Øystein Linnebo

What explains the truth of a universal generalization? Two types of explanation can be distinguished. While an ‘instance-based explanation’ proceeds via some or all instances of the generalization, a ‘generic explanation’ is independent of the instances, relying instead on completely general facts about the properties or operations involved in the generalization. This intuitive distinction is analyzed by means of a truthmaker semantics, which also sheds light on the correct logic of quantification. On the most natural version of the semantics, this analysis vindicates some claims made—without a proper defense—by Michael Dummett, Solomon Feferman, and others. Where instance-based explanations are freely available, classical logic is shown to be warranted. By contrast, intuitionistic logic remains warranted regardless of what explanations are available.

Critical views of logic are presented. These are views that are critical of logic in a sense akin to the way in which Kant is critical rather than dogmatic about traditional metaphysics. Such approaches differ from the Fregean ‘logic-first’ view. In accordance with the latter, logic is often regarded as epistemologically and methodologically fundamental. Hence, all disciplines – including mathematics – are considered as answerable to logic, rather than vice versa. In critical views of logic, by contrast, the logical principles that govern some subject matter may depend on the metaphysics of this subject matter or on the semantics of our discourse about it. Space is thereby carved out between a Fregean position on the one hand and thoroughgoing holism à la Quine on the other.

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms.

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

Ordinarily, the order in which some objects are attached to a scale does not affect the total weight measured by the scale. This principle is shown to fail in certain cases involving infinitely many objects. In these cases, we can produce any desired reading of the scale merely by changing the order in which a fixed collection of objects are attached to the scale. This puzzling phenomenon brings out the metaphysical significance of a theorem about infinite series that is well known by mathematicians but has so far eluded philosophical scrutiny.

Peacocke’s recent The Primacy of Metaphysics covers a wide range of topics. This critical discussion focuses on the book’s novel account of extensive magnitudes and numbers. First, I further develop and defend Peacocke’s argument against nominalistic approaches to magnitudes and numbers. Then, I argue that his view is more Aristotelian than Platonist because reified magnitudes and numbers are accounted for via corresponding properties and these properties’ application conditions, and because the mentioned objects have a “shallow nature” relative to the corresponding properties. The result is an asymmetric conception of abstraction, which contrasts with the neo-Fregeans’ but has important tenets in common with an approach that I have recently developed.

Analytic philosophy has traditionally been little concerned with the history of philosophy, including that of analytic philosophy itself. But in recent years the study of the early period of the analytic tradition has become an active and lively branch of Anglo-American philosophy. Early Analytic Philosophy, a collection of papers presented in honor of professor Leonard Linsky at the University of Chicago in April 1992, is an example of this. The contributors, many of them leading scholars in the field of early analytic philosophy, approach the thought of the founders of the analytic tradition in its own right and with a keen eye to how it differs from much of contemporary analytic philosophy. Many of the papers challenge received interpretations and propose alternative readings.

Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes.

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

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.

Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book attempts to develop the needed explanations by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed.

Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there surprisingly turns out to be logical space for a response to the improved challenge where no such space appeared to exist.

Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic.

Mathematics is one of the most successful human endeavors—a paradigm of precision and objectivity. It is also one of our most puzzling endeavors, as it seems to deliver non-experiential knowledge of a non-physical reality consisting of numbers, sets, and functions. How can the success and objectivity of mathematics be reconciled with its puzzling features, which seem to set it apart from all the usual empirical sciences? This book offers a short but systematic introduction to the philosophy of mathematics. Readers are introduced to all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. The book also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The exposition is always closely informed by ongoing research in the field and sometimes draws on the author’s own contributions to this research. This means that Gottlob Frege—a German mathematician and philosopher widely recognized as one of the founders of analytic philosophy—figures prominently in the book, both through his own views and his criticism of other thinkers.