Øystein Linnebo

Predicative fragments of Frege Arithmetic

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.

THE POTENTIAL HIERARCHY OF SETS

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

Identity and Discernibility in Philosophy and Logic

Questions about the relation between identity and discernibility are important both in philosophy and in model theory. We show how a philosophical question about identity and discernibility can be ‘factorized’ into a philosophical question about the adequacy of a formal language to the description of the world, and a mathematical question about discernibility in this language. We provide formal definitions of various notions of discernibility and offer a complete classification of their logical relations. Some new and surprising facts are proved; for instance, that weak discernibility corresponds to discernibility in a language with constants for every object, and that weak discernibility is the most discerning non-trivial discernibility relation.

Frege's Proof of Referentiality

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.

Bad company tamed

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.

Which Abstraction Principles are Acceptable? Some Limitative Results

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be avoided only by major departures from the existing neo-Fregean programme. 1. Introduction2. A Simple Counterexample3. A Fregean Counterexample4. The Argument 4.1. Defending step 14.2. Defending step 24.3. Defending step 34.4. Defending step 45. Concluding Remarks Introduction A Simple Counterexample A Fregean Counterexample The Argument 4.1. Defending step 14.2. Defending step 24.3. Defending step 34.4. Defending step 4 Defending step 1 Defending step 2 Defending step 3 Defending step 4 Concluding Remarks

Plurals and modals

Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.

‘Just is’-Statements as Generalized Identities

Abstract Identity is ordinarily taken to be a relation defined on all and only objects. This consensus is challenged by Agustín Rayo, who seeks to develop an analogue of the identity sign that can be flanked by sentences. This paper is a critical exploration of the attempted generalization. First the desired generalization is clarified and analyzed. Then it is argued that there is no notion of content that does the desired philosophical job, namely ensure that necessarily equivalent sentences coincide in this kind of content, and such pairs of sentences consequently are on a par with respect to metaphysical explanations.

Some Criteria for Acceptable Abstraction

where the variables α and β range over entities of some sort, and where ∼ is an equivalence relation on this sort of entity. Frege’s inconsistent Basic Law V shows that not every such principle is acceptable. A variety of criteria for acceptable abstraction have been proposed.1 In this note I analyze the logical relations between some of the proposed criteria, in particular conservativeness, irenicity, and stability. I answer some technical questions thrown open by the discovery of errors in the proofs of some crucial theorems in a deservedly influential study of these issues [Weir, 2003]. No attempt will be made in this note to assess the plausibility of the suggested criteria.2

Cardinality and Acceptable Abstraction

It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but analyze the interesting idea on which it is based, namely that an acceptable abstraction has to ‘generate’ the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.