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Profile: Richard Zach (University of Calgary)
  1. Richard Zach, Gödel’s First Incompleteness Theorem and Mathematical Instrumentalism.
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  2. Phil Serchuk, Ian Hargreaves & Richard Zach (2011). Vagueness, Logic and Use: Four Experimental Studies on Vagueness. Mind and Language 26 (5):540-573.
    Although arguments for and against competing theories of vagueness often appeal to claims about the use of vague predicates by ordinary speakers, such claims are rarely tested. An exception is Bonini et al. (1999), who report empirical results on the use of vague predicates by Italian speakers, and take the results to count in favor of epistemicism. Yet several methodological difficulties mar their experiments; we outline these problems and devise revised experiments that do not show the same results. We then (...)
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  3. Aldo Antonelli, Alasdair Urquhart & Richard Zach (2008). Mathematical Methods in Philosophy Editors' Introduction. Review of Symbolic Logic 1 (2):143-145.
  4. Jeremy Avigad & Richard Zach, The Epsilon Calculus. Stanford Encyclopedia of Philosophy.
    The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term..
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  5. Paolo Mancosu, Richard Zach & Calixto Badesa (2008). The Development of Mathematical Logic From Russell to Tarski, 1900-1935. In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.
    The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...)
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  6. Richard Zach, Matthias Baaz & Norbert Preining (2007). First-Order Gödel Logics. Annals of Pure and Applied Logic 147 (1):23-47.
    First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete (...)
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  7. Georg Moser & Richard Zach (2006). The Epsilon Calculus and Herbrand Complexity. Studia Logica 82 (1):133 - 155.
    Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator ex. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential (...)
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  8. Richard Zach (2006). Hilbert's Program Then and Now. In Dale Jacquette (ed.), Philosophy of Logic. North Holland. 5--411.
  9. Richard Zach (2005). Book Review: Michael Potter. Reason's Nearest Kin. Philosophies of Arithmetic From Kant to Carnap. [REVIEW] Notre Dame Journal of Formal Logic 46 (4):503-513.
  10. Richard Zach (2005). Critical Study of Michael Potter’s Reason’s Nearest Kin. Notre Dame Journal of Formal Logic 46:503-513.
    Critical study of Michael Potter, Reason's Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. x + 305 pages.
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  11. Richard Zach (2004). Hilbert's 'Verunglückter Beweis', the First Epsilon Theorem, and Consistency Proofs. History and Philosophy of Logic 25 (2):79-94.
    In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain ?general consistency result? due to Bernays. An analysis of the form of this so-called ?failed proof? (...)
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  12. Richard Zach (2004). Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤Ω. Journal of Philosophical Logic 33 (2):155-164.
    Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most ω, the resulting logics are decidable. (...)
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  13. Richard Zach (2004). Le quantificateur effini, la descente infinie et les preuves de consistance de Gauthier. Philosophiques 31 (1):221-224.
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  14. Richard Zach (2003). Review of G. S. Boolos, J. P. Burgess, R. C. Jeffrey, Computability and Logic. [REVIEW] Bulletin of Symbolic Logic 9 (4):520-520.
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  15. Richard Zach (2003). Boolos George S., Burgess John P., and Jeffrey Richard C.. Computability and Logic, Cambridge University Press, Cambridge, 2002. Xi+ 356 Pp. [REVIEW] Bulletin of Symbolic Logic 9 (4):520-521.
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  16. Richard Zach, Hilbert's Program. Stanford Encyclopedia of Philosophy.
    In the early 1920s, the German mathematician David Hilbert (1862-1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...)
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  17. Richard Zach (2003). The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program. Synthese 137 (1-2):211 - 259.
    After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...)
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  18. Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach (2001). 2000-2001 Spring Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 7 (3).
     
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  19. Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach (2001). The Minneapolis Hyatt Regency, Minneapolis, Minnesota May 3–4, 2001. Bulletin of Symbolic Logic 7 (3).
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  20. Richard Zach (1999). Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic. Bulletin of Symbolic Logic 5 (3):331-366.
    Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...)
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  21. Matthias Baaz, Christian G. Fermüller, Gernot Salzer & Richard Zach (1998). Labeled Calculi and Finite-Valued Logics. Studia Logica 61 (1):7-33.
    A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number (...)
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  22. Matthias Baaz & Richard Zach (1998). Note on Generalizing Theorems in Algebraically Closed Fields. Archive for Mathematical Logic 37 (5-6):297-307.
    The generalization properties of algebraically closed fields $ACF_p$ of characteristic $p > 0$ and $ACF_0$ of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that $ACF_p$ admits finite term bases, and $ACF_0$ admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some $k$ , $A(1 + \cdots + 1)$ ( $n$ 1's) is provable in $k$ steps, then $(\forall x)A(x)$ is provable.
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  23. Matthias Baaz & Richard Zach (1995). Generalizing Theorems in Real Closed Fields. Annals of Pure and Applied Logic 75 (1-2):3-23.
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  24. Petr Hajek & Richard Zach (1994). Review of Leonard Bole and Piotr Borowik: Many-Valued Logics: 1. Theoretical Foundations, Berlin: Springer, 1991. [REVIEW] Journal of Applied Non-Classical Logics 4 (2):215-220.
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