Hypothetical Reasoning and Definitional Reflection in Logic Programming

This paper describes the logical and philosophical background of an extension of logic programming which uses a general schema for introducing assumptions and thus presents a new view of hypothetical reasoning. The detailed proof theory of this system is given in [7], matters of implementation and control of the corresponding programming language GCLA with detailed examples can be found in [1, 2]. In Section I we consider the local rule-based approach to a notion of atomic consequence as opposed to the global logical approach. Section 2 describes our system and char~terises the inference schema of definitional reflection which is central for our approach. In Section 3 we motivate the computational interpretation of this system. Finally, Section 4 relates our approach to the idea of logical frameworks and the way elimination inferences for logical constants are treated therein, and thus to the notions of logic and structure. It shows that from a certain perspective, logical reasoning is nothing but a special case of reasoning in our system.