Speaker: Yifeng Ding (PKU)

 

Abstract: In the last decade, there has been lively discussion of whether relative likelihood expressions in natural language, such as `more likely than', should be given formal semantics using qualitative structures or numerical probability. In a recent paper, Marushak argues that probabilistic semantics has an advantage over a qualitative semantics discussed by Holliday and Icard, because the latter is committed to a qualitative implication of the regularity principle for probability, in the form of their Noetherian assumption on preorders. In this paper, we analyze the effect of assuming regularity and Noetherianity on the logic of relative likelihood. We suggest that a key consequence of regularity for epistemic possibilities, corresponding to what Hofweber calls the ``Minimal Constraint'' (MC) for theories of chance, may be a desirable entailment prediction for natural language.  We prove several new completeness theorems for logics of relative likelihood with and without the (MC) axiom and a related axiom (R) of regularity. In addition, we discuss whether it is possible to have a semantics that combines three desiderata: (i) validating (MC)/(R), (ii) validating additivity principles for relative likelihood, and (iii) dropping any restrictions, such as the Noetherian assumption, on the relative likelihood of epistemic possibilities. We show that although Marushak's proposed semantics fails desideratum (ii), in virtue of invalidating De Finetti's qualitative additivity principle, a slight modification of Holliday and Icard's semantics delivers all three desiderata.