Despite their controversial ontological status, the discussion on arbitrary objects has been reignited in recent years. According to the supporting views, they present interesting and unique qualities. Among those, two define their nature: their assuming of values, and the way in which they present properties. Leon Horsten has advanced a particular view on arbitrary objects which thoroughly describes the earlier, arguing they assume values according to a sui generis modality, which he calls afthairetic. In this paper, we offer a general (...) method for defining the minimal system of this modality for any given first-order theory, and possible extensions of it that incorporate further aspects of Horsten’s account. The minimal system presents an unconventional inference rule, which deals with two different notions of derivability. For this reason and the failure of the Necessitation rule, in its full generality, the resulting system is non-normal. Then, we provide conditional soundness and completeness results for the minimal system and its extensions. (shrink)

A single significant instance may support general conclusions, with possible exceptions being tolerated. This is the case in practical human reasoning (e.g. moral and legal normativity: general rules tolerating exceptions), in theoretical human reasoning engaging with external reality (e.g. empirical and social sciences: the use of case studies and model organisms) and in abstract domains (possibly mind-unrelated, e.g. pure mathematics: the use of arbitrary objects). While this has been recognized in modern times, such a process is not captured by current (...) models of supporting general conclusions. This paper articulates the thesis that there is a kind of reasoning, generic reasoning, previously unrecognized as an independent type of reasoning. A theory of generic reasoning explains how a single significant instance may support general conclusions, with possible exceptions being tolerated. This paper will adopt, as a working hypothesis, that generic reasoning is irreducible to currently recognized kinds of ‘pure’ reasoning. The aim is to understand generic reasoning, both theoretically and in its applications. (shrink)