From the Notebooks to the Investigations and Beyond
-
Ruy J.G.B. de Queiroz
Abstract
The use of the open and searchable Wittgenstein’s Nachlass (The Wittgenstein Archives at the University of Bergen (WAB)) has proved instrumental in the quest for a common thread of Wittgenstein’s view on the connections between meaning, use and consequences, going from the Notebooks to later writings (including the Philosophical Investigations) and beyond. Here we take this as the basis for a proposal for a formal counterpart of a ‘meaning-as-use’ (dialogical/game-theoretical) semantics for the language of predicate logic. In order to further consolidate this perspective, we shall need to bring in key excerpts from Wittgenstein oeuvre (including the Nachlass) and from those formal semanticists who advocate a different perspective on the connections between proofs and meaning. With this in mind we consider several passages from Wittgenstein’s published as well as unpublished writings to build a whole picture of a formal counterpart to ‘meaning is use’ based on the idea that explanations of consequences via ‘movements within language’ ought to be taken as a central aspect to Wittgenstein’s shift from ‘interpretation of symbols in a state of affairs’ to ‘use of symbols’ which underpins his ‘meaning is use’ paradigm. As in the Investigations “every interpretation hangs in the air together with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning”, as well as in a remark from his transitional period (1929–30): Perhaps one should say that the expression “interpretation of symbols” is misleading and one should instead say “the use of symbols”. Significantly, we wish the present examination of the searchable Nachlass can make a relevant step towards a formal counterpart to the “meaning is use” dictum, while highlighting an important common thread from Wittgenstein’s very early to very late writings. For this we focus here on the themes of explanation of consequences, movements within language, moving away from states of affairs, meaning versus verification.
Acknowledgements
First and foremost, we acknowledge the role of Marcos Silva (UFPE), the main organiser of VIII Brazilian Society for Analytic Philosophy Conference, 22-26 July 2024 https://sites.google.com/view/sbfa-sbpha/olinda-2024_1, Academia Santa Gertrudes, Olinda, Pernambuco, Brazil, who kindly presented us with an invitation to give a keynote talk. This served as additional stimulus to write this paper, which, in turn led us to new findings in Wittgenstein’s Nachlass. Last, but not least, thanks and much appreciation to the anonymous reviewer for the scholarly, careful and competent assessment of the paper. (Gratefully acknowledged is the free access to automated translation applications such as Google Translator and DeepL.com).
Appendix: Reduction Rules and Dialogue/Games Semantics
The renewed examination of Wittgenstein’s Nachlass presented here supports and augments the significance of Wittgenstein’s suggestion that meaning is determined by explicating immediate consequences of a statement or term. These new findings counter the claim that the so-called Gentzen-Dummett-Prawitz-Martin-Löf ‘meaning as determined by assertability conditions/introduction rules’ are properly designated as ‘Wittgensteinian views’. Instead, these passages strongly suggest connections with the “pragmatist”/“dialogical” approaches to meaning, already in Peirce’s writings on the interaction between the Interpreter and the Utterer, which likewise appear to bear upon the “game”/“dialogue” approaches to meaning (Lorenzen, Hintikka, Fraïssé). The Inversion Principle (advocated by Gentzen and Prawitz) also appears to have the same kind of “metalevel” explication of meaning as those game/dialogical approaches, including Tarski’s “metalanguage” approach. Hence in Natural Deduction, the reduction rules (which formalise the Inversion Principle) should be seen as meaning-giving, not a mere justification of the elimination rule(s) based on the introduction rule(s). The reduction rules of labelled natural deduction extends this approach to ‘Identity types’ which correspond to propositional equality. Significantly, the present examination of these materials make an important step towards a formal counterpart to the “meaning is use” dictum, while highlighting an important common thread from Wittgenstein’s very early to very late writings.
There are at least two well-established non-truth-theoretic semantics dealing with the interface of meaning, knowledge, and logic in the context of dialogues, games, or more generally interaction. One of these is also an alternative perspective on proof theory and meaning theory, advocating that Wittgenstein’s “meaning as use” paradigm, as understood in the context of proof theory, by which the so-called reduction rules (showing the workings of elimination rules on the result of introduction rules) should be seen as appropriate to formalise the explanation of the (immediate) consequences one can draw from a proposition, thus to show the function/purpose/usefulness of its main connective in the calculus of language. To recall D. Prawitz’ Natural Deduction (1965) original formulation: ‘reduction steps’ are defined as rules which operate on proofs, thus they constitute ‘metalevel’ rules rather than ‘object level’ rules. This suggests they can indeed be seen as playing a ‘semantical’ role, in analogy to several other definitions (Tarski’s truth conditions, Lorenzen’s dialogical rules, Hintikka’s game semantics rules, Abelard-Eloïse evaluation game rules.
