Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 1 of 18 Chapter 4 Expanded Categorial Grammar Chapter 4 Expanded Categorial Grammar ............................................................................................. 1 A. Expanded Categorial Syntax.......................................................................................................... 2 1. Introduction .................................................................................................................................... 2 2. What is Categorial Logic?.............................................................................................................. 2 3. Classical Logic ............................................................................................................................... 3 4. Intuitionistic Logic......................................................................................................................... 3 5. No Monotonic Logic Properly Models Grammatical-Composition.............................................. 4 6. Relevance Logic............................................................................................................................. 4 7. Linear Logic Without Identity ....................................................................................................... 5 8. Multi-Linear Logic......................................................................................................................... 6 1. Formal Presentation of System ML ............................................................................................... 6 1. Arrow-Fragment .................................................................................................................... 6 2. Syntax..................................................................................................................................... 6 3. Derivation System.................................................................................................................. 6 4. Definition of Derivation......................................................................................................... 7 5. Assumptions........................................................................................................................... 7 6. Inference-Rules ...................................................................................................................... 7 7. Cross-Arrow-Fragment of System L ..................................................................................... 8 8. Syntax..................................................................................................................................... 8 9. Derivation System [Added Rules] ......................................................................................... 8 10. Examples of Derivations........................................................................................................ 9 B. Expanded Categorial Semantics................................................................................................... 11 1. Semantic-Composition................................................................................................................. 11 2. General Composition Rule........................................................................................................... 11 2. Structural Rules............................................................................................................................ 11 1. Definition of Type-Logical Derivation................................................................................ 11 2. Assumption-Rule ................................................................................................................. 12 3. Lambda-Out ( λO) [function-application]............................................................................ 12 4. Lambda-In ( λI) [function-generation]................................................................................. 12 5. Cross-Out ( ×O) ................................................................................................................... 12 6. Cross-In ( ×I)........................................................................................................................ 12 7. Sub-Structural (Index) Rules ............................................................................................... 12 8. Lambda-Calculus ................................................................................................................. 12 3. Extra Composition Rules ............................................................................................................ 13 4. Examples of Semantic Derivations .............................................................................................. 13 1. Simple Function-Composition (Transitivity) ...................................................................... 13 2. Schönfinkel's Transform...................................................................................................... 14 3. Montague's Transform ......................................................................................................... 15 4. Generalized-Conjunction ..................................................................................................... 15 5. Fail................................................................................................................................................ 16 6. Appendix – Summary of Systems Considered ............................................................................ 18 Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 2 of 18 A. Expanded Categorial Syntax 1. Introduction In the previous chapter we observed that many sentences do not readily submit to the grammatical analysis we developed in the previous two chapters. In response to this, we propose to enlarge the rules of composition. In particular, we propose Categorial Logic , which is intended to be a calculus of grammatical- composition ,1 including syntactic-composition and semantic-composition. 2 We call it a "logic" because, in characterizing it, we utilize logical formalism, and we draw inspiration from existing logical systems – including classical logic, intuitionistic logic, relevance logic, and linear logic. According to standard categorial-grammar, grammatical-composition consists (almost) exclusively in function-application , depicted as follows. φ a function from A into B AB α an item of type A A φ(α) an item of type B B If we think of arrow as the logical if-then connective, then this procedure corresponds to the following argument form, if A then B A B which is the inference-principle known as modus ponens. Whereas standard categorial-grammar admits only one mode of composition, corresponding to modus ponens , expanded categorial-grammar admits infinitely-many modes of composition, each one corresponding to a valid-inference of categorial logic. This principle is officially presented as follows. Categorial Logic Principle A A Where 0 , …, k are types. A A A 1, …, k combine to form 0 precisely if A A ⊢ A 1 , …, k 0 Here, ‘ ⊢’ is a meta-linguistic predicate corresponding to logical entailment. 3 2. What is Categorial Logic? The obvious question then is – what logical system best models grammatical-composition? In this connection, there are several prominent extant logical systems that serve as candidates, including Classical Logic Intuitionistic Logic Relevance Logic Linear Logic which we examine in the next few sections. 1 The idea of a calculus of (pure) syntactic-composition traces to Joachim Lambek's "The Mathematics of Sentence Structure", American Mathematical Monthly , 65 (1958), 154–170. 2 Here, by syntax, we actually mean just the syntax-semantics interface. In particular, our syntactic-rules mimic our semantic-rules in being completely indifferent to the left-right structure of parse-trees. This is in marked contrast to other approaches to categorial grammar that are more narrowly syntactic in orientation. 3 Logicians use various turnstile-symbols, including ‘ ⊢’ and ‘ ⊨’, to depict various manners of judging entailment; usually ‘ ⊢’ pertains to proof-theoretic entailment, ‘ ⊨’ pertains to model-theoretic entailment. Hardegree, Compositional Semantics , Chapter 4: Expanded Categorial Grammar 3 of 18 3. Classical Logic Since it is the strongest logic of the group, 4 Classical Logic authorizes the maximum number of compositions, but unfortunately it also authorizes compositions that are grammatically implausible. For example, the following is a principle of Classical Logic. 5 (c1) A ; B ⊢ A →B Accordingly, using CL to judge grammatical-composition, the following composition-rule is authorized. S ; S ⊢ SS We read this saying that one may combine two sentences ( S) to form a sentential-adverb ( S- operator). Since this is grammatically highly implausible, categorial logic must reject (c1). In an important sense, to be explained shortly, (c1) follows from the following inference principle, which is also valid in Classical Logic. (c2) B ⊢ A →B This is an inference principle found especially obnoxious by many pioneers of alternative logical systems, including the strict-entailment logics of C.I. Lewis 6 and the relevant-entailment logics of Anderson and Belnap. 7 Grammatically-understood, (c2) is equally obnoxious, since the following instance S ⊢ SS authorizes transforming 8 a sentence into a sentential-adverb