Truth and probability in game-theoretical semantics

Journal of Philosophical Logic, 1974

We show that both Hilbert's and Brouwer's interpretations of quantification yield models of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The former yields the standard interpretation M of PA over N, which is relatively sound with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under M; and which can be viewed as circumscribing the ambit of non-finitary human reasoning about `true' arithmetical propositions. The latter yields a finitary interpretation B of PA over N, which is relatively sound with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under B; and which can be viewed as circumscribing the ambit of finitary mechanistic reasoning about `true' arithmetical propositions.

2006

of accuracy. On the other hand, the constructivist claims that to know that a particular sequence of rational numbers converges is to know how to approximate the corresponding real number to an arbitrarily given degree of accuracy. Hence the constructivist rejects the law of excluded middle. Using the law of excluded middle one can also argue that every sequence of rational numbers that does not converge must diverge. However, one can not from the knowledge that a particular sequence of rational numbers does not converge construct the corresponding witness. The example just given illustrates the direct nature of constructive existence, as opposed to the indirect nature of classical existence, and shows why the law of excluded middle is not accepted as a law of constructive logic. Finally, I would like to thank my supervisor, Per Martin-Löf, for posing the problem of investigating how the double-negation interpretation operates on derivations and not only on formulas as well as for his continued guidance of my work. Without him, this thesis would never have come into existence. Jens Brage Reynolds (1972), and Plotkin (1975) for the foundations of CPS translations and Reynolds (1993) for the early history of continuations. To my knowledge there is no interpretation of classical logic in constructive logic that makes full use of the syntactic-semantic method of constructive type theory. With this thesis I hope to fill this gap. The subject of interpretations of classical logic in constructive logic began with the double negation interpretation of classical logic in minimal logic due to Kolmogorov (1925). The double negation interpretation was then followed by the interpretation of Peano arithmetic in Heyting arithmetic due to Gödel (1933) and the interpretation of classical logic in intuitionistic logic due to Kuroda (1951). Yet it was not until Griffin (1990) showed how to extend the formulae-as-types correspondence to classical logic that significant growth took place. His solution was to include operations on the flow of control, similar to call/cc of Scheme, into the notion of computation given by a simply typed call-by-value λ-calculus. After that Parigot (1992) introduced his λµ-calculus to realize classical proofs as programs. The λµ-calculus extended the simply typed λ-calculus with operators that can be used to model operations on the flow of control. The development then took the form of CPS translations of different λµ-calculi into different λ-calculi. See Ong (1996) and Ong and Stewart (1997) for call-by-value respectively call-byname CPS translations of Parigot's λµ-calculus into the simply typed λ-calculus. See Selinger (2001, p. 24) for an informal description of the semantics of the λµcalculus.

Synthese, 2000

The theory that "consistency implies existence" was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.

This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite" in mathematics from proofs of purely nitary statements. One of the main approaches that turned out to be the most useful in pursuit of this program was that due to Gerhard Gentzen, in the 1930s, via his calculi of \sequents" and his Cut-Elimination Theorem for them. Following that we t r a c e how and why prima facie in nitary concepts, such as ordinals, and in nitary methods, such as the use of in nitely long proofs, gradually came to dominate proof-theoretical developments.

