Stephen Spielman

In spite of the fact that the Neyman-Pearson theory of testing is the official theory of statistical testing, most research publications in the social sciences use a pattern of inductive reasoning that is characteristic of Fisherian tests of significance. The exact structure and rationale of this pattern of reasoning is widely misunderstood. The goal of the paper is to describe precisely the pattern and its rationale, and to show that while it is far more cogent than Fisher's critics have realized, it does not logically sustain the inferences it sanctions

This paper outlines a formal recursive wager resolution calculus (WRC) that provides a novel conceptual framework for sentential logic via bridge rules that link wager resolution with truth values. When paired with a traditional truth-centric criterion of logical soundness WRC generates a sentential logic that is broadly truth-conditional but not truth-functional, supports the rules of proof employed in standard mathematics, and is immune to the most vexing features of their traditional implementation. WRC also supports a novel probabilistic criterion of logical soundness, the fair betting probability criterion (FBP). It guarantees that the conclusion of an FBP-valid argument is at least as credible as a conjunction of premises, and also that the conclusion is true if the premises are. In addition, WRC provides a platform for a novel non-probabilistic, computationally simpler criterion of logical soundness – the criterion of Super-validity - that issues the same logical appraisals as FBP, and hence the same guarantees.

In a recent note Roger Carlson presented a rather negative appraisal of my treatment of the logic of Fisherian significance testing in [10]. The main issue between us involves Carlson's thesis that, within the limits set by Fisher, standard significance tests are valuable tools of data analysis as they stand, i.e., without modification of the structure of the reasoning they employ. Call this the adequacy thesis. In my paper I argued that the pattern of reasoning employed by tests of significance needs to be justified in spite of its unquestioned acceptance by most researchers, The best justification offered to date—by R. A. Fisher—is seriously defective. Therefore the adequacy thesis is not justified. I proposed some alterations of the pattern of reasoning employed by tests of significance and argued that they guarantee that tests are adequate epistemological tools.

In a recent paper [2] David Lewis offered an extremely interesting and, if correct, important solution to the main unsolved problem of Carnap's program for inductive logic—the choice of an appropriate C-function. The gist of Lewis' solution is to first obtain a pilot sample from the target population and then select, on the basis of this sample, from among the immodestλ-methods. An immodest inductive method is one which estimates that the mean squared error of its estimates of population relative frequencies is less than or equal that of any other inductive method. By great ingenuity Lewis apparently shows that on any evidence e there is one and only one immodest λ-method. I will attempt to show that Lewis made a conceptual error in formulating the equations to be solved and that when it is corrected it can be easily demonstrated that all admissible λ-methods are immodest and hence that Lewis' criterion is of no value.

The theory of personal probability needs to be developed as a logic of credibility in order to provide an adequate basis for the theories of scientific inference and rational decision making. But standard systems of personal probability impose formal structures on probability relationships which are too restrictive to qualify them as logics of credibility. Moreover, some rules for conditional probability have no justification as principles of credibility. A formal system of qualitative probability which is free of these defects and which leads to a presuppositionless representation theorem is sketched.