Skip to main content
SpringerLink
Log in

Oskar Becker and the Modal Translation of Intuitionistic Logic

  • Conference paper
  • First Online:
Thinking and Calculating

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 54))

  • 215 Accesses

Abstract

We reconsider Oskar Becker’s pioneering contributions to modal logic in On the Logic of Modalities (1930), in particular Becker’s unjustly neglected anticipation of the idea of a modal interpretation of intuitionistic logic, which was realized three years later by Kurt Gödel.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Godel (1933).

  2. 2.

    Becker (1930) (here quoted according to the original pagination). The forthcoming volume Centrone and Minari (2022b) contains the first English translation of Becker’s Zur Logik der Modalitäten, together with an extensive commentary.

  3. 3.

    Godel (1931).

  4. 4.

    Likewise, no mention of Becker is found in A. S. Troelstra’s “Introductory note to An interpretation of the intuitionistic propositional calculus”, in Feferman et al. (1986, 296–299). Notice that a modal translation of intuitionistic logic was in a sense foreshadowed also by Ivan Orlov in (Orlov, 1928). The paper, written in Russian, remained however unknown outside the Soviet Union for a very long time: Orlov's contributions were indeed “rediscovered” and came to be known and discussed only in the 1990's, see Chagrov and Zakharyashchev (1992) and Došen (1992).

  5. 5.

    Actually, in Gödel’s paper the axioms are not given in schematic form, and the rule of substitution is assumed.

  6. 6.

    Thanks to Feys (1950), Prior (1955) and, in particular, Lemmon (1957).

  7. 7.

    Lewis and Langford (1932) is not mentioned by Gödel. He says that \(\mathfrak {S}\) is equivalent to “Lewis’s system of strict implication”, that is the Survey system [Lewis (1918), emended in Lewis (1920) and eventually named ‘\(\mathbf {S3}\)’ in the mentioned Appendix II), supplemented by “Becker’s axiom” \(\Box (\Box \alpha \rightarrow \Box \Box \alpha )\).

  8. 8.

    Gödel also indicates as variants \((\lnot \beta )^{\scriptscriptstyle {G}}\ :=\, \Box \lnot \Box \beta ^{\scriptscriptstyle {G}}\) and \((\beta \wedge \gamma )^{\scriptscriptstyle {G}}\, :=\, \Box \beta ^{\scriptscriptstyle {G}}\wedge \Box \gamma ^{\scriptscriptstyle {G}}\).

  9. 9.

    A syntactical proof is indeed straightforward (and tedious).

  10. 10.

    McKinsey and Tarski (1948), Theorems 5.2 and 5.3 (the latter for Gödel’s variant of the translation \((\ldots )^{\scriptscriptstyle {G}}\)), which almost immediately follow from Theorem 5.1 that states the soundness and faithfulness of their own translation \((\ldots )^*\). The two translations \((\ldots )^{\scriptscriptstyle {G}}\) and \((\ldots )^*\) are indeed easily proved to be related as follows: for all \(\mathcal {L}_I\)-formulas \(\alpha\), \(\vdash _{\mathbf {S4}} \Box \alpha ^{\scriptscriptstyle {G}}\leftrightarrow \alpha ^*\), hence \(\vdash _{\mathbf {S4}} \alpha ^{\scriptscriptstyle {G}}\Leftrightarrow \,\vdash _{\mathbf {S4}}\alpha ^*\).

  11. 11.

    E.g. the disjunction property for \(\mathbf {IPC}\), which follows from the translation theorem together with Gödel’s conjecture that \(\vdash _{\mathbf {S4}}\Box \alpha \vee \Box \beta\) implies \(\vdash _{\mathbf {S4}}\Box \alpha\) or \(\vdash _{\mathbf {S4}}\Box \beta\), later proved in McKinsey and Tarski (1948). The first “official” proof of the disjunction property for \(\mathbf {IPC}\) was given by Gerhard Gentzen in 1935, via cut-elimination (Gentzen, 1935).

  12. 12.

    Kripke (1963, 1965b).

  13. 13.

    Kripke (1965a), 92.

  14. 14.

    See Artemov and Beklemishev (2004) for a survey.

  15. 15.

    See Artemov and Fitting (2019).

  16. 16.

    Oskar Becker (Leipzig 1889–Bonn 1964) is often remembered as one of the most prominent students of Edmund Husserl. He graduated in mathematics in 1914 (Becker, 1914), and in 1922 he wrote under Husserl’s supervision his Habilitationsschrift, Contributions Toward a Phenomenological Foundation of Geometry and Its Physical Applications (Becker, 1923). In 1927 Becker published what is considered to be his masterpiece, Mathematical Existence (Becker, 1927), in the Jahrbuch für Philosophie und phänomenologische Forschung (he was, together with Martin Heidegger, Moritz Geiger, Alexander Pfänder, Adolf Reinach and Max Scheler a member of the editorial board of this journal). In 1952—when the study of modal logic was already well beyond its pioneering era—Becker would come back to this subject with the monograph Investigations on the Modal Calculus (Becker, 1952), perhaps too old-fashioned for the time, cp. (Martin, 1969). For a complete bibliography of Becker’s works see Zimny (1969).

  17. 17.

    Lewis (1920).

  18. 18.

