### Um Ceclo de Computeraçao

"... Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible ..."

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Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible before. • Many recent scientific results (e.g., in chemistry) would not have been possible without computers. • The Kepler Conjecture: no packing of congruent balls in Euclidean space has density greater than the density of the face-centered cubic packing. • Sam Ferguson and Tom Hales proved the Kepler Conjecture in 1998, but it was not published until 2006. • The Flyspeck project aims to give a formal proof of the Kepler Conjecture.

### Principia Mathematica anniversary symposiumSummary

, 2010

"... • General definition of function 1879 [14] is key to Frege’s formalisation of logic. • Self-application of functions was at the heart of Russell’s paradox 1902 [44]. • To avoid paradox Russell controled function application via type theory. • Russell [45] 1903 gives the first type theory: the Ramifi ..."

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• General definition of function 1879 [14] is key to Frege’s formalisation of logic. • Self-application of functions was at the heart of Russell’s paradox 1902 [44]. • To avoid paradox Russell controled function application via type theory. • Russell [45] 1903 gives the first type theory: the Ramified Type Theory (rtt). • rtt is used in Russell and Whitehead’s Principia Mathematica [49] 1910–1912. • Simple theory of types (stt): Ramsey [40] 1926, Hilbert and Ackermann [26] 1928. • Church’s simply typed λ-calculus λ → [11] 1940 = λ-calculus + stt. Principia Mathematica anniversary symposium 1 • The hierarchies of types (and orders) as found in rtt and stt are unsatisfactory. • The notion of function adopted in the λ-calculus is unsatisfactory (cf. [29]). • Hence, birth of different systems of functions and types, each with different functional power. • We discuss the evolution of functions and types and their use in logic, language and computation. • We then concentrate on these notions in mathematical vernaculars (as in Automath) and in logic. • Frege’s functions = Principia’s functions = λ-calculus functions (1932). • Not all functions need to be fully abstracted as in the λ-calculus. For some functions, their values are enough. Principia Mathematica anniversary symposium 2 • Non-first-class functions allow us to stay at a lower order (keeping decidability, typability, etc.) without losing the flexibility of the higher-order aspects. • Furthermore, non-first-class functions allow placing the type systems of modern theorem provers/programming languages like ML, LF and Automath more accurately in the modern formal hierarchy of types. • Another issue that we touch on is the lessons learned from formalising mathematics in logic (à la Principia) and in proof checkers (à la Automath, or any modern proof checker).

### The birth of analytic philosophy ∗

"... Analytic philosophy was, at its birth, an attempt to escape from an earlier tradition, that of Kant, and the first battleground was mathematics. Kant had claimed that mathematics is grounded neither in experience nor in logic but in the spatio-temporal structure which we ourselves impose on experien ..."

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Analytic philosophy was, at its birth, an attempt to escape from an earlier tradition, that of Kant, and the first battleground was mathematics. Kant had claimed that mathematics is grounded neither in experience nor in logic but in the spatio-temporal structure which we ourselves impose on experience. First Frege tried to refute Kant’s account in the case of arithmetic by showing that it could be derived from logic; then Russell extended the project to the whole of mathematics. Both failed, but in addressing the problems which the project generated they founded what is nowadays known as analytic philosophy or, perhaps more appropriately, as the analytic method in philosophy. What this brief summary masks, however, is that it is far from easy to say what the analytic method in philosophy amounts to. By tracing the outlines of the moment when it was born we shall here try to identify some of its distinctive features.

### Groundedness IV: Grounded Classes

, 2012

"... • The full class comprehension schema leads to paradox: witness ‘ yηy’. It is consistent, if restricted to formulae φ which do not speak of classes – but we want more. – For example, if our goal is semantics for a language with absolute generality [Linnebo, 2006]. • This situation resembles the prob ..."