Previously we have pointed out parallels between our proposal and other approaches to the semantics of logical connectives based on explaining the (immediate) consequences of the corresponding proposition, such as Lorenzen’s (1950, 1955, 1969, p. 25) dialogical games and Hintikka 1968, 1979, p. 3) semantical games. The general underlying principle is the logical Inversion Principle, uncovered by Gentzen, and later by Lorenzen: the elimination procedure is the exact inverse of the introduction, therefore all that can be asked from an assertion is what is indicated explaining the elimination procedure. One can look at Lorenzen’s dialogical games and try to show how the Inversion Principle resides in the game-approach by comparing it to the reduction-based account of meaning. The correspondence works as follows: (i) the assertion corresponds to the introduction rule(s) (and the respective constructors); (ii) the attacks correspond to the elimination rule(s) (and the respective destructors); (iii) the set of defenses correspond to the reduction rules (the effect of elimination rules on the result of introduction rules); (iv) the relation specifying for each attack the corresponding defense(s) are defined by the result of reduction rules.
Taking from the rules of reduction between proofs in a system of Natural Deduction enriched with terms alongside formulas, such as in the so-called Curry-Howard interpretation (Howard (1980)) of which Martin-Löf’s type theory can be seen as an instance, and isolating the terms corresponding the derivations, one can draw the following parallels: Looking at the conclusion of reduction inference rules, one can take the destructor as being the Attack (or ‘Nature’, its counterpart in the terminology of Hintikka’s Game-Theoretical Semantics), and the constructor as being the Assertion (or Hintikka’s ‘Myself’). This way the game-theoretic explanations of logical connectives find direct counterpart in the functional interpretation with the semantics of convertibility:
∧-β-reduction
Associated rewritings:
| Assertion/Introd. | Attack/Elim. | ↠ β | Defense |
| Conjunction (‘∧’): | |||
| A ∧ B | L? | A | |
| A ∧ B | R? | B | |
| ⟨a, b⟩ : A ∧ B | FST(⟨a, b⟩) | ↠ β | a : A |
| ⟨a, b⟩ : A ∧ B | SND(⟨a, b⟩) | ↠ β | b : B |
∨-β-reduction
Associated rewritings:
| Assertion/Introd. | Attack/Elim. | ↠ β | Defense |
| Disjunction (‘∨’): | |||
| A ∨ B | ? | A | |
| A ∨ B | ? | B | |
| inl(a) : A ∨ B | CASE(inl
|
↠ β | a : A, f(a/x) : C |
| inr(b) : A ∨ B | CASE(inr
|
↠ β | b : B, g(b/y) : C |
Whilst in both FST(⟨a, b⟩) and SND(⟨a, b⟩) the destructors allow access to either of the conjuncts, in CASE(inl(a), υx.f(x), υy.g(y)) and CASE(inr(b), υx.f(x), υy.g(y)) the destructor is not given access to the either disjunct but must ask for whichever disjunct comes from the introduction.
→ -β-reduction
Associated rewriting:
| Assertion/Introd. | Attack/Elim. | ↠ β | Defense |
| Implication (‘ → ’): | |||
| A → B | A ? | B | |
| a : A | |||
| λx.b(x) : A → B | APP(λx.b(x), a) | ↠ β | b(a/x) : B |
∀-β-reduction
Associated rewriting:
∃-β-reduction
Associated rewriting:
| Assertion/Introd. | Attack/Elim. | ↠ β | Defense |
| Universal Quantifier (‘∀’): | |||
| ∀x D .P(x) | s : D ? | P(s) | |
| s : D | |||
| Λx.f(x) : ∀x D .P(x) | |||
| EXTR(Λx.f(x), s) | ↠ β | f(s/x) : P(s) | |
| Existential Quantifier (‘∃’): | |||
| ∃x D .P(x) | ? | s : D, P(s) | |
| s : D, f(s) : P(s) | |||
| ɛx.(f(x), s) : ∃x D .P(x) | |||
| INST(ɛx.(f(x), s), σg.σt.d(g(t), t)) | ↠ β | d(f(s)/g(t), s/t) : C |
Whilst in EXTR(Λx.f(x), s) the destructor has access to the witness (can use a generic element) s, in INST(ɛx.(f(x), s), σg.σt.d(g(t), t)) the only option is to eliminate over the witness s which is inside the term ɛx.(f(x), s) built with the constructor because it was chosen by the introduction.
