Analyzing Extraction. N.B.: Each type has its own set of variables (Church typing).

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1 Analyzing Extraction Carl Pollard Linguistics Mar. 8, 2007 (1) TLC Revisited To motivate our approach to analyzing nonlocal phenomena, we first look at a reformulation TLC of that makes explicit some aspects of hypothesis management that were left implicit before. (2) Review of Implicative TLC Implicative TLC is Positive TLC without T and. The set of Implicative TLC terms is recursively defined as follows: a. (Hypotheses) If x is a variable of type A, then x : A; b. (Nonlogical Axioms) if c is a nonlogical constant of type A, then c : A; c. (Modus Ponens) if f : A B and a : A, then f(a) : B; and d. (Hypothetical Proof) if x is a variable of type A and b : B, then λ x b : A B. N.B.: Each type has its own set of variables (Church typing). (3) Contexts a. A context is a string of variable/type pairs, written to the left of the turnstile at each node of the proof tree. b. The contexts keep track of the undischarged hypotheses. c. Contexts are strings (not just sets or lists) because we track i. the order of hypotheses ii. multiple occurrences of the same hypothesis. d. We make explicit the structural rules that allow contexts to be restructured. e. Instead of typed variables (Church typing), we use a fixed stock of general-purpose variables and let the contexts track what types are assigned to the variables in a given proof (Curry typing). f. No context can assign distinct types to the same variable. g. We use capital Greek letters as metavariables over contexts. 1

2 (4) Implicative TLC Reformulated Using Contexts a. (Hypotheses) x : A x : A; b. (Nonlogical Axioms) c : A (c a nonlogical constant of type A); c. (Modus Ponens) if Γ f : A B and a : A, then Γ, f(a) : B; d. (Hypothetical Proof) if x : A, Γ b : B, then Γ λ x b : A B. e. (Weakening) if Γ, b : B, then Γ, x : A, b : B; f. (Permutation) if Γ, x : A, y : B, c : C, then Γ, y : B, x : A, c : C; and g. (Contraction) if Γ, x : A, x : A, b : B, then Γ, x : A, b : B. (5) Note Especially Hypothetical Proof: If x : A, Γ b : B, then Γ λ x b : A B Another name for this is Implication Introduction This is the usual way to prove a conditional: hypothesize the antecedent, prove the consequent, then withdraw the hypothesis. (6) Structural Rules The last three rules in (4 are called structural rules. Weakening lets you add a redundant hypothesis, which can be subsequently discharged (vacuous lambda astraction), to prove a conditional where the antecedent is irrelevant to the consequent. Permutation means the order of the hypotheses can be ignored. Contraction means duplicate hypotheses can be ignored. Usually hypotheses are thought of as a set not a string, so Permutatation and Contraction are implicit. But the string way generalizes to our situation, where there are hypotheses (for nonlocal dependencies) for multiple flavors of implication, each of which admits a different subset of structural rules. 2

3 (7) Substructural Implicative Logics The propositional logic (PL) axiomatized by Implicative TLC is Implicative Intuitionistic PL (IIPL). We get different substructural PLs by discarding different subsets of structural rules, e.g. Relevant PL: discard Weakening Ticket PL: discard Weakening and Permutation Linear PL: discard Weakening and Contraction Lambek Calculus has left and right flavors of implication each with its own its own Modus Ponens and Hypothetical Proof, and no structural rules. (8) When Hypotheses Come in Flavors Extended to handle nonlocal phenomena, our framework will have multiple flavors of implication each of which manages hypotheses in different ways, so we will partition the contexts to track different flavors of hypotheses. In this course we only have time to look at one flavor of nonlocal implication, the slash flavor used for extraction-type dependencies (Ā-Movement). Other nonlocal implications will be introduced in Syntax III. (9) Review of Our Syntax Rules so Far Complement Fusion: If a : A and b : B then a b : A B Subject Merge: If a : A and f : A subj B then ( subj a f) : B Complement Merge: If f : A comp B and a : A then (f a comp ) : B Specifiee Merge: If f : A spec B and a : A then (f a spec ) : B Subtype Embedding: If a : A Bool and b : A a then embed a (b) : A Coordination (Simplified): If a, c : A and b : Conj, then κ A (a, b, c) : A Nonlogical unary rules: for - s, bare plural and bare mass NPs, predicative NPs, predicative locative PPs, etc. 3

4 (10) Reformulating the Syntax Rules with Contexts For now we are concerned only with slash-flavored hypotheses. None of the unary rules affect contexts, i.e. the context of the conclusion (mother) is the same as for the premiss (daughter). The remaining rules are reformulated as follows: Complement Fusion: If Γ a : A and b : B then Γ, a b : A B Subject Merge: If Γ a : A and f : A subj B then Γ, ( subj a f) : B Complement Merge: If Γ f : A comp B and a : A then Γ, (f a comp ) : B Specifiee Merge: If Γ f : A spec B and a : A then Γ, (f a spec ) : B Coordination: If Γ a : A, b : Conj, and Γ c : A, then Γ κ A (a, b, c) : A (11) Coordination is Special All rules except Coordination pass up the slash hypotheses of the daughters in the order in which they are temporally realized (just like slash-inheritance in G/HPSG). Coordination differs from all other rules in requiring that the contexts of the coordinated daughters coincide with the context of the whole coordinate phrase. In linear logic jargon, coordination is additive relative to slash, and the other rules are multiplicative. (12) Across the Board (ATB) Condition Empirically, the additivity of Coordination corresponds to the Across-the-Board (ATB) Condition on extraction from coordinate phrases: a. BAGELS i, Kim likes t i and Sandy hates t i. b. *BAGELS i, Kim likes muffins and Sandy hates t i. c. *BAGELS i, Kim likes t i and Sandy hates muffins. 4

5 (13) Additional Rules for slash Finite Move: If x : A, Γ s : S fin then Γ λ slash x s : A slash S fin Infinitive Move: If x : Acc, Γ v : B subj S inf then Γ λ slash x v : Acc slash (B subj S inf ) Topicalization: If Γ a : A and b : A slash S fin then Γ, τ(a, b) : S top Contraction: If Γ, x : A, x : A, b : B then Γ, x : A, b : B (14) Analysis of Topicalization An instance of Finite Move allows binding of a trace (undischarged slash-hypothesis) of type NP acc in a finite S (e.g. Kim likes t), to form an Acc slash S fin. Topicalization can combine an NP acc (e.g. muffins) with this to form the S top Muffins, Kim likes. We can t actually complete the analysis until we introduce the Hypotheses schema for slash. That must be done with care to avoid allowing traces any old place. Properly formulated, this rule plays the same role in our framework as the ECP did in GB theory. Once we have that rule, the analysis of bagels, Kim likes is: τ(bagels, λ slash t ( subj Kim (likes t comp ))) Other rules similar to Topicalization form finite consituent questions, finite relative clauses, cleft clauses, and pseudocleft clauses. 5