Abstract
As a celebration of the Tractatus 100th anniversary it might be worth revisiting its relation to the later writings. From the former to the latter, David Pears recalls that “everyone is aware of the holistic character of Wittgenstein’s later philosophy, but it is not so well known that it was already beginning to establish itself in the Tractatus” (The False Prison, 1987). From the latter to the former, Stephen Hilmy’s (The Later Wittgenstein, 1987) extensive study of the Nachlass has helped removing classical misconceptions such as Hintikka’s claim that “Wittgenstein in the Philosophical Investigations almost completely gave up the calculus analogy.” Hilmy points out that even in the Investigations one finds the use of the calculus/game paradigm to the understanding of language, such as “in operating with the word” (Part I, §559) and “it plays a different part in the calculus”. Hilmy also quotes from a late (1946) unpublished manuscript (MS 130) “this sentence has use in the calculus of language”, which seems to be compatible with “asking whether and how a proposition can be verified is only a particular way of asking ‘How do you mean?’” Central in this back and forth there is an aspect which seems to deserve attention in the discussion of a semantics for the language of mathematics which might be based on (normalisation of) proofs and/or Hintikka/Lorenzen game-dialogue: the explication of consequences. Such a discussion is substantially supported by the use of the open and searchable The Wittgenstein Archives at the University of Bergen. These findings are framed within the discussion of the meaning of logical constants in the context of natural deduction style rules of inference.
Acknowledgements
First and foremost, we acknowledge the role of Anderson Nakano (PUC-SP) and Marcos Silva (UFPE), the organisers of the online meeting 100 anos do Tractatus Logico-Philosophicus (14–17 Sept 2021), who kindly accepted the offer of a talk. This served as additional stimulus to write this paper, which, in turn led us to new findings in Wittgenstein’s Nachlass. Equally important was the very careful and very positive work done by the anonymous reviewer who has significantly contributed to improve the text.
References
Dummett, M. 1977. Elements of Intuitionism. Oxford: Clarendon Press.Search in Google Scholar
Dummett, M. 1991. The Logical Basis of Metaphysics. Cambridge: Harvard University Press.Search in Google Scholar
Gabbay, D. M., and R. J. G. B. de Queiroz. 1992. “Extending the Curry-Howard Interpretation to Linear, Relevant and Other Resource Logics.” The Journal of Symbolic Logic 57 (4): 1319–65. https://doi.org/10.2307/2275370.Search in Google Scholar
Gentzen, G. 1935. “Untersuchungen über das logische Schließen. I.” Mathematische Zeitschrift 39: 176–210. https://doi.org/10.1007/BF01201353.Search in Google Scholar
Henkin, L. 1961. “Some Remarks on Infinitely Long Formulas.” In Infinitistic Methods (Proc. Sympos. Foundations of Math.), 167–83. Warsaw: Pergamon, Oxford.Search in Google Scholar
Hermes, H. 1959. “Zum Inversionsprinzip der operativen Logik.” In Constructivity in Mathematics, edited by A. Heyting, 62–8. Amsterdam: North-Holland.Search in Google Scholar
Hilmy, S. 1987. The Later Wittgenstein: The Emergence of a New Philosophical Method. Oxford: Blackwell.Search in Google Scholar
Hintikka, J. 1968. “Language-games for Quantifiers.” In Studies in Logical Theory, edited by N. Rescher, 46–72. Oxford: Blackwell.Search in Google Scholar
Hintikka, J. 1979. “Quantifiers vs. Quantification Theory.” In Game-Theoretical Semantics. Synthese Language Library, Vol. 5, edited by E. Saarinen. Dordrecht: Springer.Search in Google Scholar
Hintikka, M., and J. Hintikka. 1986. Investigating Wittgenstein. Oxford: Basil Blackwell.Search in Google Scholar
Hodges, W. 2001. “A Sceptical Look.” Proceedings of the Aristotelian Society Supplementary 75 (1): 17–32. https://doi.org/10.1111/1467-8349.00076.Search in Google Scholar
Howard, W. 1980. “The Formulae-As-Types Notion of Construction.” In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism, edited by H. Curry, J. R. Hindley, and J. Seldin. Cambridge: Academic Press.Search in Google Scholar
Howard, W. 2014. Wadler’s Blog. Also available at https://wadler.blogspot.com/2014/08/howard-on-curry-howard.html.Search in Google Scholar
Keiff, L. 2009. “Dialogical Logic.” In Stanford Encyclopedia of Philosophy. Stanford. Also available at https://plato.stanford.edu/entries/logic-dialogical/.Search in Google Scholar
Krabbe, E. 2001. “Dialogue Logic Restituted.” In Dialogue Foundations, edited by W. Hodges, and E. C. W. Krabbe, 33–49. Hoboken: Wiley.10.1111/1467-8349.00077Search in Google Scholar
Lorenzen, P. 1950. “Konstruktive Begr’́undung der Mathematik.” Mathematische Zeitschrift 53 (2): 162–202. https://doi.org/10.1007/bf01162411.Search in Google Scholar
Lorenzen, P. 1955. Einführung in die operative Logik und Mathematik. Die Grundlehren der mathematischen Wissenschaften, Vol. 78, VII + 302. Berlin: Springer-Verlag.10.1007/978-3-662-01539-1Search in Google Scholar
Lorenzen, P. 1969. Normative Logic and Ethics, Series B.I-Hochschultaschenbücher. Systematische Philosophie, Vol. 236*. Mannheim: Bibliographisches Institut.Search in Google Scholar
Martin-Löf, P. 1984. Intuitionistic Type Theory. (Notes taken by G. Sambin). Bibliopolis, Napoli.Search in Google Scholar
Martin-Löf, P. 1987. “Truth of a Proposition, Evidence of a Judgement, Validity of a Proof.” Synthese 73: 407–20. https://doi.org/10.1007/bf00484985.Search in Google Scholar
Martin-Löf, P. 2013. “Verificationism Then and Now.” In Judgement and the Epistemic Foundation of Logic, Logic, Epistemology, and the Unity of Science, Vol. 31. Chapter 1, edited by M. van der Schaar. Springer.10.1007/978-94-007-5137-8_1Search in Google Scholar
Martin-Löf, P. 2019. “Logic and Ethics.” In Proof-Theoretic Semantics: Assessment and Future Perspectives. Proceedings of the Third Tübingen Conference on Proof-Theoretic Semantics, 27–30 March 2019, edited by T. Piecha, and P. Schroeder-Heister, 227–35. Univ Tübingen.Search in Google Scholar
Martínez-Rivillas, D. O. 2022. “Towards a Homotopy Domain Theory.” PhD thesis. Recife: CIn-UFPE (November 2022). Centro de Informática, Universidade Federal de Pernambuco. . Also available at https://repositorio.ufpe.br/handle/123456789/49221.Search in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2022a. “∞-Groupoid Generated by an Arbitrary Topological λ-Model.” Logic Journal of IGPL 30 (3): 465–88. https://doi.org/10.1093/jigpal/jzab015.Search in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2022b. “Towards a Homotopy Domain Theory.” Archive for Mathematical Logic. https://doi.org/10.1007/s00153-022-00856-0.Search in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2023a. “The Theory of an Arbitrary Higher λ-Model.” To appear in Bulletin of the Section of Logic. arXiv:2111.07092.10.18778/0138-0680.2023.11Search in Google Scholar
Martínez-Rivillas, D. O., and de Queiroz, R. J. G. B. 2023b. “Solving Homotopy Domain Equations”. (Submitted for publication.) arXiv:2104.01195Search in Google Scholar
Moore, G. E. 1955. “Wittgenstein’s Lectures in 1930–33.” Mind 54 (253): 1–27. https://doi.org/10.1093/mind/lxiv.253.1.Search in Google Scholar
de Oliveira, A. G., and R. J. G. B. de Queiroz. 1999. “A Normalization Procedure for the Equational Fragment of Labelled Natural Deduction.” Logic Journal of IGPL 7 (2): 173–215. https://doi.org/10.1093/jigpal/7.2.173.Search in Google Scholar
de Oliveira, A. G., and R. J. G. B. de Queiroz. 2005. “A New Basic Set of Proof Transformations.” In We Will Show Them! Essays in Honour of Dov Gabbay, Vol. 2, edited by S. Artemov, H. Barringer, A. Garcez, L. Lamb, and J. Woods, 499–528. London: College Publications.Search in Google Scholar
