Towards a minimal logic for minimalist grammars: a transformational use of Lambek calculus

Many convergence points have been observed during the recent years between the Minimalist Program and the program of Categorial Grammar, above all since the formalization of minimalist ideas by E. Stabler. For instance, the Merge-operation is exactly like functional application. Moreover the fundamental operation of feature-checking, which is at the basis of the Move-operation, can really be depicted as a resource consumption procedure, something familiar to so called resource conscious logics. This makes rise a deep interest in looking for a logical formulation of minimalist grammars. Such an enterprise is not done for the sake of spurious formalization. If we take the chomskyan framework seriously, it seems natural to assume that UG consists in a very general set of principles that must be expressed in the most condensed way, and that derivations are made of steps of a few different sorts exactly like it is the case in a logic. 8.0.1 The convergence of the minimalist program and categorial grammar Both generative grammar and categorial grammar postulate the existence of a universal grammar. In generative grammar, this universal grammar is supposed to be provided by a set of principles, now reduced to a small number, among which are: the structure dependence principle, the Merge and Move operations, the Binding principles, the Head Movement Constraint... Particular languages are obtained by assigning values to so called parameters. Because of the evolution of the theory, and notably the fact that moves are assumed to be always triggered by the operation of feature checking, these parameters are simply boolean values assigned to the strength of features. On the categorial grammar side, e.g. in Moortgat setting [8, 9], it is assumed that UG is provided by a base logic, which puts together several connective families •i, /i, \i, connected by communication postulates, and that only this set of postulates may change according to the language to be learnt: the ground logic remains invariant, and is thus supposed to capture all the universal principles. From our viewpoint, Chomsky’s conception is very appealing because of the simple nature of the parameters which are postulated. The features by means of which communications are established between the sentence constituants may be 83 Proceedings of Formal Grammar 1999 Geert-Jan M. Kruijff & Richard T. Oehrle (editors). Chapter 8, Copyright c ©1999, A. Lecomte & C. Retoré. Towards a minimal logic for minimalist grammars: A. Lecomte & C. Retoré /84 seen as nodes in a net, and their relative strengths as connection weights, thus evoking some connectionnist aspects, but Universal Grammar remains unclear. On the contrary, the conception of UG in categorial grammar is very clear and appealing: it is in fact natural to assume that a so general and abstract device may be seen as a kind of logic, simply because logics study abstract symbolic systems, but language variation seen as a changing set of postulates is not entirely satisfactory: it seems hard to assume that a language is learnt by learning abstract postulates of this kind. This motivates the attempts to conciliate the two approaches. What we aim to find is a very limited set of rules that would be sufficient in order to give an account of Merge and Move, and that would be sufficiently restricted in order to incorporate at least some of the principles (like economy constraints), in such a way that there would be no need for their independent formulation. We assume in this paper that each item of the lexicon consists in a set of features, which are divided as follows: • Phonetic features for example /speaks/,/linguist/,/some/, . . . • Semantic features for example (speaks),(linguist),(some), . . . • Syntactic or formal features: – categorial features (categories) involved in merge: BASE = {c, t, v, d, n, . . . } – functional features involved in move: FUN = {k, K, wh, . . . } 8.0.2 Stabler’s minimalist grammars In his paper Derivational Minimalism [10] and in the subsequent Remnant movement and structural complexity [11], Stabler provides formal grammars which realize the view of grammar expressed in the Minimalist Program. Lexical entries are ordered sequences in: LABEL = SELECT∗ (LICENSORS) SELECT∗ BASE LICENSEES P ∗ I∗ • P phonetic features • I semantic features • SELECT = {=b,=B, B=|b ∈ BASE} select a category • LICENSEES = {−x|x ∈ FUN} needs a move feature • LICENSORS = {+x,+X|x ∈ FUN} provides a move feature The structures he is using are binary trees, with internal nodes labelled < or > which indicates where the head of the (sub)tree is to be found. These trees can be interpreted as ’T-markers’, that term being adapted from Chomsky (1955/75) — 85\ Formal Grammar 1999 structures which record the history of a derivation, including the transformational phase (see also Cornell [4]). In Stabler grammars, there are two kinds of merge, and one kind of movement, and they are completely determined by the sequence of features at the head of the T-marker. Merge is defined between two T-markers u and t the head of u starting with =x and the head t starting with x with x ∈ BASE; let u′ (resp. t′) denote u (resp. t) in which the =x (resp. x) feature starting the head is cancelled. • if u is a lexical item then the resulting tree is u′ < t′ (so u′ is the head and is on the left) • otherwise the resulting tree is t′ > u′ (so u′ is the head in this case as well, but it is on the right) Roughly speaking, movement is defined as follows: assume that at the leftmost position (spec∗ position) we have a +x and that at the rightmost (comp+ position) we have a −x: then the movement takes the whole constituent having −x as a head and moves it to the leftmost position (spec∗ position). 8.0.3 Why do we want a logical formalization? Before stating the advantages that would bring a logical formalization, let us observe its naturality. There is a striking similarity between categorial grammar and minimalist grammars: • both merge and move, in the minimalist syntax, are governed by resource consumption, which suggest a formulation within resource sensitive logics • there is a clear parallel between function and head (or between argument and non-head). The first advantage of a logical formalization is a simplification which can be stated as a radical lexicalization: both merge and move make reference to the tree structure and in particular to the head of the T-marker while a logical formulation of rules should only depend on the roots of the trees: all structure under the root can be erased. Logic is also attractive because trees that we obtain in logic, which represent proofs in a Natural Deduction system, can be easily translated into logical forms, an objective that is pursued as well in generative grammar as in categorial grammar. The Curry-Howard homomorphism has often be used in this perspective. Indeed, in our opinion, “pure” resource logic alone cannot describe language: language-specific notions and the logic-based computing device have to be integrated as “harmoniously” as possible. Towards a minimal logic for minimalist grammars: A. Lecomte & C. Retoré /86 8.0.4 Minimalist grammars within the Lambek calculus We must here warn the reader that although we use Lambek calculus [7], we do not use Lambek grammars (Lambek grammars are defined by m1 . . . mn ∈ L iff for each i there exists a type Ti of mi such that T1 . . . Tn S). In our presentation, we will derive the start symbol of the grammar (here c) from the types of the words, but we depart from the ordinary use of Lambek grammars in viewing types as closed deductions rather than hypotheses1. There are therefore no hypotheses, nor any order on hypotheses which depict word order. Actually, word order is computed by means of labels propagation plus a small device (like an automaton) which can erase phonological content after copying. Let us start, for an example, from the following translation of Stabler types into Lambek formulas: entry label type every =n d −k every ((d ⊗ k)/n) some =n d −k some ((d ⊗ k)/n) language n language n linguist n linguist n speaks =d +k =d v speaks ((k\(d\v))/d) (tense) =v +k t ((k\t)/v) (comp) =t c (t\c) where strings (like language, speaks and so on) represent labels for deductions of the corresponding Lambek formulae. For instance, to say that ((d ⊗ k)/n) is associated with some is to say that we assume the closed deduction: