A Substructural Solution to the Lottery and Preface Paradoxes

The lottery (Kyburg, Probability and inductive logic. Macmillan, 1970) and the preface (Makinson, Analysis 25(6):205–207, 1965) paradoxes challenge a fundamental principle of rationality: if two propositions A and B are rationally acceptable, so is their conjunction, “A and B”. The paradoxes show that there are cases in which it seems legitimate to endorse each proposition of a finite list, but it is not rational to endorse them all together. A prominent solution to the paradoxes uses substructural logics to distinguish these two senses of the universal quantifier (Zardini, \(\forall \) and \(\omega \) . In Quantifiers, quantifiers, and quantifiers: Themes in logic, metaphysics, and language (pp. 489–526). Springer, 2015; Paoli, Philos Stud 124(3):313–330, 2005). This paper is twofold: first, we develop this idea by giving a detailed analysis of why a logic without contraction such as linear logic is suitable for analysing the paradoxes, identifying the elements present in the paradoxes that classical logic fails to capture, and linear logic nicely formalises in its vocabulary. Second, we reinterpret Paoli and Zardini’s solution pragmatically, arguing that the classical universal quantifier correctly captures the literal meaning of “all” but that linear logic captures two pragmatically enriched senses which are the default reading in some contexts, such as in the lottery of preface scenarios.