A Constructive Mathematic approach for Natural Language formal grammars

A mathematical description of natural language grammars has been proposed first by Leibniz. After the definition given by Frege of unsaturated expression and the foundation of a logical grammar by Husserl, the application of logic to treat natural language grammars in a computational way raised the interest of linguists, for example applying Lambek's categorial calculus. In recent years, the most consolidated formal grammars (e.g., Minimalism, HPSG, TAG, CCG, Dependency Grammars) began to show an interest in giving a strong psychological interpretation to the formalism and hence to natural language data on which they are applied. Nevertheless, no one seems to have paid much attention to cognitive linguistics, a branch of linguistics that actively uses concepts and results from cognitive sciences. Apparently unrelated, the study of computational concepts and formalisms has developed in pair with constructive formal systems, especially in the branch of logic called proof theory, see, e.g., the Curry-Howard isomorphism and the typed functional languages. In this paper, we want to bridge these worlds and thus present our natural language formalism, called Adpositional Grammars (AdGrams), that is founded over both cognitive linguistics and constructive mathematics.