Pears, D. 1987. The False Prison. A Study of the Development of Wittgenstein’s Philosophy, Vol. I. Oxford: Clarendon Press.Search in Google Scholar
Pears, D. 1990. “Wittgenstein’s Holism.” Dialectica 44 (2): 165–73. https://doi.org/10.1111/j.1746-8361.1990.tb01657.x.Search in Google Scholar
Peirce, C. S. 1932. Collected Papers of Charles Sanders Peirce, Volumes I and II: Principles of Philosophy and Elements of Logic, edited by C. Hartshorne, and P. Weiss. Cambridge: Harvard Univ. Press.Search in Google Scholar
Pietarinen, A. V. 2014. “Logical and Linguistic Games from Peirce to Grice to Hintikka.” Teorema 33 (2): 121–36.Search in Google Scholar
von Plato, J. 2017. Saved from the Cellar. Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics. Berlin: Springer.10.1007/978-3-319-42120-9Search in Google Scholar
Prawitz, D. 1965. Natural Deduction: A Proof-Theoretical Study. Stockholm: Acta Universitatis Stockholmiensis, Stockholm Studies in Philosophy No. 3. Almqvist & Wiksell.Search in Google Scholar
Prawitz, D. 1977. “Meaning and Proofs: On the Conflict between Classical and Intuitionistic Logic.” Theoria XLIII: 2–40. https://doi.org/10.1111/j.1755-2567.1977.tb00776.x.Search in Google Scholar
Prawitz, D. 2019. “Validity of Inferences Reconsidered.” In Proof-Theoretic Semantics: Assessment and Future Perspectives. Proceedings of the Third Tübingen Conference on Proof-Theoretic Semantics, 27–30 March 2019, edited by T. Piecha, and P. Schroeder-Heister, 213–26. Tübingen: Univ. Tübingen.Search in Google Scholar
de Queiroz, R. J. G. B. 1987. “Note on Frege’s Notions of Definition and the Relationship Proof Theory vs. Recursion Theory (Extended Abstract).” In Abstracts of the VIIIth International Congress of Logic, Methodology and Philosophy of Science. Vol. 5, 69–73. Moscow: Part I, Institute of Philosophy of the Academy of Sciences of the USSR.Search in Google Scholar
de Queiroz, R. J. G. B. 1988a. “A Proof-Theoretic Account of Programming and the Role of Reduction Rules.” Dialectica 42 (4): 265–82. https://doi.org/10.1111/j.1746-8361.1988.tb00919.x.Search in Google Scholar
de Queiroz, R. J. G. B. 1988b. “The Mathematical Language and its Semantics: To Show the Consequences of a Proposition Is to Give its Meaning.” In Reports of the Thirteenth International Wittgenstein Symposium 18, edited by P. Weingartner, G. Schurz, E. Leinfellner, R. Haller, and A. Hübner, 259–66.Search in Google Scholar
de Queiroz, R. J. G. B. 1989. “Meaning, Function, Purpose, Usefulness, Consequences – Interconnected Concepts (Abstract).” In Abstracts of Fourteenth International Wittgenstein Symposium (Centenary Celebration, 1989). Symposium held in Kirchberg/Wechsel, 20.Search in Google Scholar
de Queiroz, R. J. G. B. 1991. “Meaning as Grammar Plus Consequences.” Dialectica 45 (1): 83–6. https://doi.org/10.1111/j.1746-8361.1991.tb00979.x.Search in Google Scholar
de Queiroz, R. J. G. B. 1992. “Grundgesetze Alongside Begriffsschrift (Abstract).” In Abstracts of Fifteenth International Wittgenstein Symposium, 1992. Symposium held in Kirchberg/Wechsel, 15–6.Search in Google Scholar
de Queiroz, R. J. G. B. 1994. “Normalisation and Language Games.” Dialectica 48 (2): 83–123. https://doi.org/10.1111/j.1746-8361.1994.tb00107.x.Search in Google Scholar
de Queiroz, R. J. G. B. 2001. “Meaning, Function, Purpose, Usefulness, Consequences – Interconnected Concepts.” Logic Journal of IGPL 9 (5): 693–734. https://doi.org/10.1093/jigpal/9.5.693.Search in Google Scholar
de Queiroz, R. J. G. B. 2008. “On Reduction Rules, Meaning-As-Use, and Proof-Theoretic Semantics.” Studia Logica 90: 211–47. https://doi.org/10.1007/s11225-008-9150-5.Search in Google Scholar
