Coherence and the axioms of confirmation

TLDR

Carnap noted that probability is an equivocal term used in two senses—degree of rational belief given evidence and relative frequency—and this paper focuses on the former, distinguishing quantitative and comparative concepts of confirmation with corresponding formal expressions. The study focuses exclusively on the concept of confirmation as the degree of rational belief given evidence, avoiding the frequency interpretation.

Abstract

It has been pointed out by Carnap that ‘probability’ is an equivocal term, which is used currently in two senses: (i) the degree to which it is rational to believe a hypothesis h on specified evidence e , and (ii) the relative frequency (in an indefinitely long run) of one property of events or things with respect to another. This paper is concerned only with the first of these two senses, which will be referred to as ‘the concept of confirmation,’ in order to avoid equivocation. We may distinguish a quantitative and a comparative concept of confirmation. The general form of statements involving the former is The degree of confirmation of the proposition h , given the proposition e as evidence, is r (where r is a real number between 0 and 1). In this paper the notation for a statement of this form is The general form of statements involving the comparative concept is The proposition h is equally or less confirmed on e than is the proposition h′ on e′ , which may be symbolized as The following abbreviations will also be used in discussing comparative confirmation: The former of these two abbreviations means intuitively The proposition h is less confirmed on evidence e than h′ is on e′ . The latter means The proposition h is confirmed on evidence e to the same degree that h′ is on e′ .