Id-β-reduction
Associated rewriting:
| Assertion/Introd. | Attack/Elim. | ↠ β | Defense |
| Propositional Equality (‘Id A (u, v)’): | |||
| Id A (u, v) | ? | u = r v : A | |
| r(u, v) : Id A (u, v) | REWR(r(u, v), σt.d(t)) | ↠ β | d(r/t) : C |
In REWR(r(u, v), σt.d(t)) the destructor has no choice regarding the ‘reason’ for u being equal to v, since r will have been chosen by the time of the assertion, i.e. the application of the introduction rule.
Drawing these parallels is intended to exhibit a connection with the “pragmatist”/“dialogical” approaches to meaning as revealed in Peirce’s writings on the interaction between the Interpreter and the Utterer which seem to pertain to the “game”/“dialogue” approaches to meaning (Lorenzen, Hintikka, Ehrenfeucht-Fraïssé.
References
Dummett, D. 1975. “The Philosophical Basis of Intuitionistic Logic.” In Logic Colloquium ’73, Vol. 80, edited by H. E. Rose, and J. C. Shepherdson, 5–40. Studies in Logic and the Foundations of Mathematics. Elsevier.10.1016/S0049-237X(08)71941-4Suche in Google Scholar
Dummett, D. 1977. Elements of Intuitionism. Oxford University Press.Suche in Google Scholar
Dummett, M. 1991. The Logical Basis of Metaphysics. Cambridge (Mass.): Harvard University Press.Suche 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, Part I, Institute of Philosophy of the Academy of Sciences of the USSR, Moscow, 69–73.Suche 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.Suche 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, Vol. 18, edited by P. Weingartner, G. Schurz, E. Leinfellner, R. Haller, and A. Hübner, 259–66.Suche 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), 20. Symposium held in Kirchberg/Wechsel, August 13–20 1989.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
de Queiroz, R. J. G. B. 1992. “Grundgesetze Alongside Begriffsschrift (Abstract).” In Abstracts of Fifteenth International Wittgenstein Symposium, 15–6. Symposium held in Kirchberg/Wechsel, August 16–23 1992.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
de Queiroz, R. J. G. B., and D. M. Gabbay. 1997. “The Functional Interpretation of Modal Necessity.” In Advances in Intensional Logic, Applied Logic Series, Kluwer, edited by M. de Rijke, 61–91.10.1007/978-94-015-8879-9_3Suche 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. https://doi.org/10.1007/978-94-011-4574-9_10.Suche 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): 173–215.10.1093/jigpal/7.2.173Suche 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.Suche 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.Suche 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.Suche 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.Suche in Google Scholar
de Queiroz, R. J. G. B., de Oliveira, A. G., and Gabbay, D. M. 2011. The Functional Interpretation of Logical Deduction. Vol. 5 of Advances in Logic series. Imperial College Press/World Scientific, Oct 2011.10.1142/9789814360968Suche 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, Vol. 39, edited by L. Pereira, E. Haeusler, and V. de Paiva, 63–91. Trends in Logic. Dordrecht: Springer. https://doi.org/10.1007/978-94-007-7548-0_4.Suche in Google Scholar
de Queiroz, R. J. G. B., A. G. de Oliveira, and A. F de Ramos. 2016. “Propositional Equality, Identity Types, and Computational Paths.” South American Journal of Logic 2 (2): 245–96. Suche in Google Scholar
de Queiroz, R. J. G. B. 2023. “From Tractatus to Later Writings and Back – New Implications from Wittgenstein’s Nachlass.” SATS 24 (2): 167–203. https://doi.org/10.1515/sats-2022-0016.Suche 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.2021.16370Suche 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.” In South American J. of Logic. https://www.sa-logic.org/aaccess/ruy.pdf. Also arXiv:1906.09105.Suche in Google Scholar
de Veras, T. M. L., A. F. Ramos, R. J. G. B. de Queiroz, and A. G. de Oliveira. 2023b. “Computational Paths – A Weak Groupoid.” Journal of Logic and Computation. https://doi.org/10.1093/logcom/exad071. Also arXiv: 2007.07769.Suche 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.” Journal of Symbolic Logic 57 (4): 1319–65. https://doi.org/10.2307/2275370.Suche in Google Scholar
