In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

The paper provides a proof theoretic characterization of the Russellian theory of definite descriptions (RDD) as characterized by Kalish, Montague and Mar (KMM). To this effect three sequent calculi are introduced: LKID0, LKID1 and LKID2. LKID0 is an auxiliary system which is easily shown to be equivalent to KMM. The main research is devoted to LKID1 and LKID2. The former is simpler in the sense of having smaller number of rules and, after small change, satisfies cut elimination but fails to satisfy the subformula property. In LKID2 an additional analysis of different kinds of identities leads to proliferation of rules but yields the subformula property. This refined proof theoretic analysis leading to fully analytic calculus with constructive proof of cut elimination is the main contribution of the paper.

Jean Nicod (1893–1924) is a French philosopher and logician who worked with Russell during the First World War. His PhD, with a preface from Russell, was published under the title La géométrie dans le monde sensible in 1924, the year of his untimely death. The book did not have the impact he deserved. In this paper, I discuss the methodological aspect of Nicod’s approach. My aim is twofold. I would first like to show that Nicod’s definition of various notions of equivalence between theories anticipates, in many respects, the (syntactic and semantic) model-theoretic notion of interpretation of a theory into another. I would secondly like to present the philosophical agenda that led Nicod to elaborate his logical framework: the defense of rationalism against Bergson’s attacks.

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations $\Omega ^{n}(\bar {\xi })$ defined with the help of numeral exponents. Since $Num$ (x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with “insuperable difficulties.” This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that the N-operator $N(\bar {\xi })$ and recursively defined arithmetic operators $\Omega ^{n}(\bar {\xi })$ are not different in kind, we can address Ramsey’s problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction.

In this paper, I criticize Structured Propositionalism, the most widely held theory of the nature of propositions according to which they are structured entities with constituents. I argue that the proponents of Structured Propositionalism have paid insufficient attention to the metaphysical presuppositions of the view – most egregiously, to the notion of propositional constituency. This is somewhat ironic, since the friends of structured propositions tend to argue as if the appeal to constituency gives their view a dialectical advantage. I criticize four different approaches to providing a metaphysics of propositional constituency: set-theoretic, mereological, hylomorphic, and structure-making. Finally, I consider the option of taking constituency in a deflationary, metaphysically ‘lightweight’ sense. I argue that, though invoking constituency in a lightweight sense may be useful for avoiding the ontological problems that plague the ‘heavyweight’ conception, it no longer proffers a dialectical advantage to Structured Propositionalism.

Moritz Schlick's work shaped logical empiricism and thereby an important part of philosophy in the first half of the twentieth century. A continuous thread that runs through his work is a philosophical diagnosis of the ‘great errors in philosophy’: philosophers assume that there is intuitive knowledge or knowledge by acquaintance. Yet, acquaintance is not knowledge, but an evaluative attitude. In this paper I will reconstruct Schlick's arguments for this conclusion in the light of his early practical philosophy and his reading of Schopenhauer's The World as Will and Idea. Having the historical roots of Schlick's dichotomy between acquaintance (intuition) and knowledge in view will put us in a position to explore and question its presuppositions.

The paradox that appears under Burali–Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be–absurdly–an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali–Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali–Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic.

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.

There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is $\Pi _1^1$-complete. But third, it is only granted the assumption of countability that the class of tautologies is ${\Sigma _1}$-definable in set theory.

Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.

Frege explained the notion of generality by stating that each its instance is a fact, and added only later the crucial observation that a generality can be inferred from an arbitrary instance. The reception of Frege’s quantifiers was a fifty-year struggle over a conceptual priority: truth or provability. With the former as the basic notion, generality had to be faced as an infinite collection of facts, whereas with the latter, generality was based on a uniformity with a finitary sense: the provability of an arbitrary instance.

Mark Nelson thinks that Bertrand Russell's well-known criticisms of St. Thomas Aquinas turn on Russell's acceptance of a highly implausible epistemic principle (DAM), and that my previous objection to this claim depends upon the attribution to Russell of an even more implausible Insincerity Objection. While I agree that Russell's criticisms do not turn on the Insincerity Objection, I argue that my previous rejection of the attribution of (DAM) to Russell is well-justified: there is a plausible reading of Russell that requires neither (DAM) nor the Insincerity Objection.