In this paper it is shown that the Verification Thesis (all truths are knowable) is only susceptible to Fitch’s Paradox if one conflates the de re and de dicto interpretation of knowability. A formalisation shows that if one treats knowability as a complex second-order predicate, then the paradox falls apart.

We consider the consistency proof for a weak fragment of arithmetic published by von Neumann in 1927. This proof is rather neglected in the literature on the history of consistency proofs in the Hilbert school. We explain von Neumann’s proof and argue that it fills a gap between Hilbert’s consistency proofs for the so-called elementary calculus of free variables with a successor and a predecessor function and Ackermann’s consistency proof for second-order primitive recursive arithmetic. In particular, von Neumann’s proof is the first rigorous proof of the consistency of an axiomatization of the first-order theory of a successor function.

In this article, I consider the status of several statements analogous to the Church–Turing thesis that assert that some definition of algorithmic randomness captures the intuitive conception of randomness. I argue that we should not only reject the theses that have appeared in the algorithmic randomness literature, but more generally that we ought not evaluate the adequacy of a definition of randomness on the basis of whether it captures the so-called intuitive conception of randomness to begin with. Instead, I argue that a more promising alternative is to evaluate the adequacy of a definition of randomness on the basis of whether it captures what I refer to as a “notion of almost everywhere typicality.” In support of my main claims, I will appeal to recent work in showing the connection between of algorithmic randomness and certain “almost everywhere” theorems from classical mathematics.

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.

In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment.

We prove that any equational basis that defines representable relation algebras (RRA) over weakly representable relation algebras (wRRA) must contain infinitely many variables. The proof uses a construction of arbitrarily large finite weakly representable but not representable relation algebras whose “small” subalgebras are representable.

A new logic is presented without predicates—except equality. Yet its expressive power is the same as that of predicate logic, and relations can faithfully be represented in it. In this logic we also develop an alternative for set theory. There is a need for such a new approach, since we do not live in a world of sets and predicates, but rather in a world of things with relations between them.

Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.

A logical constant is weakly disharmonious if its elimination rules are weaker than its introduction rules. Substructural weak disharmony is the weak disharmony generated by structural restrictions on the eliminations. I argue that substructural weak disharmony is not a defect of the constants which exhibit it. To the extent that it is problematic, it calls into question the structural properties of the derivability relation. This prompts us to rethink the issue of controlling the structural properties of a logic by means of harmony. I argue that such a control is possible and desirable. Moreover, it is best achieved by global tests of harmony.

This paper presents a proof system for discourse representation theoretic reasoning and dynamic predicate logical inference. It gives a sound and complete characterization of the dynamic declaration of discourse referents and the essentially indexical means to refer back to them. The indexical outlook upon discourse reference is argued to further our understanding of some issues deemed relevant both theoretically (philo-logically) and practically (computationally).

We correct two typos and an error in The Arithmetic of the Even and the Odd, and provide an axiom system for a weak arithmetic in which the fact that a square root of an integer is either irrational or an integer can be proved.