Leon Horsten

Williamson has forcefully argued that Fitch's argument shows that the domain of the unknowable is non-empty. And he exhorts us to make more inroads into the land of the unknowable. Concluding his discussion of Fitch's argument, he writes: " Once we acknowledge that [the domain of the unknowable] is non-empty, we can explore more effectively its extent. … We are only beginning to understand the deeper limits of our knowledge. " I shall formulate and evaluate a new argument concerning the domain of the unknowable. It is an argument about knowability. More specifically, it is an argument about what we can know about the natural numbers. Since the domain of discourse will be the natural numbers structure, the notion of knowability can for the purposes of the argument be identified with a priori knowability or – which amounts to the same thing – absolute provability .Suppose, for a reductio, that there exists a property θ of natural numbers such that it is provable that for some natural number n, θ is true but unprovable. …

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. 1 Introduction2 The Limits of Classical Probability Theory2.1 Classical probability functions2.2 Limitations2.3 Infinitesimals to the rescue?3 NAP Theory3.1 First four axioms of NAP3.2 Continuity and conditional probability3.3 The final axiom of NAP3.4 Infinite sums3.5 Definition of NAP functions via infinite sums3.6 Relation to numerosity theory4 Objections and Replies4.1 Cantor and the Archimedean property4.2 Ticket missing from an infinite lottery4.3 Williamson’s infinite sequence of coin tosses4.4 Point sets on a circle4.5 Easwaran and Pruss5 Dividends5.1 Measure and utility5.2 Regularity and uniformity5.3 Credence and chance5.4 Conditional probability6 General Considerations6.1 Non-uniqueness6.2 InvarianceAppendix

The difficulties with formalizing the intensional notions necessity, knowability and omniscience, and rational belief are well-known. If these notions are formalized as predicates applying to (codes of) sentences, then from apparently weak and uncontroversial logical principles governing these notions, outright contradictions can be derived. Tense logic is one of the best understood and most extensively developed branches of intensional logic. In tense logic, the temporal notions future and past are formalized as sentential operators rather than as predicates. The question therefore arises whether the notions that are investigated in tense logic can be consistently formalized as predicates. In this paper it is shown that the answer to this question is negative. The logical treatment of the notions of future and past as predicates gives rise to paradoxes due the specific interplay between both notions. For this reason, the tense paradoxes that will be presented are not identical to the paradoxes referred to above