Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-29T19:36:56.285Z Has data issue: false hasContentIssue false

# 4 - Defeasible Consequence Relations

Published online by Cambridge University Press:  18 July 2009

## Summary

This chapter deals with the definition of a relation of defeasible consequence based on the notion of general extension introduced in Chap. 3, as well as some alternative developments. The basic notion of defeasible consequence is introduced in Section 4.1: The main result is that the relation so defined satisfies Gabbay's three desiderata (Theorem 4.1.6). Section 4.2 presents alternative developments, namely: (1) the important case of seminormal default theories (which turn out to have a unique minimal general extension); (2) defeasible consequence as based on extensions that are nonminimal (“optimal” in the sense of Manna and Shamir); and (3) a variant of general extensions that, at the cost of a slight complication, avoids certain somewhat counterintuitive results. In Section 4.3 we draw conclusions and comparisons to other approaches, and in Section 4.4 we sketch how to give a “transfinite” version of the present approach. Proofs of selected theorems can be found in Section 4.5.

DEFEASIBLE CONSEQUENCE

As we set out to define a relation of defeasible consequence, it will be convenient to introduce an abbreviated notation for extensions: We will use boldface uppercase Greek letters to stand for triples of sets of defaults, as in Γ = (Γ+, Γ, Γ*). Similarly, given sequences of sets of defaults and we write Γn for If Γn is a sequence of sets of defaults (n ≤ 0), we write limGn for the unique Γ = (Γ+, Γ, Γ*) such that Finally, to simplify notation still further, we write just W + ϕ in place of W ∪ {ϕ}.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2005

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

# Save book to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

# Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

# Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×