Gentzen, G. 1935. “Untersuchungen über das logische Schließen. I.” Mathematische Zeitschrift 39: 176–210. https://doi.org/10.1007/BF01201353. (English translation in J. von Plato (2017)).Suche in Google Scholar
Hintikka, J. 1968. “Language-games for Quantifiers.” In Studies in Logical Theory, edited by N. Rescher, 46–72. Blackwell.Suche 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.Suche 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. Academic Press.Suche in Google Scholar
Lorenzen, P. 1950. “Konstruktive Begründung der Mathematik.” Mathematische Zeitschrift 53 (2): 162–202. https://doi.org/10.1007/bf01162411.Suche in Google Scholar
Lorenzen, P. 1955. Einführung in die operative Logik und Mathematik. Die Grundlehren der mathematischen Wissenschaften, Vol. 78, VII + 298. Berlin-Göttingen-Heidelberg: Springer-Verlag.10.1007/978-3-662-01539-1Suche in Google Scholar
Lorenzen, P. 1969. Normative Logic and Ethics, Series B.I-Hochschultaschenbücher. Systematische Philosophie, Vol. 236*. Mannheim/Zürich: Bibliographisches Institut.Suche in Google Scholar
Martin-Löf, P. 1984. Intuitionistic Type Theory. (Notes taken by G. Sambin). Napoli: Bibliopolis.Suche 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.Suche 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, edited by M. van der Schaar. Springer. Chapter 1. https://doi.org/10.1007/978-94-007-5137-8_1.Suche 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.Suche in Google Scholar
Martin-Löf, P. 2022. “Corrections of Assertion and Validity of Inference.” In Transcript of a Lecture Given on 26 October 2022 at the Rolf Schock Symposium in Stockholm. https://pml.flu.cas.cz/uploads/PML-Stockholm26Oct22.pdf.Suche in Google Scholar
Martínez-Rivillas, D. O. 2022. Towards a Homotopy Domain Theory. PhD thesis, CIn-UFPE (November 2022). Recife, Brazil: Centro de Informática, Universidade Federal de Pernambuco. https://repositorio.ufpe.br/handle/123456789/49221.Suche 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. Also arXiv:1906.05729.Suche in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2022b. “Towards a Homotopy Domain Theory.” Archive for Mathematical Logic 62: 559–79. https://doi.org/10.1007/s00153-022-00856-0Suche in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2023a. “The Theory of an Arbitrary Higher λ-Model.” Bulletin of the Section of Logic 52 (1): 39–58. https://doi.org/10.18778/0138-0680.2023.11. Also arXiv:2111.07092.Suche in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2023b. “Solving Homotopy Domain Equations.” (Submitted for publication) arXiv:2104.01195.Suche in Google Scholar
Martínez-Rivillas, D. O., and R. J. G. B. de Queiroz. 2025. “The K∞ Homotopy λ-Model.” (Submitted for publication) arXiv:22505.07103.Suche in Google Scholar
Peirce, C. S. 1932. Collected Papers of Charles Sanders Peirce, Volumes I and II: Principles of Philosophy and Elements of Logic. C. Hartshorne and P. Weiss (ed.). Harvard Univ Press.Suche in Google Scholar
von Plato, J. 2017. Saved from the Cellar. Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics. Springer.10.1007/978-3-319-42120-9Suche in Google Scholar
Prawitz, D. 1965. Natural Deduction: A Proof-Theoretical Study. Acta Universitatis Stockholmiensis, Stockholm Studies in Philosophy no. 3. Stockholm, Göteborg, and Uppsala: Almqvist & Wiksell.Suche in Google Scholar
Prawitz, D. 1977. “Meaning and Proofs: On the Conflict between Classical and Intuitionistic Logic.” Theoria (Sweden) XLIII: 2–40. https://doi.org/10.1111/j.1755-2567.1977.tb00776.x.Suche in Google Scholar
Prawitz, D. 2019a. “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. Univ Tübingen.Suche in Google Scholar
Prawitz, D. 2019b. “The Fundamental Problem of General Proof Theory.” In General Proof Theory. Special Issue of Studia Logica, Vol. 107, edited by T. Piecha, and P. Schroeder-Heister, 11–29. https://doi.org/10.1007/s11225-018-9785-9.Suche in Google Scholar
