Hypothesis finding with proof theoretical appropriateness criteria

For two given formulae B and E with BE, hypothesis finding means to produce a formula H such that . Hypothesis finding, or variants thereof, is central to various types of inference, e.g., abductive inference, inductive inference, machine learning, and machine discovery. Clarifying the nature of hypothesis finding is still in its infancy, a situation similar to the establishment of logical foundations of inference related to induction and discovery. Although trivial solutions to hypothesis finding are easy to give, finding appropriate hypotheses still remains as a great challenge. A central role in this context plays the question, what it means for a hypothesis to be appropriate? In this paper we propose an answer to this question, which is based on proof theoretical criteria. This is in contrast to most previous approaches where appropriateness of hypotheses was based on concepts of semantical weakness in classical logic. More precisely, we use provability in Relevance Logic instead of classical semantical entailment, we demand utmost exploitation of the inferential potential (deductive content) inherent in B→E and we demand H to be a minimal deductive supplement to B→E. Along these lines we developed the concept of a minimized residue hypothesis which also constitutes an interesting trade-off between ‘logical smallness’ and ‘syntactical smallness’.