Language Syntax

The concept of probability to be defined here and which will lead to a resolution of the conflicts and confusions adumbrated in the last chapter will be a logical one. Most logical interpretations of probability, Carnap’s, for example, have been offered only for very simple languages. These interpretations are therefore inappropriate for languages rich enough for the expression of ordinary statistical theories and hypotheses. The claim has been, of course, that we must start somewhere, and that it is easier to start with something simple rather than with something complicated. But this is not always the case. It would be unreasonable, for example, to attempt to provide a theory of analysis exclusively in terms of integers on the grounds that integers are simpler than real numbers or sets. Somewhat analogously, I would claim that we cannot even define probability at all except in languages rich enough for the expression of statistical hypotheses, just as we cannot define a derivative except in a language rich enough to talk about real numbers. Statistical hypotheses come first; probabilities come after; and statistical inference is a matter both of statistical hypotheses and of probabilities.