2015

I would first like to thank my advisor David DeVidi, who both encouraged and, when necessary, restrained my efforts, and whose own work inspired so much of the subject matter of this thesis. I would also like to thank my committee, from the Waterloo philosophy department, Tim Kenyon, Doreen Frazer and especially Gerry Callaghan, who valiantly stepped in at the last minute; Steve Furino, from the Waterloo faculty of mathematics; and my external examiner Greg Lavers for his careful reading and helpful comments. In addition, I must thank all the Waterloo Philosophy Department Faculty and Staff. Especially Debbie Detrich who from my first arrival at UW has advised me, and helped me navigate the UW system, looking out for me as she does all the graduate students in the Philosophy Department, we are very lucky to have her. I should also thank my graduate colleagues at Waterloo, who created such a wonderful and welcoming environment. Especially my year's Ph.D. cohort: Mike McEwan and Eric Hochstein. They made living in Waterloo tolerable with their friendship, discussions in various uptown establishments, and the Saturday breakfasts that followed. In a similar vein I would be remiss if I did not thank the Univeristy of Western Ontario philosophers, and Ivey folk, who made living in London bearable. In addition to those mentioned above, I would also like to thank the people who I encountered when I started my journey in philosophy, at the University of Ottawa-especially since I forgot to include an acknowledgement's section in my master's thesis. Thanks, first, to Mathieu Marion, my master's supervisor, who first introduced me to non-classical logics; and to Sophie Rietti and Patrice Philie, friends and now colleagues at Ottawa, for their friendship and insight. Thanks also are due to my former graduate student colleagues, in particular: Dan McArthur, who encouraged me to return to academia; Mark Young, whose friendship and loyalty can only be described as supererogatory; Shane Ralston, who I am proud to say made it through his Ph.D. at Ottawa without me getting him arrested once; and Dean Lauer whose wonderful companionship kept us out late so many nights, often until closing hours. Thanks to my wonderful gigantic family, both the Rowes and the Mulvihills, and now, since I made the best decision of my life and married my wonderful wife, the Mazutises and Trences. Please know that I am aware of how lucky I am to have so many wonderful and supportive people in my life. A special thank you is due to my amazing parents, Bob and Helen, who have supported me through the years, both morally and financially, and to my sister, Shauna Mulvihill, whom the fates have punished with me as a sibling. Finally, to my brilliant and beautiful wife Daina Mazutis and my daughter Emilija Mulvihill; I know I could not have finished this without you Daina. Thank you for your support and the most wonderful daughter in the world. Thank you all. vii Conclusion 181 7 Conclusion 183

The study of the foundations of mathematics is in a crisis: the failure of logicism has invited a satisfactory alternative. This should be a better proof theory; Gentzen's natural deduction came to provide this but had a limited success. The discussion on this practically faded out. Solomon Feferman and his coworkers propose tentatively a synthesis that relies on achievements from diverse directions. Presenting sympathetically ideas of Hilbert, Brouwer, Gentzen, Weyl, and Bishop, they hope to change in the standard textbooks of logic. This requires a more problem-oriented presentatlon of the background material that should make the situation more accessible to the ordinary philosophy faculty. 'Ihk invites supplementary discussions of achievements that they are thus far sadly overlooked, especially those of Popper, Robinson, and Lakatos.

Abstract: The thesis that any mathematical proof can be formalized was advocated by (among others) Hilbert and Gentzen around a century ago. The Hilbert-Gentzen thesis says: “Every real proof can be represented by a formal proof.”[1] This article will go through several possible interpretations of this thesis, to be able to understand and discuss it fairly. To be able to fully understand the reasoning that separates the different interpretations, there will be a summarisation of the history and arguments for formalisation of mathematics, and a discussion on the different arguments for why formalisation has the status it has within mathematics. The H-G thesis has held a firm place in the foundations and philosophy of mathematics since these thoughts were first expressed a century ago. However the recent “informal proofs debate” gives us several arguments against the thesis, arguing that it is false and it should be rejected. This article will (1) define the thesis with help from the context in which it was introduced, (2) give the main arguments for accepting it, (3) give the main arguments for refuting it, using arguments from M. Marfori, C. Cellucci, K. Gödel, B. Larvor, and F. Tanswell, and then (4) evaluate the strength and scope of the arguments and counter arguments. This will be a summarization and discussion on the current status of the Hilbert-Gentzen thesis, and the conclusion will be that the thesis has overestimated the value of formalization and that the thesis should be rejected. Keywords: Hilbert-Gentzen thesis, mathematical proofs, real proofs, informal proofs, derivationism, formalisation [1] Cellucci, C. (2008) “Why proof? What is a proof?” Deduction, Computation, Experiment. Exploring the Effectiveness of Proof, 1-27

Proceedings of the 2008 International Conference on …, 2008