Probabilism, Assertion, and Higher-Order Vagueness

Hartry Field has recently suggested that a non-standard probability calculus better represents our beliefs about vague matters. His theory has two notable features: (i) that your attitude to P when you are certain that P is higher-order borderline ought to be the same as your attitude when you are certain that P is simply borderline, and (ii) that when you are certain that P is borderline you should have no credence in P and no credence in ~. This chapter rejects both elements of this view and advocates instead for the view that when you are in possession of all the possible evidence, and it is borderline whether P is borderline, it is borderline whether you should believe P. Secondly, it argues for probabilism: the view that your credences ought to conform to the probability calculus. To get a handle on these issues, the chapter looks at Dutch book arguments and comparative axiomatizations of probability theory.

Introduction: All that Counts – Modernism, Fiction, Mathematics

The introduction establishes the theoretical grounds for the analysis of mathematics and modernism and situates the book within the critical contexts of modernism and science studies and literature and science studies. It sets out the unique epistemological status of mathematics, key stages in its historical development, and charts the new territory that the book opens up by examining literary engagements with mathematics and modernism. With reference to texts and theories by Herbert Mehrtens, Jeremy Gray, Leo Corry and Moritz Epple, the introduction establishes the concept of modernist mathematics, the role of the so-called ‘foundational crisis of mathematics’, and competing logicist, formalist and intuitionist positions, particularly as represented by David Hilbert and L.E.J. Brower. The introduction also sets out how the issues at stake in mathematics feed into the modernist revaluation of rationality and Enlightenment values and echo the sense of crisis in other areas. A particular focus is on theories that reflect on mathematics as a human construct and deliberately created fiction, including texts by Friedrich Nietzsche, Ernst Cassirer, Oswald Spengler, Hans Vaihinger, Hartry Field and Alain Badiou.

Fictionalism and the problem of universals in the philosophy of mathematics

Many long-standing problems pertaining to contemporary philosophy of mathematics can be traced back to different approaches in determining the nature of mathematical entities which have been dominated by the debate between realists and nominalists. Through this discussion conceptualism is represented as a middle solution. However, it seems that until the 20th century there was no third position that would not necessitate any reliance on one of the two points of view. Fictionalism, on the other hand, observes mathematical entities in a radically different way. This is reflected in the claim that the concepts being used in mathematics are nothing but a product of human fiction. This paper discusses the relationship between fictionalism and two traditional viewpoints within the discussion which attempts to successfully determine the ontological status of universals. One of the main points, demonstrated with concrete examples, is that fictionalism cannot be classified as a nominalist position (despite contrary claims of authors such as Hartry Field). Since fictionalism is observed as an independent viewpoint, it is necessary to examine its range as well as the sustainability of the implications of opinions stated by their advocates.

Making Full Use of Heterogeneous Reference Potential

The aim of this paper is to develop an account of semantic reference initially put forward by Philip Kitcher that should be of interest to scientific realists. I discuss the motivations behind Kitcher’s proposal and especially how it faces a retrospection problem viz. the beliefs used to fix reference are selected retrospectively but not by the historical actors at the time. In order to sidestep this problem, I look at the idea of partial reference as put forward by Hartry Field and further advanced by Christina McLeish. I discuss the merits of these proposals and will argue that McLeish’s development of the idea leads to a deeply confused semantics for realism. In the last section, Kitcher’s account of heterogeneous reference potential is further developed in light of the issues raised in relation to partial reference and the retrospection problem. I think this raises the prospect of filling in a descriptivist picture of reference for theoretical terms of science.

OUR RELIABILITY IS IN PRINCIPLE EXPLAINABLE

Non-skeptical robust realists about normativity, mathematics, or any other domain of non-causal truths are committed to a correlation between their beliefs and non-causal, mind-independent facts. Hartry Field and others have argued that if realists cannot explain this striking correlation, that is a strong reason to reject their theory. Some consider this argument, known as the Benacerraf–Field argument, as the strongest challenge to robust realism about mathematics (Field 1989, 2001), normativity (Enoch 2011), and even logic (Schechter 2010). In this article I offer two closely related accounts for the type of explanation needed in order to address Field's challenge. I then argue that both accounts imply that the striking correlation to which robust realists are committed is explainable, thereby discharging Field's challenge. Finally, I respond to some objections and end with a few unresolved worries.

On Artifacts and Truth-Preservation

In Saving Truth from Paradox, Hartry Field presents and defends a theory of truth with a new conditional. In this paper, I present two criticisms of this theory, one concerning its assessments of validity and one concerning its treatment of truth-preservation claims. One way of adjusting the theory adequately responds to the truth-preservation criticism, at the cost of making the validity criticism worse. I show that in a restricted setting, Field has a way to respond to the validity criticism. I close with some general considerations on the use of revision-theoretic methods in theories of truth.

Expresabilidad, validez y recursos lógicos

El objetivo de este artículo es investigar diversos resultados limitativos acerca del concepto de validez. En particular, argumento que ninguna teoría lógica de orden superior con semántica estándar puede tener recursos expresivos suficientes como para capturar su propio concepto de validez. Además, muestro que la lógica de la verdad transparente que Hartry Field desarrolló recientemente conduce a resultados limitativos similares.

TRUTH, LOGICAL VALIDITY AND DETERMINATENESS: A COMMENTARY ON FIELD’SSAVING TRUTH FROM PARADOX

We consider notions oftruthandlogical validitydefined in various recent constructions of Hartry Field. We try to explicate his notion ofdeterminate truthby clarifying thepath-dependent hierarchiesof hisdeterminateness operator.