## Reports on Mathematical Logic

### No. 28 (1994)

Haydee Werneck POUBEL and Luiz Carlos P. D. PEREIRA

A CATEGORICAL APPROACH TO HIGHER-LEVEL INTRODUCTION AND ELIMINATION RULES

A b s t r a c t. A natural extension of Natural Deduction was defined by Schroder-Heister where not only formulas but also rules could be used as hypotheses and hence discharged. It was shown that this extension allows the definition of higher-level introduction and elimination schemes and that the set $\{ \vee, \wedge, \rightarrow, \bot \}$ of intuitionist sentential operators forms a {\it complete} set of operators modulo the higher level introduction and elimination schemes, i.e., that any operator whose introduction and elimination rules are instances of the higher-level schemes can be defined in terms of $\{ \vee, \wedge, \rightarrow, \bot \}$.
The aim of the present work is to extend the well-known connections between Proof Theory and Category Theory to higher-level Natural Deduction. To be precise, we will show how an adjointness between cartesian closed categories with finite coproducts can be associated, in a systematic way, with any operator $\phi$ defined by the higher-level schemes. The objects in the categories will be rules instead of formulas.