A Logical Theory of Nonmonotonic Inference and Belief Change

by Bochman, Alexander

• ISBN13: 9783540417668
• ISBN10: 3540417664
• Format: Hardcover
• Copyright: 2001-08-01
• Publisher: Springer-Verlag New York Inc

Summary

This monograph provides logical foundations and a uniform description for nonmonotonic reasoning and belief change. The approach to both these subjects is based on a powerful notion of an epistemic state that subsumes both existing models for nonmonotonic inference and current models for belief change. Many results and constructions in the book are completely new and have not appeared earlier in the literature. The book is primarily intended for experts in Artificial Intelligence and Knowledge Representation who are interested in tools for describing commonsense reasoning tasks as well as in representation capabilities of such tools. It is also of interest to general logicians.

Table of Contents

Introduction	1	(12)
Part I. The Framework
Consequence Relations	13	(30)
Scott consequence relations	13	(9)
Basic definitions	13	(3)
Representation theorem	16	(2)
Compact sets of theories	18	(1)
Right compactness	19	(1)
Determinate and linear consequence relations	20	(2)
Tarski consequence relations	22	(3)
Supraclassicality	25	(6)
Supraclassical Tarski consequence relations	25	(1)
Classical consequence relations	26	(1)
Semi-classical consequence relations	27	(2)
Supraclassical Scott relations	29	(2)
Grounded Scott consequence relations	31	(4)
Base-generated consequence relations	35	(8)
Union-closure and strong base-generation	38	(2)
Base-generated Tarski consequence relations	40	(3)
Epistemic States	43	(34)
General epistemic states	43	(7)
Main kinds of epistemic states	50	(9)
Standard epistemic states	50	(1)
Monotonicity and its kinds	51	(2)
Persistent epistemic states	53	(1)
Pure epistemic states	53	(3)
Homogeneous and worlds-based states	56	(2)
Kinds of preference order	58	(1)
Base-generated epistemic states	59	(3)
General and normal base-generation	60	(2)
Prioritized base-generation	62	(6)
Combining preference relations	63	(1)
Default priorities	64	(4)
Flocks	68	(2)
Nonmonotonic inference and its kinds	70	(7)
Poole's abductive framework	70	(1)
Skeptical inference	71	(2)
Contraction inference	73	(1)
Credulous inference	74	(1)
A digression: careful and adventurous inference	75	(2)
Similarity and Decomposition	77	(34)
Similarity	77	(8)
Similarity of pure bases	79	(1)
Union-closure and base-generation	80	(1)
Similarity of prioritized bases	81	(2)
Persistence and flock-generation	83	(2)
Decomposition of epistemic states	85	(4)
Normal similarity	89	(4)
Language-restricted epistemic states	93	(3)
Equivalence	96	(9)
Static equivalence	100	(1)
Skeptical equivalence	101	(2)
Belief equivalence	103	(2)
Decomposition of inference relations	105	(1)
Syntactic characterization of inference relations	106	(5)
Epistemic Entrenchment	111	(26)
Expectation relations	111	(2)
Entrenchment, plausibility and dependence	113	(6)
Entrenchment relations	113	(2)
Plausibility relations	115	(1)
Dependence relations	115	(1)
Correspondences	116	(2)
Comparative probability	118	(1)
Regular expectation relations	119	(3)
Dependence orders, consequence relations and partial epistemic entrenchment	122	(2)
Qualitative expectation relations	124	(4)
Modular and linear expectation relations	128	(3)
Belief sets and underlying logics of expectation relations	131	(6)
The internal logic of dependence orders	131	(1)
Belief sets of dependence orders	132	(1)
Belief set and logic of qualitative entrenchment	133	(4)
Part II. Nonmonotonic Inference
The Basic Inference Relation	137	(26)
Introduction	137	(2)
Postulates	139	(3)
Basic inference via expectations	142	(4)
Regular inference relations	146	(5)
Regularity and strict expectation relations	147	(2)
Duality	149	(2)
The internal logic of an inference relation	151	(5)
Safe inference	156	(3)
Extending the basic relation	159	(4)
Skeptical Inference Relations	163	(50)
Preferential inference relations	163	(2)
Preferential inference via expectations	165	(1)
Preferential inference as skeptical inference	166	(3)
