Reasoning with Polarity in Categorial Type Logic

ix I Categorial Type Logics 1 1 The Logical Approach in Linguistics 3 1.1 Rule-Based Categorial Grammars . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Classical Categorial Grammar . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Combinatory Categorial Grammar . . . . . . . . . . . . . . . . . 6 1.2 A Logic of Types: Lambek 1958 . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Parsing as Deduction . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Logical Rules and Structural Rules . . . . . . . . . . . . . . . . . 11 1.2.3 Structural Constraints . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 The Composition of Meaning . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Semantic Types and Typed Lambda Terms . . . . . . . . . . . . . 15 1.3.2 Interpretations for the Sample Grammar . . . . . . . . . . . . . . 18 1.4 Putting Things Together . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.1 Rule-Based Approach vs. Deductive Approach . . . . . . . . . . . 21 1.4.2 Curry-Howard Correspondence . . . . . . . . . . . . . . . . . . . 22 1.5 Key Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 The Mathematical Structure of CTL 27 2.1 Capturing Residuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.1 The Logic of Residuation NL . . . . . . . . . . . . . . . . . . . . . 28 2.1.2 The Residuated Unary Operators NL(3) . . . . . . . . . . . . . . 30 2.1.3 Kripke Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Displaying Residuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 Binary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.2 Unary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 35