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  1. added 2018-10-10
    Induction and Predicativity.Peter Smith - unknown
    I am interested in the philosophical prospects of what is called ‘predicativism given the natural numbers’. And today, in particular, I want to critically discuss one argument that has been offered to suggest that this kind of predicativism can’t have a stable philosophical motivation. Actually you don’t really need to know about predicativism to find some stand-alone interest in the theme I will be discussing. But still, it’s worth putting things into context. So I’m going to start by spending a (...)
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  2. added 2018-09-20
    Exploring Predicativity.Laura Crosilla - 2018 - In Proof and Computation. pp. 83-108.
    Prominent constructive theories of sets as Martin-Löf type theory and Aczel and Myhill constructive set theory, feature a distinctive form of constructivity: predicativity. This may be phrased as a constructibility requirement for sets, which ought to be finitely specifiable in terms of some uncontroversial initial “objects” and simple operations over them. Predicativity emerged at the beginning of the 20th century as a fundamental component of an influential analysis of the paradoxes by Poincaré and Russell. According to this analysis the paradoxes (...)
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  3. added 2018-09-20
    Predicativity and Feferman.Laura Crosilla - 2017 - In Feferman on Foundations. pp. 423-447.
    Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell's type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predica-tivity have since taken a life of their own, (...)
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  4. added 2018-09-04
    Predicativity and Structuralism in Dedekind’s Construction of the Reals.Audrey Yap - 2009 - Erkenntnis 71 (2):157-173.
    It is a commonly held view that Dedekind's construction of the real numbers is impredicative. This naturally raises the question of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind's philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects of the solution: one is to look more closely at his methodological views, and in particular, the places in which (...)
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  5. added 2018-02-17
    Cantor’s Proof in the Full Definable Universe.Laureano Luna & William Taylor - 2010 - Australasian Journal of Logic 9:10-25.
    Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...)
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  6. added 2017-11-04
    Predicativity.Solomon Feferman - 2005 - In Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press. pp. 590-624.
    What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...)
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  7. added 2017-01-20
    Predicative Arithmetic.Edward Nelson - 1986 - Princeton University Press.
  8. added 2017-01-19
    Challenges to Predicative Foundations of Arithmetic.Solomon Feferman - manuscript
    This is a sequel to our article “Predicative foundations of arithmetic” (1995), referred to in the following as [PFA]; here we review and clarify what was accomplished in [PFA], present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by [PFA] was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George and Daniel (...)
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  9. added 2017-01-15
    Logic in the Tractatus.Max Weiss - 2017 - Review of Symbolic Logic 10 (1):1-50.
    I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably (...)
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  10. added 2016-02-25
    The Strength of Abstraction with Predicative Comprehension.Sean Walsh - 2016 - Bulletin of Symbolic Logic 22 (1):105–120.
    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 (...)
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  11. added 2015-06-18
    On the Self-Predicative Universals of Category Theory.David Ellerman - manuscript
    This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...)
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  12. added 2015-03-23
    Predicative Frege Arithmetic and ‘Everyday’ Mathematics.Richard G. Heck - 2014 - Philosophia Mathematica 22 (3):279-307.
    The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.
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  13. added 2015-03-17
    Predicative Foundations of Arithmetic.Solomon Feferman & Geoffrey Hellman - 1995 - Journal of Philosophical Logic 24 (1):1 - 17.
  14. added 2014-04-01
    The Consistency of Predicative Fragments of Frege's Grundgesetze der Arithmetik.Richard Heck - 1996 - History and Philosophy of Logic 17 (1):209-220.
    As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell?s Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege?s Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is (...)
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  15. added 2014-03-23
    On the Untenability of Nelson's Predicativism.St Iwan - 2000 - Erkenntnis 53 (1-2):147-154.
    By combining some technical results from metamathematicalinvestigations of systems of Bounded Arithmetic, I will givean argument for the untenability of Nelson 's finitistic program,encapsulated in his book Predicative Arithmetic.
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  16. added 2014-03-23
    Proof-Theoretic Reduction as a Philosopher's Tool.Thomas Hofweber - 2000 - Erkenntnis 53 (1-2):127-146.
    Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert (...)
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  17. added 2014-03-21
    Predicative Fragments of Frege Arithmetic.Øystein Linnebo - 2004 - Bulletin of Symbolic Logic 10 (2):153-174.
    Frege Arithmetic (FA) is the second-order theory whose sole non-logical 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 one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order 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 second-order logic—and (...)
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  18. added 2014-03-04
    Ramified Frege Arithmetic.Richard Heck - 2011 - Journal of Philosophical Logic 40 (6):715-735.
    Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.
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  19. added 2011-08-16
    On Constructive Interpretation of Predicative Mathematics.Charles Parsons - 1990 - Garland.
  20. added 2011-06-18
    Relationships Between Constructive, Predicative and Classical Systems of Analysis.Solomon Feferman - unknown
    Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative" de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to (...)
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  21. added 2011-05-31
    A Defence of Predicativism as a Philosophy of Mathematics.Tim Storer - unknown
    A specification of a mathematical object is impredicative if it essentially involves quantification over a domain which includes the object being specified (or sets which contain that object, or similar). The basic worry is that we have no non-circular way of understanding such a specification. Predicativism is the view that mathematics should be limited to the study of objects which can be specified predicatively. There are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. (...)
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  22. added 2009-04-07
    Review of S. Feferman's in the Light of Logic. [REVIEW]Andrew Arana - 2005 - Mathematical Intelligencer 27 (4).
    We review Solomon Feferman's 1998 essay collection In The Light of Logic (Oxford University Press).
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  23. added 2008-12-31
    Predicativism as a Philosophical Position.Geoffrey Hellman - 2004 - Revue Internationale de Philosophie 3:295-312.
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