Consequence-based inference relations	169	(4)
Semi-classical translation	170	(1)
Basic propositions in nonmonotonic inference	171	(2)
Syntactic characterizations	173	(8)
Consequence relations generating a set of conditionals	173	(3)
Classical reduction	176	(1)
Belief and knowledge core of conditional bases	177	(4)
Inferential equivalence	181	(2)
Model transformations and representative classes of epistemic states	183	(6)
Epistemic states and possible worlds models	183	(2)
Purification of general epistemic states	185	(4)
Base-generated inference relations	189	(8)
Injectivity and pure base-generation	197	(9)
Syntactic characterizations	203	(3)
Weakly rational inference	206	(1)
Linearity and rational inference	207	(6)
Defeasible Entailment	213	(38)
Nonmonotonic meta-reasoning	213	(2)
Desiderata for defeasible entailment	215	(7)
Conditional entailment	222	(7)
Local priorities	229	(5)
Local dominance	233	(1)
Preferential inheritance	234	(17)
Inheritance argumentation frameworks	235	(4)
Expanding extensions	239	(7)
The representation	246	(5)
Credulous Nonmonotonic Inference	251	(20)
Credulous inference relations	252	(2)
Credulous inference and expectation relations	253	(1)
Semantic representations	254	(6)
Possible worlds semantics	258	(2)
Permissive inference	260	(4)
Safe credulous inference	264	(1)
Some other kinds of brave inference	265	(2)
Interrelations	267	(4)
Credulous conditionals as default assumptions	268	(3)
Contraction Inference	271	(50)
Contraction inference relations	271	(13)
Postulates of contraction inference	272	(3)
The internal logic of contractions and the success postulate	275	(2)
Contraction and expectation relations	277	(2)
Contraction and preferential inference	279	(1)
Inference-equivalent contractions	280	(2)
Simple, conservative and cautious contractions	282	(2)
Completeness	284	(9)
Representation of conservative, cautious and general contractions	288	(3)
A refined representation	291	(1)
Possible worlds semantics	292	(1)
Contraction inference in standard and pure epistemic states	293	(3)
Determination and vacuity	296	(2)
Rational contractions	298	(12)
Linear contractions	305	(2)
Severe contractions	307	(3)
Withdrawals	310	(2)
Rational, linear and severe withdrawals	312	(1)
Recovering contraction relations	312	(9)
Rational recovering contractions	316	(5)
Part III. Belief Change
Belief Change and Its Problems	321	(24)
The problem of belief change	321	(6)
Epistemic states as a framework of belief change	327	(9)
AGM states and base-generated states	327	(2)
Determination, union-closure, monotonicity, and standardness	329	(2)
Coherence versus foundations	331	(4)
Changing epistemic states	335	(1)
Sums and contractions	336	(5)
Sums of epistemic states	337	(1)
Contraction	338	(3)
Merge and expansion	341	(1)
Derived changes	342	(3)
Contractions of Epistemic States	345	(34)
Definition and properties	345	(5)
Contractions of bases and flocks	347	(2)
Abstract contraction systems	349	(1)
Contractions of pure epistemic states	350	(8)
Contractions of consequence relations	351	(3)
Contraction dynamics	354	(3)
Conditional contractions	357	(1)
Belief contraction functions	358	(6)
Postulates	359	(1)
Failure and success	360	(4)
Kinds of belief contractions	364	(12)
Determinate contractions	365	(1)
Withdrawals	366	(1)
Recovering contractions	367	(1)
Pure belief contractions	368	(4)
Rational contractions	372	(4)
Refined representations	376	(3)
Merge and Expansion	379	(24)
Merging epistemic states	380	(8)
Pure merge	384	(2)
Prioritized merge	386	(2)
Basic expansions	388	(10)
Belief expansions	390	(2)
Pure expansion	392	(4)
Prioritized expansion	396	(2)
Knowledge expansion	398	(1)
Flat expansions	399	(4)
Compound and Derived Changes	403	(22)
Consolidation and revision	403	(2)
Belief revision and nonmonotonic inference	405	(4)
Completeness results	407	(2)
Inference and entrenchment change	409	(8)
Inference change generated by expansions	409	(3)
Prioritized inference change	412	(2)
Flat inference change	414	(2)
Knowledge-based inference change	416	(1)
Constructibility and natural classes	417	(8)
Natural kinds of epistemic states	419	(6)
References	425	(6)
Index	431