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ROA:500
Title:Entailed Ranking Arguments
Authors:Alan Prince
Comment:
Length:117
Abstract:An 'elementary ranking condition' (ERC) embodies the kind of restrictions imposed by a comparison between a desired optimum and a single competitor. All entailments between elementary ranking conditions can be ascertained through three simple formal rules; one of them introduces a method of argument combination -- 'fusion' -- shown to have the same sense as in relevance logic. Fusion is also central to detecting inconsistency in a set of ERCs; inconsistency and entailment are closely related here, much as in ordinary logic. Fusion therefore plays a key role in the definition of Recursive Constraint Demotion (RCD: Tesar & Smolensky 1994, 1998). When ERCs are hierarchized by the ranking of the constraints that crucially evaluate them, their entailment and fusional relations are seen to correlate with aspects of ranking structure. RCD and the Minimal Stratified Hierarchy it produces also figure prominently in an efficient procedure for calculating entailments. Harmonic bounding, both simple and collective, leads to the existence of entailment relations, and removal of entailment dependencies from a set of ERCs eliminates harmonic bounding in its underlying candidate set. The logic of entailment in OT is seen to be the implication-negation fragment of RM (Sobocinski 1952, Parks 1972) and the logic of OT in general is shown by a semantical argument to be precisely RM itself. When the logic is extended from ERCs to constraints, it allows for a direct representation of the notion of a strict domination hierarchy using only the connectives of the logic; various ranking restrictions are shown to follow when logical relations exist between constraints.

Contents
Prefatory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
0. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1. L-retraction and W-extension. . . . . . . . . . . . . . . . .5
2. Fusion. . . . . . . . . . . . . . . . . . . . . . . . . . . .8
3. Fusion and Conjunction. . . . . . . . . . . . . . . . . . . 15
4. Entailment, Ranking, and the Minimal Stratified Hierarchy . 21
5. Finding Entailments . . . . . . . . . . . . . . . . . . . . 31
6. Entailment and Harmonic Bounding. . . . . . . . . . . . . . 35
7. The Logic of Optimality Theory. . . . . . . . . . . . . . . 47
7.1 OT as Logic, and v.v. . . . . . . . . . . . . . . . . 47
7.2 Beyond VS to RM. . . . . . . . . . . . . . . . . . . 54
7.3 Ordered Polyvaluation as RM Semantics. . . . . . . . 59
7.4 RM as the logic of OT. . . . . . . . . . . . . . . . 62
7.5 From RM to PC. . . . . . . . . . . . . . . . . . . . 70
7.6 Systems of Ordered Polyvaluations. . . . . . . . . . 71
7.7 Syntactical Manipulations of Paramount Interest. . . 78
8. Constraint Logic. . . . . . . . . . . . . . . . . . . . . . 81
9. The Arithmetic of Optimality Theory . . . . . . . . . . . .100

Appendix 1. Functional Characterization of Constraints . . . .102
Appendix 2. Direct Implication Checking and RCD. . . . . . . .104
Appendix 3. Entailment & Nonentailment btw Fusions and Fusands106
Appendix 4. A Kripke-Style Semantics for OT. . . . . . . . . .107
Appendix 5. Axioms for S and RM. . . . . . . . . . . . . . . .109

References . . . . . . . . . . . . . . . . . . . . . . . . . .110
Type:Paper/tech report
Area/Keywords:Formal Analysis
Article:Version 1