Studied Flexibility
Herman Hendriks

Abstract:
In theories of formal grammar it has become customary to assume that 
linguistic expressions belong to syntactic {\em categories}, whereas their 
interpretations inhabit semantic {\em types}. {\em Studied Flexibility} is 
an exploration of the consequences of this twofold assumption. Its starting 
point are the basic ideas of logical syntax and semantics as they are found 
in categorial grammar and lambda calculus, and it focuses on their 
convergence in theories of linguistic syntax and semantics.

In Chapter 1, `Flexible Montague Grammar', it is argued that adoption of 
flexible type assignment in Montuage grammar leads to an elegant account of
natural language scope ambiguities which arise in the pesence of 
quantifying and coordinating expressions. Whereas Montague's original 
fragments resort to the syntactic device of quantifying-in for representing
quantifier scope ambiguities, Cooper's alternative mechanism of 
semantically storing quantifiers avoids the `unintuitive' syntactic aspects 
of Montague's proposal -- at the expense, however, of complicating the 
semantic component. Hence Cooper's conclusion that `wide scope quantification
seems to involve somewhat unpalatable principles either in the syntax or in 
the semantics.' Flexible interpretation is an alternative which avoids the
unintuitive syntactic and semantic features of quantifying-in and storage.
This alternative involves giving up Montague's strategy of uniformly 
assigning {\em all} members of a certain category the most complicated type 
that is needed for {\em some} expression in that category. This strategy of
generalizing to the worst case fails, not because the worst case cannot 
always be generalized to, but simply because there {\em is} no such case. 
Instead, a reverse strategy is proposed which generalizes to the `best case'
on the lexical level. Generalized syntactic/semantic rules permit the 
compounding of all `mutually fitting' translations, type-shifting rules 
produce derived translations out of lexical and compound ones, and the 
recursive nature of these rules reflects the empirical fact that there is no
worst case. The proposal is formalized as a fully explicit fragment of 
flexible Montague grammar, which is shown to allow one to represent scope
ambiguities without special syntactic or semantic devices and, thus, to 
involve a more adequate division of labour between the syntactic and semantic
component.

Chapter 2, `Compositionality and Flexibility', is concerned with determining 
whether the flexible Montague grammar of Chapter 1 observes the principle of 
compositionality. A detailed consideration of the implications of the 
principle of compositionaliy for the organization of grammar fragments in 
general leads to a formalization of the principle which differs from the one 
presented by Janssen. It is argued that this formalization can be motivated 
and applied more easily, and that it avoids some technical complications 
inherent in Janssen's approach. The flexible Montague grammar of Chapter 1 
turns out to be compositional under the `most intuitive' interpretation of 
the principle, provided that the type-shifting derivation of translations is 
explicitly incorporated into the grammar.

Chapter 3, `Lambek Semantics', deals with semantic interpretation in the 
Lambek calculus {\bf L}, of which Lambek established the syntactic 
decidability. It presents and motivates an alternative, equivalent 
formulation of the Van Benthem/Moortgat semantics for {\bf L}. In this 
semantics, the interpretations of a grammatical expression are directly 
determined by the proofs of its validity in the syntactic calculus. The 
alternative formulation is used in a straightforward semantic version of 
Lambek's {\em Cut} elimination theorem which entails that {\bf L} is 
semantically decidable as well: the result of applying Lambek's {\em Cut} 
elimination algorithm is a derivation which is semantically equivalent to 
the original derivation. Moreover, it is shown that the calculus {\bf L} can 
be further normalized to a calculus {\bf L*} that offers a solution to the 
so-called `spurious ambiguity problem' -- the problem that different proofs 
of a given sequent may yield one and the same semantic interpretation. In 
{\bf L*}, each interpretation of a sequent corresponds to exactly one proof. 
This solution is compared with (an explicit elaboration of) proposals by 
Moortgat and Roorda, and applied in an extension of an encoding result of 
Ponse.