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On specifying truthconditions
 The Philosophical Review
, 2008
"... Consider a committalist—someone who believes that assertions of a sentence like ‘the number of the planets is 8 ’ carry commitment to numbers—and a noncommittalist—someone who believes that all it takes for assertions of this sentence to be correct is that there be eight planets. The committalist wa ..."
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Consider a committalist—someone who believes that assertions of a sentence like ‘the number of the planets is 8 ’ carry commitment to numbers—and a noncommittalist—someone who believes that all it takes for assertions of this sentence to be correct is that there be eight planets. The committalist wants to know more about the noncommittalist’s view. She understands what the noncommittalist thinks is required of the world in order for assertions of simple sentences like ‘the number of the planets is 8 ’ to be correct, but she wants to know how the proposal is supposed to work in general. Could the noncommittalist respond by supplying a recipe for translating each arithmetical sentence into a sentence that wears its ontological innocence on its sleave? Surprisingly, there is a precise and interesting sense in which the answer is ‘no’. We will see that when certain constraints are in place, it is impossible to specify an adequate translationmethod. Fortunately, there is a technique for specifying truthconditions that is not based on translation, and can be used to explain to the committalist what the noncommittalist thinks is required of the world in order for arithmetical assertions to be correct. I call it ‘the φ(w)technique’. A shortcoming of the φ(w)technique is that it is of limited dialectical
Logicism Reconsidered
 In Shapiro
, 2005
"... This paper is divided into four sections. The first two identify different logicist theses, and show that their truthvalues can be conclusively established on minimal assumptions. Section 3 sets forth a notion of ‘contentrecarving ’ as a possible constraint on logicist theses. Section 4—which is l ..."
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Cited by 2 (2 self)
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This paper is divided into four sections. The first two identify different logicist theses, and show that their truthvalues can be conclusively established on minimal assumptions. Section 3 sets forth a notion of ‘contentrecarving ’ as a possible constraint on logicist theses. Section 4—which is largely independent from the rest of the paper—is a discussion of ‘NeoLogicism’. 1 Logicism 1.1 What is Logicism? Briefly, logicism is the view that mathematics is a part of logic. But this formulation is imprecise because it fails to distinguish between the following three claims: 1. LanguageLogicism The language of mathematics consists of purely logical expressions.
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
1.1 SecondOrder Languages
, 2013
"... Frege (1892) famously set forth the claim that the concept horse is not a concept. Here I will articulate a view of concepts and objects according to which this claim is essentially correct. I will not, however, be concerned with Frege scholarship. 1 Although I’m hoping to develop a view that is bro ..."
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Frege (1892) famously set forth the claim that the concept horse is not a concept. Here I will articulate a view of concepts and objects according to which this claim is essentially correct. I will not, however, be concerned with Frege scholarship. 1 Although I’m hoping to develop a view that is broadly Fregean in spirit, I will not attempt argue that the view is
Inferentialism, Logicism, Harmony, and a Counterpoint by
, 2007
"... Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truthconditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is cont ..."
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Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truthconditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is contrasted with the more recent quietist inferentialism of Brandom. Various other issues are highlighted for inferentialism in general, by reference to which different kinds of inferentialism can be characterized. Inferentialism for the logical operators is explained, with special reference to the Principle of Harmony. The statement of that principle in the author’s book Natural Logic is finetuned here in the way obviously required in order to bar an interesting wouldbe counterexample furnished by Crispin Wright, and to stave off any more of the same.
In The Foundations of Arithmetic and The Basic Laws of Arithmetic, Frege held the view that number
, 2002
"... terms refer to objects. 1 Later in his life, however, he seems to have been open to other possibilities: Since a statement of number based on counting contains an assertion about a concept, in a logically perfect language a sentence used to make such a statement must contain two parts, first a sign ..."
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terms refer to objects. 1 Later in his life, however, he seems to have been open to other possibilities: Since a statement of number based on counting contains an assertion about a concept, in a logically perfect language a sentence used to make such a statement must contain two parts, first a sign for the concept about which the statement is made, and secondly a sign for a secondorder concept. These secondorder concepts form a series and there is a rule in accordance with which, if one of these concepts is given, we can specify the next. But still we do not have in them the numbers of arithmetic; we do not have objects, but concepts. How can we get from these concepts to the numbers of arithmetic in a way that cannot be faulted? Or are there simply no numbers in arithmetic? Could the numbers help to form signs for these secondorder concepts, and yet not be signs in their own right? 2 To illustrate Frege’s point, let us consider the numberstatement ‘there are three cats’. It might be paraphrased in a firstorder language as: 3 (1) (∃3x)[Cat(x)]. If its logical form is to be taken at face value, (1) can be divided into two main logical components: first, the predicate ‘Cat(...)’, which for Frege refers to the (firstorder) concept cat; and, second, the
Frege’s Other Program
"... Abstract Frege’s logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the “neologicist ” approach of Hale & Wright. Less attention has been given to Frege’s extensionalist program, according to which arithmetic is to be reconstructed in term ..."
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Abstract Frege’s logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the “neologicist ” approach of Hale & Wright. Less attention has been given to Frege’s extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of secondorder logic augmented with a predicate representing the fact that an object x is the extension of a concept C, together with extralogical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the socalled Hume’s Principle and its connections to the root of the contradiction in Frege’s system. 1
What is the Purpose of NeoLogicism?
"... This paper introduces and evaluates two contemporary approaches of neologicism. Our aim is to highlight the differences between these two neologicist programmes and clarify what each projects attempts to achieve. To this end, we first ..."
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This paper introduces and evaluates two contemporary approaches of neologicism. Our aim is to highlight the differences between these two neologicist programmes and clarify what each projects attempts to achieve. To this end, we first