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Amending Frege’s Grundgesetze der Arithmetik
, 2002
"... Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgeset ..."
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Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
DOI 10.1007/s112290119883y What are numbers?
"... Abstract This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs existnwise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the socalled Caesar objection will be answered by ex ..."
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Abstract This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs existnwise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the socalled Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m and n, if there are exactly mFs and also there are exactly nFs, then m = n.
Introduction to Absolute Generality
, 2006
"... absolutely everything there is. A cursory look at philosophical practice reveals numerous instances ..."
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absolutely everything there is. A cursory look at philosophical practice reveals numerous instances
(HP) (V) (H 2 P)
"... 2 How groundedness might help • What extensions there are depends on what concepts there are. • What concepts there are depends on what extensions there are. A heavyhanded proposal Require that an abstraction principle be predicative in the sense that its RHS not quantify over the sort of objects t ..."
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2 How groundedness might help • What extensions there are depends on what concepts there are. • What concepts there are depends on what extensions there are. A heavyhanded proposal Require that an abstraction principle be predicative in the sense that its RHS not quantify over the sort of objects to which its LHS refers. A more general idea • Abstraction as a process by which more and more abstracts are ‘introduced’. • Each step of the process ‘presupposes ’ only what was available at the previous step Old news? Asymmetry The righthand side of an abstraction principle must not ‘presuppose ’ any of the objects to which the lefthand side refers. Quantification incurs ‘presupposition ’ (QIP) A quantified statement ‘presupposes ’ every object in the range of its quantifiers. Conclusion Impredicative abstraction principles are impermissible. 1 Can we resist QIP by analyzing ‘presupposition’? • Following Kripke, we consider monotonic operators J • J(X) = Y formally represents the philosophical idea that the elements of Y ‘depend on X’, or X ‘suffices for ’ each element of Y, or the elements of Y ‘presuppose ’ at most X. • Need to spell out clearly what philosophical idea J is meant to track. Two key questions Q1. What are the sets on which J operates (e.g. sets of abstracta, (representations of) identity facts, etc.)? Q2. What is the operator J (e.g. strong Kleene, supervaluational, or Leitgebstyle)? 3 Option 1: Leitgeb on grounded abstraction (Leitgeb, ming) Consider an abstraction principle: nx.F x = nx.Gx ↔ Φ[F, G] We wish to restrict this to the instances that are grounded (in a sense to be articulated). Presumably, this means accept only instances nx.φ(x) = nx.ψ(x) ↔ Φ[φ/F, ψ/G] whose RHS is grounded. The ingredients • A domain Dom of preobjects. • A domain Con ⊆ ℘(Dom) of [pre]concepts on these preobjects. • A [pre] abstraction operator N: Con → Dom, which is a bijection. The recipe • Build up identity and distinctness facts on Dom by iterating a Leitgebjump
Was Wittgenstein an epistemic relativist *
"... Relativists and antirelativists alike are nowadays mostly united in considering Wittgenstein an epistemic relativist. 1 Accordingly, 2 there could be, either in principle or as a matter of fact, different epistemic systems, none of which would be intrinsically correct; each of them would be, from a ..."
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Relativists and antirelativists alike are nowadays mostly united in considering Wittgenstein an epistemic relativist. 1 Accordingly, 2 there could be, either in principle or as a matter of fact, different epistemic systems, none of which would be intrinsically correct; each of them would be, from a metaphysical point of view, as good as any other one, and would certify as (true and) justified different propositions. As a consequence, knowledge – that is, justified true belief –, if and when attained, would always be situated: what counts as knowledge within one system of justification may not be so within another. Moreover, should alternative epistemic systems compete with each other, the choice couldn’t be based on rational considerations, for it is only within each system that reasons and justifications are produced. Hence, the passage from one epistemic system to another would always be a form of conversion or persuasion, reached through arational means. Relativist readings of Wittgenstein’s thought base their interpretation mostly on his claim, in On Certainty, that at the foundations of our language games and, in particular, of our epistemic ones – those in which we provide reasons for and against certain propositions or theories, and are interested in assessing their truth – lie propositions which are neither true nor false; grounded or
On ‘Average’
, 2009
"... This paper investigates the semantics of sentences that express numerical averages, focusing initially on cases such as ‘The average American has 2.3 children. ’ Such sentences have been used by both linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, N ..."
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This paper investigates the semantics of sentences that express numerical averages, focusing initially on cases such as ‘The average American has 2.3 children. ’ Such sentences have been used by both linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and Stephen Yablo have used them to provide evidence that apparent singular reference need not be taken as ontologically committing. We develop a fully general and independently justified compositional semantics in which such constructions are assigned truth conditions that are not ontologically problematic, and show that our analysis is superior to all extant rivals. Our analysis provides evidence that a good semantics yields a sensible ontology. It also reveals that natural language involves genuine singular term reference to numbers. 1.
Existence and Number
"... The FregeRussell view is that existence is a secondorder property rather than a property of individuals. One of the most compelling arguments for this view is based on the premise that there is an especially close connection between existence and number. The most promising version of this argument ..."
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The FregeRussell view is that existence is a secondorder property rather than a property of individuals. One of the most compelling arguments for this view is based on the premise that there is an especially close connection between existence and number. The most promising version of this argument is by C.J.F Williams (1981, 1992). In what follows, I argue that this argument fails. I then defend an account according to which both predications of number and existence attribute properties to individuals.
Philos Stud DOI 10.1007/s1109801197791 Reference to numbers in natural language
"... objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reve ..."
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objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects ’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted.