Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with (...) other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). (shrink)

Conciliationism holds that disagreement of apparent epistemic peers often substantially undermines rational confidence in our opinions. Uniqueness principles say that there is at most one maximally rational doxastic response to any given batch of total evidence. The two views are often thought to be tightly connected. This paper distinguishes two ways of motivating conciliationism, and two ways that conciliationism may be undermined by permissive accounts of rationality. It shows how conciliationism can flourish under certain strongly permissive accounts of rationality. This (...) occurs when the motivation for conciliationism does not come from taking disagreement as evidence of one's own rational failings. However, divorcing the motive for conciliating from worries about rationality does not remove a feature of conciliationism that some find troubling: that conciliationism can lead to cases of “rational toxicity,” in which the most rational response to one's evidence involves violating some rational ideal. (shrink)