Øystein Linnebo

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articulating and defending the idea of such objects, and outlines some problems that these strategies face.

I defend the view that our ontology divides into categories, each with its own canonical way of identifying and distinguishing the objects it encompasses. For instance, I argue that natural numbers are identified and distinguished by their positions in the number sequence, and physical bodies, by facts having to do with spatiotemporal continuity. I also argue that objects belonging to different categories are ipso facto distinct. My arguments are based on an analysis of reference, which ascribes to reference a richer structure than it is normally taken to have

In the context of discussions about the nature of ‘identical particles’ and the status of Leibniz’s Principle of the Identity of Indiscernibles in Quantum Mechanics, a novel kind of physical discernibility has recently been proposed, which we call witness-discernibility. We inquire into how witness-discernibility relates to known kinds of discernibility. Our conclusion will be that for a wide variety of cases, including the intended quantum-mechanical ones, witness-discernibility collapses extensionally to absolute discernibility, that is, to discernibility by properties

Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege's writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well-informed overview provided by the first of its three chapters. Specialists will find the book an indispensable reference and an invaluable source of insights and new results. Although Frege is widely regarded as the father of analytic philosophy, his work on the foundations of mathematics was for a long time rather peripheral to ongoing research. The main reason for this is no doubt Russell's discovery in 1901 that the paradox now bearing his name can be derived in Frege's logical system. But recent decades have seen a huge surge of interest in Fregean approaches to the foundations of mathematics. A variety of consistent theories have been discovered that can be salvaged from Frege's inconsistent system, and foundational and philosophical claims have been made on behalf of many of these theories. Burgess claims quite plausibly that the significance of any such modified Fregean theory will in large part depend on how much of ordinary mathematics it enables us to develop. His book is accordingly 'a survey of various modified Fregean systems, attempting to determine the scope and limits of each'. The book's agenda is thus predominantly technical, and its spirit open-minded and experimental.

It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal part of this water? According to Quine, you must at least implicitly be operating with some criterion of identity that informs you when two sightings of water count as sightings of the same referent. For unless you have at least an implicit grasp of what is required for your intended referent to be identical with another object with which you are presented, you have not succeeded in singling out a unique object for reference.

I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which it may successfully be applied.

The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts there are.

where ‘aa’ is a plural term, and ‘F’ a plural predicate. Following George Boolos (1984) and others, many philosophers and logicians also think that plural expressions should be analysed as not introducing any new ontological commitments to some sort of ‘plural entities’, but rather as involving a new form of reference to objects to which we are already committed (for an overview and further details, see Linnebo 2004). For instance, the plural term ‘aa’ refers to Alice, Bob and Charlie simultaneously, and the plural predicate ‘F’ is true of some things just in case these things cooperate. A natural question that arises is whether the step from the singular to the plural can be iterated. Are there terms that stand to ordinary plural terms the way ordinary plural terms stand to singular terms? Let’s call such terms superplural. A superplural term would thus, loosely speaking, refer to several ‘pluralities’ at once, much as an ordinary plural term refers to several objects at once.1 Further, let’s call a predicate superplural if it can be predicated of superplural terms. It is reasonably straightforward to devise a formal logic of superplural terms, superplural predicates, and even superplural quantifiers (see Rayo 2006). But does this formal logic reflect any features of natural languages? In particular, does ordinary English contain superplural terms and predicates? The purpose of this article is to address these questions. We examine some earlier arguments for the existence of superplural expressions in English and find them to be either..

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects