A Logic for Subjective Belief

We regard logic as a language for communicating beliefs, and adopt the principle that a proposition is defined not by assigning objective conditions for its truth but by associating with its assertion a definite commitment. For a prime proposition, the commitment is to pay $1 should a ‘trial’ yield a negative outcome. (The possibility of trial is assumed, but no truth value is postulated and we do not assume that repetitions of a trial always yield the same outcome.) With prime propositions alone it is not possible to express partial beliefs. This possibility enters with the definition of the logical connective → (implies): following Lorenzen, we say the assertion of Ρ→Q entails a commitment to assert Q if the ‘opponent’ asserts P. If P and Q are prime this expresses the belief that Q is ‘at least as true as’ P; more complicated propositions represent more elaborate beliefs. The other connectives are defined in a similar way. The assertion of a proposition by the ‘proponent’ ℙ now leads, via a dialogue, to a position in which each speaker is committed to certain prime propositions. A certain set of these positions is regarded as acceptable by ℙ; making weak assumptions of rationality of ℙ we arrive at an axiom concerning this set and thence construct a risk function which uniquely characterizes it. In general, the risk function is subadditive (on a certain group) but, in the case of a “confident” speaker (when the set of acceptable positions is maximal and the risk function is minimal) it is additive and directly determines a probability assignment on the prime propositions. Using these constructions, we show finally that every rational speaker behaves (in a certain exact sense) as if he believes that every prime proposition has a definite objective probability of which he has only partial information. Thus we obtain a justification, in subjective terms, for a belief in objective probability.