Defaults in Update Semantics

Defaults in Update Semantics - FRANK VELTMAN DEFAULTS IN...

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FRANK VELTMAN DEFAULTS IN UPDATE SEMANTICS ABSTRACT. The aim of this paper is twofold: (i) to introduce the framework of update semantics and to explain what kind of phenomena may successfully be analysed in it; (ii) to give a detailed analysis of one such phenomenon: default reasoning. KEY WORDS: dynamic semantics, defaults, epistemic modalities. 1. INTRODUCTION : THE FRAMEWORK OF UPDATE SEMANTICS The standard definition of logical validity runs as follows: An argument is valid if its premises cannot all be true without its conclusion being true as well. Most logical theories developed so far have taken this definition of validity as their starting point. Consequently, the heart of these theories consists in a specification of truth conditions. The heart of the theories developed in this paper does not consist in a specification of truth condi- tions. The slogan ‘You know the meaning of a sentence if you know the conditions under which it is true’ is replaced by this one: ‘You know the meaning of a sentence if you know the change it brings about in the infor- mation state of anyone who accepts the news conveyed by it’. 1 Thus, meaning becomes a dynamic notion: the meaning of a sentence is an operation on information states. To define an update semantics for a language L, one has to specify a set Σ of relevant information states, and a function [ ] that assigns to each sentence φ an operation [ φ ] on Σ . The resulting triple L , Σ , [ ] is called an update system. If σ is a state and φ a sentence, we write σ ]’ to denote the result of updating σ with φ . Since [ φ ] is the function and σ the argument, it would have been more in line with common practice to write ‘[ φ ]( σ )’, but postfix notation is more convenient for dealing with texts. Now we can write ‘ σ 1 ]...[ ψ n ]’ for the result of updating σ with the se- quence of sentences ψ 1 ,..., ψ n . An important notion is the notion of acceptance . Let σ be any state and φ be any sentence. Consider the state σ ]. This state will in most cases be different from σ , but every now and then it may happen that σ ] = σ .
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2 FRANK VELTMAN If so, the information conveyed by φ is already subsumed by σ . In such a case we write σ ||- φ and we say that φ is accepted in σ . 1.1 Constraints that Do Not Aways Hold The phrase ‘update semantics’ might be misleading in that it suggests that all you have to do in order to update your information state with φ is to add the informational content of φ to the information you already have. DEFINITION 1.1. An update system L, Σ , [ ] is additive iff there exists a state 0 , the minimal state, in Σ and a binary operation + on Σ such that (i) the operation + has all the properties of a join operation: 0 + σ = σ ; σ + σ = σ ; σ + τ = τ + σ ; ( ρ + σ ) + τ = ρ + ( σ + τ ).
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