de Queiroz, R. J. G. B., and D. M. Gabbay. 1994. “Equality in Labelled Deductive Systems and the Functional Interpretation of Propositional Equality.” In Proceedings of the 9th Amsterdam Colloquium 1994, edited by P. Dekker, and M. Stokhof, 547–6. ILLC/Department of Philosophy, University of Amsterdam.Search in Google Scholar
de Queiroz, R. J. G. B., and D. M. Gabbay. 1995. “The Functional Interpretation of the Existential Quantifier.” Bull of the IGPL 3 (2–3): 243–90. https://doi.org/10.1093/jigpal/3.2-3.243.Search in Google Scholar
de Queiroz, R. J. G. B., and Gabbay, D. M. 1997. “The Functional Interpretation of Modal Necessity”. In Advances in Intensional Logic, edited by M. de Rijke, 61–91. Applied Logic Series, Kluwer.10.1007/978-94-015-8879-9_3Search in Google Scholar
de Queiroz, R. J. G. B., and D. M. Gabbay. 1999. “Labelled Natural Deduction.” In Logic, Language and Reasoning. Trends in Logic, Vol. 5, edited by H. J. Ohlbach, and U. Reyle. Dordrecht: Springer.10.1007/978-94-011-4574-9_10Search in Google Scholar
de Queiroz, R. J. G. B., and T. S. E. Maibaum. 1990. “Proof Theory and Computer Programming.” Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36: 389–414. https://doi.org/10.1002/malq.19900360505.Search in Google Scholar
de Queiroz, R. J. G. B., and T. S. E. Maibaum. 1991. “Abstract Data Types and Type Theory: Theories as Types.” Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 37: 149–66. https://doi.org/10.1002/malq.19910370904.Search in Google Scholar
de Queiroz, R. J. G. B., and A. G. de Oliveira. 2011. “The Functional Interpretation of Direct Computations.” Electronic Notes in Theoretical Computer Science 269: 19–40. https://doi.org/10.1016/j.entcs.2011.03.003.Search in Google Scholar
de Queiroz, R. J. G. B., and A. G. de Oliveira. 2014. “Natural Deduction for Equality: The Missing Entity.” In Advances in Natural Deduction. Pages 63–91. Trends in Logic, Vol. 39, edited by L. Pereira, E. Haeusler, and V. de Paiva. Dordrecht: Springer.10.1007/978-94-007-7548-0_4Search in Google Scholar
de Queiroz, R. J. G. B., A. G. de Oliveira, and D. M. Gabbay. 2011. The Functional Interpretation of Logical Deduction. Vol. 5 of Advances in Logic Series. Singapore: Imperial College Press/World Scientific.10.1142/8215Search in Google Scholar
de Queiroz, R. J. G. B., A. G. de Oliveira, and A. F. Ramos. 2016. “Propositional Equality, Identity Types, and Computational Paths.” South American Journal of Logic 2 (2): 245–96.Search in Google Scholar
Ramos, A. F. 2018. “Explicit Computational Paths in Type Theory.” PhD thesis. Recife: CIn-UFPE (August 2018). Centro de Informática, Universidade Federal de Pernambuco. (Abstract in: Ramos, A. (2019). Explicit Computational Paths in Type Theory. Bulletin of Symbolic Logic, 25(2):213-214. 10.1017/bsl.2019.2). Also available at https://repositorio.ufpe.br/handle/123456789/32902.10.1017/bsl.2019.2Search in Google Scholar
Ramos, A. F., R. J. G. B. de Queiroz, and A. G. de Oliveira. 2017. “On the Identity Type as the Type of Computational Paths.” Logic Journal of IGPL 25 (4): 562–84. https://doi.org/10.1093/jigpal/jzx015.Search in Google Scholar
Ramos, A. F., R. J. G. B. de Queiroz, A. G. de Oliveira, and T. M. L. de Veras. 2018. “Explicit Computational Paths.” South American Journal of Logic 4 (2): 441–84.Search in Google Scholar
Ramos, A. F., R. J. G. B. de Queiroz, and A. G. de Oliveira. 2021a. “Computational Paths and the Fundamental Groupoid of a Type.” In Encontro de Teoria da Computação (ETC), 6, Evento Online, 22–5. Porto Alegre: Sociedade Brasileira de Computação.10.5753/etc.2021.16371Search in Google Scholar
Ramos, A. F., R. J. G. B. de Queiroz, and A. G. de Oliveira. 2021b. “Conversão de Termos, Homotopia, e Estrutura de Grupóide.” In Workshop Brasileiro de Lógica (WBL), 2, Evento Online, 33–40. Porto Alegre. Sociedade Brasileira de Computação.10.5753/wbl.2021.15776Search in Google Scholar
