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2016 | OriginalPaper | Chapter

13. Equational CTD

Authors : Dov M. Gabbay, Karl Schlechta

Published in: A New Perspective on Nonmonotonic Logics

Publisher: Springer International Publishing

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Abstract

The strategy described here is an illustration of the fact that one may describe graded information by either a relation on the information or the semantics itself, as in preferential structures, or by a relation on the truth values. Here the authors take the latter approach to solve problems with contrary-to-duty obligations.

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Footnotes
1
B for Brouwer, because we are going to use Brouwer’s fixed point theorem to show that theories always have models.
 
2
Note that the facts are sets of actual nodes. We can take the conjunction of the actual nodes as a formula faithfully representing the set of facts. Later on in this chapter we will look at an arbitrary formula \(\phi \) as generating the set of facts \(\{y| y\) is either \(x_i\) or \(\lnot x_i\), unique for each i, such that \(\phi |- y\}\).
According to this definition, \(\phi = x_1 \vee x_2\), generates no facts. We will, however, find it convenient later in the chapter, (in connection with solving the Miner’s Paradox, Remark 13.4.2 below) to regard a disjunction as generating several possible sets of facts, one for each disjunct. See also Remark 13.4.1 below.
 
3
Remember when we substitute a fact we split the graph into two and so the equations change. We are not just substituting values into equations (in which case the order simultaneous or not does not matter), we are also changing the equations.
Recall what you do in Physics: If we have, for example, the equation \(y=\sin x\) and we substitute for x a very small positive value, then we change the equation to \(y=x\).
 
Metadata
Title
Equational CTD
Authors
Dov M. Gabbay
Karl Schlechta
Copyright Year
2016
Book
A New Perspective on Nonmonotonic Logics

Print ISBN: 978-3-319-46815-0

Electronic ISBN: 978-3-319-46817-4

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-46817-4_13

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