Qualitative probability as an intensional logic

Abstract

The main aim of this paper is to study the logic of a binary sentential oper- ator ‘z=‘, with the intended meaning ‘is at least as probable as’. The object language will be simple; to an ordinary language for truth-functional connec- tives we add ‘&’ as the only intensional operator. Our choice of axioms is heavily dependent of a theorem due to Kraft et al. [8], which states necessary and sufficient conditions for an ordering of the elements of a finite subset algebra to be compatible with some probability measure. Following a con- struction due to Segerberg [ 121, we show that these conditions can be trans- lated into our language. When giving semantical models of this formal language we will start from a universe U, interpreted as the set of possible worlds. We then assume that the probabilities of other worlds, from the standpoint of a given world, can be evaluated. We do this by associating with each world a probability measure on the universe. When a standard truth-valuation of propositional letters is added, we can then define the set of worlds where a given formula is true. The central clause here is that the truth-value at a world x of a formula of the form ‘A + B’ is determined from the probabilities at x of the sets of worlds where ‘A’ and ‘B’ are true. This mechanism also provides a simple way of handling higher order prob- ability statements, i.e. statements about probabilities of probabilities. As we will see, some tentative restrictions on the probability measures in the models will reflect different approaches to the logic of higher order probability sentences. For the semantics of classical one-place modal operators, the alternative- ness relation on the universe has proven to be useful and clarifying. However, as soon as one turns to more complicated intensional operators as e.g. counter- factuals, preferences, conditional obligations and probabilities, such a relation between points in‘the universe seems as mysterious as the operators themselves In our opinion, a more fruitful way to deal with these operators semantically