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What does it mean to say that logic is formal
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
Predicative Fragments of Frege Arithmetic
, 2003
"... Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply al ..."
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Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply all of secondorder Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying secondorder logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. 1
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
Success by Default? †
"... I argue that NeoFregean accounts of arithmetical language and arithmetical knowledge tacitly rely on a thesis I call [Success by Default]—the thesis that, in the absence of reasons to the contrary, we are justified in thinking that certain stipulations are successful. Since NeoFregeans have yet to ..."
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I argue that NeoFregean accounts of arithmetical language and arithmetical knowledge tacitly rely on a thesis I call [Success by Default]—the thesis that, in the absence of reasons to the contrary, we are justified in thinking that certain stipulations are successful. Since NeoFregeans have yet to supply an adequate defense of [Success by Default], I conclude that there is an important gap in NeoFregean accounts of arithmetical language and knowledge. I end the paper by offering a naturalistic remedy. 1. Stipulation We are in charge of our own linguistic behavior. Given a predicate and a set of sentences, we are free to stipulate that the predicate is to be used in such a way that each of the sentences turns out to be true. Of course, the stipulation might fail: there is, in general, no guarantee that it will have the effect of rendering its definiendum meaningful in such a way that every sentence in the set turns out to be true. Consider, for example, the stipulation that the predicate ‘... is a zapp’ is to be used in such a way that both ‘Jones is a zapp ’ and ‘Jones is not a zapp ’ turn out to be true. 1 No choice of extension for ‘... is a zapp’ would allow it to satisfy the constraint imposed by our stipulation. The stipulation is therefore unsuccessful. We attempted to endow ‘... is a zapp’ with meaning in such a way that both ‘Jones is a zapp ’ and ‘Jones is not a zapp ’ turn out to be true, and we failed. Sometimes it is up to the world to determine whether a stipulation is successful. Suppose we stipulate that the predicate ‘... is a yapp ’ is to be used in such a way that both ‘everyone in room 102 is a yapp ’ and ‘my † Many thanks to Bob Hale, Stewart Shapiro, Crispin Wright and two anonymous referees.
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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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.
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"... Bob Hale and Crispin Wright have gathered together fifteen of their papers, two of them jointly authored, on abstraction, logicism, neoFregeanism and Hume's Principle (HP): 1 #xF{x) = #xG{x) f * 3R(R maps the Fs 11 onto the Gs). ..."
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Bob Hale and Crispin Wright have gathered together fifteen of their papers, two of them jointly authored, on abstraction, logicism, neoFregeanism and Hume's Principle (HP): 1 #xF{x) = #xG{x) f * 3R(R maps the Fs 11 onto the Gs).