay in classical modal logic and room numbering the final head - she was whose effect it tipped.

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Deontic vs Alethic Modal Logic

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