Epistemic Modals, Qualitative Probability, and Nonstandard Probability

Yalcin [23] shows that Kratzer’s model [5] does not validate some intuitively valid inferences and validates some intuitively invalid ones. He adopts a model based directly on a probability measure. However, as Kratzer [6] says, ‘Our semantic knowledge alone does not give us the precise quantitative notions of probability and desirability that mathematicians and scientists work with’, Yalcin’s model seems to be unnatural as a model for comparative epistemic modals. The aim of this paper is to propose a new version of complete logic—modal-qualitative-probability logic (MQPL)—the model of the language of which has the following four merits: (i) The model reflects Kratzer’s intuition above in the sense that the model is not based directly on a probability measure, but based on a qualitative probability ordering. (ii) The model does not cause Yalcin’s problem. (iii) The model has no limitation of the size of the domain. (iv) The model can deal with the two-dimensional geometric probability that Kolmogorov probability theory cannot.