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• The full class comprehension schema leads to paradox: witness ‘ yηy’. It is consistent, if restricted to formulae φ which do not speak of classes – but we want more. – For example, if our goal is semantics for a language with absolute generality [Linnebo, 2006]. • This situation resembles the problem of finding the right instances of the T-schema. We saw how to identify its grounded instances – how about an analogous response to the class-theoretic paradoxes? • Restrict class comprehension to grounded formulae – below, we will consider different ways of spelling out this idea. Remark 1. Any such attempt faces the objection that semantical and class-theoretic paradoxes are importantly different [Frank P. Ramsey, 1931], and ought not to be treated alike. In response, we may connect with the view that unlike sets, classes are individuated by conditions (see remark 7 below). Therefore, we may argue, the question of what

### Draft as of 1/26/05

"... Conceptual role semantics (CRS) is the view that the meanings of expressions of a language (or other symbol system) or the contents of mental states are determined or explained by the role of the expressions or mental states in thinking. The theory can be taken to be applicable to language in the or ..."

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Conceptual role semantics (CRS) is the view that the meanings of expressions of a language (or other symbol system) or the contents of mental states are determined or explained by the role of the expressions or mental states in thinking. The theory can be taken to be applicable to language in the ordinary sense, to mental representations, conceived of either as symbols in a “language of thought ” or as mental states such as beliefs, or to certain other sorts of symbol systems. CRS rejects the competing idea that thoughts have intrinsic content that is prior to the use of concepts in thought. According to CRS, meaning and content derive from use, not the other way round. CRS is thus an attempt to answer the question of what determines or makes it the case that representations have particular meanings or contents. The significance of this question can be seen by considering, for example, theories of mind that postulate a language of thought. Such theories presuppose an account of what makes it the case that a symbol in the language of thought has a particular meaning. Some conceptual role theorists

### unknown title

"... Hochberg presuppose the existence of facts when they respond to Bradley’s regress – in the former case allowing for the possibility of infinitely many facts, in the latter case assuming that there are facts that fall within the range of the relevant description operator – (“ip”) – both Russell and ..."

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Hochberg presuppose the existence of facts when they respond to Bradley’s regress – in the former case allowing for the possibility of infinitely many facts, in the latter case assuming that there are facts that fall within the range of the relevant description operator – (“ip”) – both Russell and Hochberg fail to resolve the difficulty for relations that Bradley identified. Of course this concern constitutes no more than a challenge to what Hochberg has said. The text of Appearance and Reality no doubt admits of several interpretations. And independently of historical details, it is notoriously difficult to pin down the structure or adjudicate upon the significance of regress arguments. Nevertheless, Hochberg’s treatment of Bradley’s argument and the counter I have suggested serves to typify what so often occurs when one engages with the details of Hochberg’s arguments. Not only does one discover a novel and distinctive perspective on themes that are perhaps all too familiar. One is also driven back to consider afresh the basic ontological problems that inspired the early analytic philosophers.*

### Minimalism, Deflationism, and Paradoxes ∗

, 2009

"... This paper argues against a broad category of deflationist theories of truth. It does so by asking two seemingly unrelated questions. The first is about the well-known logical and semantic paradoxes: Why is there no strengthened version of Russell’s paradox, as there is a strengthened version of the ..."

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This paper argues against a broad category of deflationist theories of truth. It does so by asking two seemingly unrelated questions. The first is about the well-known logical and semantic paradoxes: Why is there no strengthened version of Russell’s paradox, as there is a strengthened version of the Liar paradox? Oddly, this question is rarely asked. It does have a fairly standard answer, which I shall not dispute for purposes of this paper. But I shall argue that asking it ultimately leads to a fundamental challenge to some popular versions of deflationism. The challenge comes about by pairing this question with a second question: What is the theory of truth about? For many theorists, there is an obvious answer to this question: the theory of truth is about truth bearers and what makes them true. But this answer appears to bring with it a commitment to a substantial notion of truth, which deflationists cannot bear. Deflationists might prefer a very different answer: the theory of truth is not really about anything. There is no substantial property of truth, so there is no domain which the theory of truth properly describes. Not all positions under the name ‘deflationism ’ subscribe to this view, but I shall argue that the important class of so-called minimalist views do. I shall argue that this sort of deflationist answer is untenable, and thus argue in broad strokes against minimalism. I shall argue by way of a comparison of the theory of truth with the theory of sets, and consideration of where paradoxes, especially strengthened versions of the paradoxes, may arise in each. This will bring the two seemingly unrelated questions together to form an antideflationist argument. I shall show that deflationist positions that accept the idea that truth is This is a revised and expanded version of my “Minimalism and Paradoxes ” Synthese 135 (2003): 13–36. Thanks