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