We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)

In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in theCritique of Pure Reasonand in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically precise (...) account of Kant’s transcendental theory of time—the most precise account to date.Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in theMetaphysical Foundations of Natural Science.While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others. (shrink)

In this paper we introduce a game semantics for System P, one of the most studied axiomatic systems for non-monotonic reasoning, conditional logic and belief revision. We prove soundness and completeness of the game semantics with respect to the rules of System P, and show that an inference is valid with respect to the game semantics if and only if it is valid with respect to the standard order semantics of System P. Combining these two results leads to a new (...) completeness proof for System P with respect to its order semantics. Our approach allows us to construct for every inference either a concrete proof of the inference from the rules in System P or a countermodel in the order semantics. Our results rely on the notion of a witnessing set for an inference, whose existence is a concise, necessary and sufficient condition for validity of an inferences in System P. We also introduce an infinitary variant of System P and use the game semantics to show its completeness for the restricted class of well-founded orders. (shrink)