Grammar as Logic

The notion of parsing as deduction appears to presuppose that of grammar as logic though in fact the former has largely involved embedding grammars in logics using a fragment conducive to theorem proving It is well known that categorial grammars take the form of implicational logics in this case the grammar is not embedded but simply is is a logic The paper argues that many further operations may be added to such a grammar the result hav ing close relations with linear logic Thus categorial grammar emerges as the implicational fragment of a much more general logical grammar A Prolog implementation illustrates ap plications to polymorphism optionality intensionality bounded and unbounded extraction and coordination reduction Implication The notion of parsing as deduction appears to presuppose that of grammar as logic though in fact the former has largely involved embedding grammars in powerful logics using a fragment conducive to theorem proving e g embedding context free grammars in the Horn clause fragment of rst order logic cf Pereira and Warren and references therein In categorial grammars the slash connectives behave directly as implication The terms of the language of categories are de ned starting from a set of basic categories as follows a If X is a basic category then X is a category b If X and Y are categories then X Y and XnY are categories A term X Y represents expressions which apply to expressions of category Y on their right to form expressions of category X a term XnY represents expressions which apply to expressions of category Y on their left to form expressions of category X Thus the following will be valid a X Y Y X b Y XnY X Note that a uniform orientation notation is used with arguments appearing on the same side of their value irrespective of directionality The inference from X Y and Y to X is analogous to the inference from Y X and Y to X Consider in particular the product free Associative Lambek Calculus ALC which has a Gentzen style sequent axiomatisation as follows Lambek X X X Z Y L X Y Z X Z Y nL XnY Z