Ramos, A. F. 2018. Explicit Computational Paths in Type Theory. PhD thesis, CIn-UFPE (August 2018). Recife, Brazil: Centro de Informática, Universidade Federal de Pernambuco. https://repositorio.ufpe.br/handle/123456789/32902. (Abstract in: Ramos, A. (2019). Explicit Computational Paths in Type Theory. Bulletin of Symbolic Logic, 25(2):213-214. https://doi.org/10.1017/bsl.2019.2).10.1017/bsl.2019.2Suche 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.Suche 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.2021.16371Suche 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.2021.15776Suche 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.Suche 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.Suche in Google Scholar
Schroeder-Heister, P. 2018. “Proof-Theoretic Semantics.” In Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/proof-theoretic-semantics/.Suche in Google Scholar
Wittgenstein, L. 1922. Tractatus Logico-Philosophicus, Translated by C.K. Ogden. Kegan Paul, Trench, Trubner & Co., Ltd. New York: Harcourt, Brace & Company, Inc.Suche in Google Scholar
Wittgenstein, L. 1953. Philosophical Investigations, G.E.M. Anscombe and R. Rhees (eds.), G.E.M. Anscombe (trans.). Oxford: Blackwell. Revised fourth edition (2009), Translated by G. E. M. Anscombe, P. M. S. Hacker and Joachim Schulte).Suche in Google Scholar
Wittgenstein, L. 1956. Remarks on the Foundations of Mathematics, G.H. von Wright, R. Rhees and G. E. M. Anscombe (eds.), G. E. M. Anscombe (trans.). Oxford: Blackwell. revised edition 1978.Suche in Google Scholar
Wittgenstein, L. 1961. Notebooks 1914–1916, G. H. von Wright and G. E. M. Anscombe (eds.). Oxford: Blackwell.Suche in Google Scholar
Wittgenstein, L. 1964. Philosophical Remarks, R. Rhees (ed.), R. Hargreaves and R. White (trans.). Oxford: Blackwell.Suche in Google Scholar
Wittgenstein, L. 1969. On Certainty, G. E. M. Anscombe and G. H. von Wright (eds.), G.E.M. Anscombe and D. Paul (trans.). Oxford: Blackwell.Suche in Google Scholar
Wittgenstein, L. 1971. ProtoTractatus–An Early Version of Tractatus Logico- Philosophicus, B. F. McGuinness, T. Nyberg, G. H. von Wright (eds.), D. F. Pears and B. F. McGuinness (trans.). Ithaca: Cornell University Press.Suche 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.Suche in Google Scholar
Wittgenstein, L. 1974b. Lectures and Conversations on Aesthetics, Psychology and Religious Belief, 1966, C. Barrett (ed.). Oxford: Blackwell.Suche in Google Scholar
Wittgenstein, L. 1974c. Philosophical Grammar, R. Rhees (ed.), A. Kenny (trans.). Oxford: Blackwell.Suche in Google Scholar
Wittgenstein, L. 1980. Remarks on the Philosophy of Psychology, vol. 1, G. E. M. Anscombe and G. H. von Wright (eds.), G. E. M. Anscombe (trans.), vol. 2, G. H. von Wright and H. Nyman (eds.), C. G. Luckhardt and M.A.E. Aue (trans.). Oxford: Blackwell.Suche in Google Scholar
Wittgenstein, L. 1982. Last Writings on the Philosophy of Psychology, vol. 1, 1982, vol. 2, 1992, G.H. von Wright and H. Nyman (eds.), trans. C.G. Luckhardt and M.A.E. Aue (trans.). Oxford: Blackwell.Suche 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. http://wab.uib.no/ (accessed October 04, 2024).Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Articles
- Why Does Sovereignty Matter, Professor Kant?
- Communication of Uncertainty: The Concept of Probability in Government Press-Conferences during the Early Covid-19 Crisis in Denmark
- How Movies About the Climate Crisis Work to Make Us Forget About the Climate Crises
- From the Notebooks to the Investigations and Beyond
- Linguistic Mistakes and Semantic Rules
- What Logicians Do (and What They Ought to do)
Artikel in diesem Heft
- Frontmatter
- Articles
- Why Does Sovereignty Matter, Professor Kant?
- Communication of Uncertainty: The Concept of Probability in Government Press-Conferences during the Early Covid-19 Crisis in Denmark
- How Movies About the Climate Crisis Work to Make Us Forget About the Climate Crises
- From the Notebooks to the Investigations and Beyond
- Linguistic Mistakes and Semantic Rules
- What Logicians Do (and What They Ought to do)
