Øystein Linnebo

According to Frege’s context principle, we must never to ask for the meaning of a word in isolation but only in the context of a sentence. The context principle poses some very hard interpretive challenges; not only are the ideas themselves hard, but relevant parts of Frege’s view change in the course of his career. This principle plays a crucial role in the _Grundlagen_, especially in Frege’s answer to the question of how numbers and other logical objects are ‘given to us’. By contrast, the context principle figures less prominently in the _Grundgesetze_. This article argues that, appearances to the contrary, the context principle survives in Frege’s _magnum opus,_ where it still has an important role to play. Reflection on this role sheds important new light on the principle. More specifically, I argue that the context principle must be disentangled from the problematic idea of ‘recarving of content’; that the principle is fully compatible with Frege’s compositional semantics; and that it suggests a more ‘lightweight’ conception of mathematical objects than is typically associated with Frege.

Potentialism holds that certain objects are successively generated in an incompletable process. While it is natural to analyze this view modally, there are theorems that connect the resulting modal analysis of potentialism with the non-modal languages of ordinary mathematics. By extending this approach to plural languages, this article proves a far stronger result about definitional equivalence. This opens the door to a new and entirely non-modal explication of potentialism, using a restricted plural logic. Some advantages of this “demodalized” explication are discussed. It is certainly simpler and more user-friendly than the extant modal analysis, as illustrated by an application to potentialist set theory. More ambitiously, I suggest that the explication might also enable potentialists to sidestep the tricky question of which mathematical objects are actual.

Which comprehension axioms of higher-order logic are acceptable? That is, under what conditions does a formula define a concept or circumscribe some objects? It is well known that unrestricted higher-order comprehension is incompatible with unrestricted reification of higher-order entities. In search of a response to this conflict, an argument against all forms of impredicative comprehension is formulated; for example, when defining a concept, we may not quantify over a totality to which this concept belongs. Although this predicativist argument is ultimately rejected, a careful analysis of it points the way to some milder logical restrictions, which suffice to resolve the conflict.

Frege’s Basic Law V says that two concepts have the same extension just in case they are coextensive. This chapter challenges the orthodoxy that the “law” must be rejected because of Russell's paradox. It is argued that pressure remains to accept something like Basic Law V, and that this pressure can be accommodated without inconsistency by adopting a richer logical framework than usual. This controlled use of Basic Law V is shown to open up new approaches to set theory and the logical paradoxes.

I articulate and defend the thesis that first-order logic is algebraic, in the sense that it does not have a single canonical application to reality. If correct, this thesis supports a flexible conception of ontology. Suppose we have a first-order language that purports to talk about certain objects, say Fs, whose conditions for correct use are expressed in terms that are already assumed to be in good standing and thus, in particular, involve no talk about Fs. Suppose further that these assertability conditions respect the laws of logic. Finally, suppose that the assertability conditions involve a criterion of identity that licenses the use of non-trivial identity statements. Then, according to the flexible conception, there really are Fs. I also defend this algebraic conception of logic, and the conception of ontology to which it gives rise, against the objection that this approach involves a problematic form of anti-realism.

The notion of definiteness has played a fundamental role in the early developments of set theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different forms of definiteness. First, a condition can be definite in the sense that, given any object, either the condition applies to that object or it does not. We call this intensional definiteness. Second, a condition or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns whether an intension applies to objects considered one by one, extensional definiteness concerns the totality of objects to which the intension applies. We discuss how these two forms of definiteness admit of precise mathematical analyses. We argue that two main types of explication of extensional definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation. We submit that the two notions of extensional definiteness that emerges from our investigation enable us to identify and understand some of the most important fault lines in the philosophy and foundations of mathematics.

How should a layperson respond to learning that the experts on a given topic disagree amongst themselves? This paper argues that, epistemically, the appropriate response to an expert disagreement depends greatly on what explains the disagreement, and that there are several quite different types of explanations for a given disagreement. Accordingly, expert disagreement calls for different epistemic responses in different circumstances. However, the paper also supplements this pluralist account of how to respond epistemically to expert disagreement with a unifying account of how laypeople should response ‘zetetically’ to expert disagreement – that is, how they should direct their inquiry concerning the contested claim. Indeed, it is precisely because of the plurality of appropriate epistemic responses to expert disagreements that laypeople often have strong reasons to inquire into what explains the presence of disagreement among experts.

In Thin Objects, I articulated a thin conception of objecthood according to which the existence of objects need not make any substantive demands on reality. Agustín Rayo has developed and defended an even less demanding conception of objecthood, which drops the requirement of informative criteria of identity. The nature and extent of the disagreement is clarified, and two clusters of advantages of the thin conception are presented. In essence, by invoking criteria of identity we obtain more robust notions of reference and objecthood. In particular, the availability of multiple perspectives on a single object is key to an important form of objectivity.

While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is “made true” at some such stage. The natural logic of this strict form of potentialism is semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic.

Aristotle argued that paradoxes of the infinite can be avoided only by insisting that all infinities are potential, not actual. There is a long tradition of thinking that a Judeo-Christian God would collapse potential infinities to actual ones, thus removing the Aristotelian guard-rail against paradox. After all, does not God know all numbers, regardless of whether they are actual or merely potential? We analyze the Aristotelian guard-rail of potentiality, as well as challenges to it due to Augustine, Burley, Scotus, and Cantor. At the heart of the debate we find the metaphysical question of how the temporal or modal “dimension” that harbors potential infinities compares to ordinary spatial dimensions. Certain theological assumptions underwrite a strong analogy. The best Aristotelian response, we suggest, is to reject the analogy by refusing to analyze modality extensionally as a point-space of possible worlds. Our analysis thus reveals lessons for atheists as well as theists.

The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one:. The principal aim of this article is to use the notion of grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction.

Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and prove that it fails to characterize the domain uniquely. Thus, we conclude, there is no easy road to impredicative definabilism.

According to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.