Effective Validity: A Generalized Logic for Stable Approximate Inference

Abstract

The traditional approach in philosophy of using logic to reconstruct scientific theories and methods operates by presenting or representing a scientific theory or method in a specialized formal language. The logic of such languages is deductive, which makes this approach effective for those aspects of science that use deductive methods or for which deductive inference provides a good idealization. Many theories and methods in science, however, use non-deductive forms of approximation. Approximate inferences, which produce approximately correct conclusions and do so only under restricted conditions before becoming unreliable, behave in a fundamentally different way. In the interest of developing accurate models of the structure of inference methods in scientific practice, the focus of this paper, we need conceptual tools that can faithfully represent the structure and behaviour of inference in scientific practice. To this end I propose a generalization of the traditional notion of logical validity, called effective validity, that captures the form of approximate inferences typically used in applied mathematics and computational science. I provide simple examples of approximate inference in mathematical modeling to show how a logic based on effectively valid inference can directly, faithfully represent a wide variety of the forms of inference used in scientific practice. I conclude by discussing how such a generalized logic of scientific inference can provide a richer understanding of problem-solving and mathematical modeling processes.