Schroeder-Heister, P. 2006. “Validity Concepts in Proof-Theoretic Semantics.” Synthese 148 (3): 525–71. https://doi.org/10.1007/s11229-004-6296-1.Search in Google Scholar
Schroeder-Heister, P. 2018. Proof-Theoretic Semantics in Stanford Encyclopedia of Philosophy. Stanford University. Also available at https://plato.stanford.edu/entries/proof-theoretic-semantics/.Search in Google Scholar
Vänäänen, J. 2022. “The Strategic Balance of Games in Logic.” arXiv:2212.01658.Search in Google Scholar
de Veras, T. M. L., A. F. Ramos, R. J. G. B. de Queiroz, and A. G. de Oliveira. 2021. “Calculation of Fundamental Groups via Computational Paths.” In Encontro de Teoria da Computação (ETC), 6, 2021, Evento Online, 17–21. Porto Alegre. Sociedade Brasileira de Computação.10.5753/etc.2021.16370Search in Google Scholar
de Veras, T. M. L., A. F. Ramos, R. J. G. B. de Queiroz, and A. G. de Oliveira. 2023a. “A Topological Application of Labelled Natural Deduction.” To appear in South American J. of Logic. arXiv:1906.09105.Search in Google Scholar
de Veras, T. M. L., Ramos, A. F., de Queiroz, R. J. G. B., Silva, T. D. O., and de Oliveira, A. G., 2023b. “Computational Paths – A Weak Groupoid.” arXiv: 2007.07769. (Submitted for publication).10.1093/logcom/exad071Search in Google Scholar
Wittgenstein, L. 1922. Tractatus Logico-Philosophicus, Translated by C. K. Ogden, and K. Paul, Trench, New York, Harcourt: Trubner & Co., Ltd., Brace & Company, Inc.Search in Google Scholar
Wittgenstein, L. 1953. Philosophical Investigations, edited by G. E. M. Anscombe, and R. Rhees, G. E. M. Anscombe (trans.) Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1956. Remarks on the Foundations of Mathematics, edited by G. H. von Wright, R. Rhees, and G. E. M. Anscombe, G. E. M. Anscombe (trans.) Oxford: Blackwell, Revised Edition 1978.Search in Google Scholar
Wittgenstein, L. 1961. Notebooks 1914–1916, edited by G. H. von Wright, and G. E. M. Anscombe. Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1964. Philosophical Remarks, edited by R. Rhees, R. Hargreaves and R. White (trans.) Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1969. On Certainty, edited by G. E. M. Anscombe, and G. H. von Wright, G. E. M. Anscombe and D. Paul (trans.) Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1971. ProtoTractatus–An Early Version of Tractatus Logico- Philosophicus, edited by B. F. McGuinness, T. Nyberg, and G. H. von Wright, D. F. Pears and B. F. McGuinness (trans.) Ithaca: Cornell University Press.Search in Google Scholar
Wittgenstein, L. 1974a. Letters to Russell, Keynes and Moore, Ed. with an Introd. by G. H. von Wright, (assisted by B. F. McGuinness). Oxford: Basil Blackwell.Search in Google Scholar
Wittgenstein, L. 1974b. Lectures and Conversations on Aesthetics, Psychology and Religious Belief, 1966, edited by C. Barrett. Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1974c. Philosophical Grammar, edited by R. Rhees, A. Kenny (trans.). Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1980. Remarks on the Philosophy of Psychology, Vol. 1 and 2, edited by G. E. M. Anscombe, G. H. von Wright, and H. Nyman, G. E. M. Anscombe, C. G. Luckhardt, and M. A. E. Aue (trans.). Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 1982. Last Writings on the Philosophy of Psychology, Vol. 1, 1982, Vol. 2, 1992, edited by G. H. von Wright, and H. Nyman, trans. C. G. Luckhardt and M. A. E. Aue. Oxford: Blackwell.Search in Google Scholar
Wittgenstein, L. 2016. Wittgenstein, Ludwig: Interactive Dynamic Presentation (IDP) of Ludwig Wittgenstein’s Philosophical Nachlass [wittgensteinonline.no]. Edited by the Wittgenstein Archives at the University of Bergen (WAB) under the Direction of Alois Pichler. Bergen: Wittgenstein Archives at the University of Bergen 2016. Also available at http://wab.uib.no/wab_BEE.page (accessed May 13, 2021).Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
