1,352 research outputs found

    The Strength of Abstraction with Predicative Comprehension

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    Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title from previous version, at request of referee

    A Quasi-Fregean Solution to ‘The Concept Horse’ Paradox

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    In this paper I offer a conceptually tighter, quasi-Fregean solution to the concept horse paradox based on the idea that the unterfallen relation is asymmetrical. The solution is conceptually tighter in the sense that it retains the Fregean principle of separating sharply between concepts and objects, it retains Frege’s conclusion that the sentence ‘the concept horse is not a concept’ is true, but does not violate our intuitions on the matter. The solution is only ‘quasi’- Fregean in the sense that it rejects Frege’s claims about the ontological import of natural language and his analysis thereof

    The bearable lightness of being

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    How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories-such as object, property, and relation-are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical form: for example, the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach-including Frege's notorious paradox of the concept horse-are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments

    Defining Original Presentism

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    It is surprisingly hard to define presentism. Traditional definitions of the view, in terms of tensed existence statements, have turned out not to to be capable of convincingly distinguishing presentism from eternalism. Picking up on a recent proposal by Tallant, I suggest that we need to locate the break between eternalism and presentism on a much more fundamental level. The problem is that presentists have tried to express their view within a framework that is inherently eternalist. I call that framework the Fregean nexus, as it is defined by Frege’s atemporal understanding of predication. In particular, I show that the tense-logical understanding of tense which is treated as common ground in the debate rests on this very same Fregean nexus, and is thus inadequate for a proper definition of presentism. I contrast the Fregean nexus with what I call the original temporal nexus, which is based on an alternative, inherently temporal form of predication. Finally, I propose to define presentism in terms of the original temporal nexus, yielding original presentism. According to original presentism, temporal propositions are distinguished from atemporal ones not by aspects of their content, as they are on views based on the Fregean nexus, but by their form—in particular, by their form of predication

    Non-Naturalism and Reference

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    Metaethical realists disagree about the nature of normative properties. Naturalists think that they are ordinary natural properties: causally efficacious, a posteriori knowable, and usable in the best explanations of natural and social sciences. Non-naturalist realists, in contrast, argue that they are sui generis: causally inert, a priori knowable and not a part of the subject matter of sciences. It has been assumed so far that naturalists can explain causally how the normative predicates manage to refer to normative properties, whereas non-naturalists are unable to provide equally satisfactory metasemantic explanations. This article first describes how the previous non-naturalist accounts of reference fail to tell us how the normative predicates could have come to refer to the non-natural properties rather than to the natural ones. I will then use the so-called qua-problem to show how the causal theories of reference of naturalists also fail to fix the reference of normative predicates to unique natural properties. Finally, I will suggest that, just as naturalists need to rely on the non-causal mechanism of reference magnetism to solve the previous problem, non-naturalists, too, can rely on the very same idea to respond to the pressing metasemantic challenges that they face concerning reference

    Logicism, Ontology, and the Epistemology of Second-Order Logic

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    In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms

    Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe

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    Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting.Comment: Forthcoming in The Journal of Symbolic Logi
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