    Thus “\(-\alpha\)”, “\(\sim \! \alpha\)”, “\(\alpha \times \beta\)”, “\(\alpha = \beta\)” correspond, respectively, to “\(\lnot \alpha\)”, “\(\lnot \lozenge \alpha\)”, “\(\alpha \wedge \beta\)”, “\(\Box (\alpha \leftrightarrow \beta)\)” in the now current notation. The other logical boolean and “strict” operators, in particular “\(\subset\)” (material implication), “<” (strict implication) and “\(+\)” (boolean disjunction) are instead defined in the expected way.

  19. 19.

    See Lewis and Langford (1932, 136 ff).

  20. 20.

    Introduced in Lemmon (1957).

  21. 21.

    Parry (1939).

  22. 22.

    The 21 positive (resp. negative) irreducible modalities—after Parry’s result—are indeed not linearly ordered w.r. to \(\rightarrowtail\).

  23. 23.

    It looks like Becker was supposing that (i), possibly together with (ii), would also imply the existence of a decision procedure for the extended calculus.

  24. 24.

    Here Becker is acknowledged for having introduced the characteristic schema of the system. In fn. 1, p. 492, the Authors also mention a letter sent by M. Wajsberg to Lewis in 1927, containing “the outline of a system of Strict Implication with the addition of the postulate later suggested in Becker’s paper”.

  25. 25.

    Becker refers explicitly to Heyting (1930), which contains the first (complete) presentation of intuitionistic logic as a formalized calculus. The paper was published in the same year of On the Logic of Modalities, but was circulating since 1928.

  26. 26.

    Becker (1930, 17).

  27. 27.

    This is “Becker’s additional axiom” mentioned in Godel (1933).

  28. 28.

    In his Review, Gödel indeed remarked that “it is nowhere shown that the [...] systems set up really differ from one another and from Lewis’s system (in other words, that the additional axioms are not in fact equivalent and do not follow from Lewis’s); nor, furthermore, that the six, or ten, basic modalities obtained cannot be still further reduced.” (Godel 1931, 201).

  29. 29.

    In §5 of Part I, Becker (1930, 25–30), entitled “On the Calculus of Modalities with least Requirements, which still yields a Linear Rank Order”.

  30. 30.

    See Centrone and Minari (2022b) for a detailed analysis and discussion.

  31. 31.

    These rules have not been correctly interpreted and formalized in Churchman (1938), the first (and unique, as far as we know) paper where this experiment by Becker is detailedly analyzed. Incidentally, notice that the inference rules

    $$\frac{\Box (\alpha \rightarrow \beta )}{\Box (\Box \alpha \rightarrow \Box \beta )}\quad \text {and}\quad \frac{\Box (\alpha \rightarrow \beta )}{\Box (\lozenge \alpha \rightarrow \lozenge \beta )}$$

    known also in the current literature as Becker’s rules, were given this name in Churchman (1938) because (uncorrectly) regarded as specific instances of one of the rules in \(\mathcal {R}\). For a detailed discussion, see Centrone and Minari (2022a).

  32. 32.

    Churchman (1938, 78 ff). The claim is indeed correct, although Churchman’s proof thereof is not, because he did not formalize the system \(\mathbf {SM}\) as Becker really intended it.

  33. 33.

    Becker (1930, 30–35).

  34. 34.

    Becker is aware of Glivenko’s double negation translation of \(\mathbf {CPC}\) into \(\mathbf {IPC}\), (Glivenko, 1929), and mentions this paper.

  35. 35.

    Here we use a convenient mix of Lewis’s symbolism and the current symbolism.

  36. 36.

    Hacking (1963). Hacking’s proof-theoretical demonstration, which makes use of a normalization theorem for a suitable natural deduction presentation of \(\mathbf {S3}\), is rather convoluted. A much simpler semantical proof, obtained by exploiting a conjecture in Oakes (1999), can be found in Centrone and Minari (2022b).

  37. 37.

    Gödel’s translation \((\ldots )^{\scriptscriptstyle {G}}\) is instead not sound with respect to \(\mathbf {S3}\).

  38. 38.

    In this regard, Feys (1937) and Feys (1938) should at least be added to the already mentioned works by Gödel, Lewis and Langford, Churchman and Parry.

References

  • Artemov, S., & Beklemishev, L. (2004). Provability logic. In D. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 13, pp. 229–403). Kluwer.

    Google Scholar 

  • Artemov, S., & Fitting, M. (2019). Justification logic: Reasoning with reasons. Cambridge University Press.

    Book  Google Scholar 

  • Becker, O. (1914). Über die Zerlegung eines Polygons in exclusive Dreiecke auf Grund der ebenen Axiome der Verknüpfung und Anordnung. F. A. Brockhaus.

    Google Scholar 

  • Becker, O. (1923). Beiträge zur phänomenologischen Begründung der Geometrie und ihrer physikalischen Anwendung. In E. Husserl (Ed.), Jahrbuch für Philosophie und phänomenologische Forschung VI (pp. 385–560). Max Niemeyer Verlag.

    Google Scholar 

  • Becker, O. (1927). Mathematische Existenz. Untersuchungen zur Logik und Ontologie mathematischer Phänomene. In E. Husserl (Ed.), Jahrbuch für Philosophie und phänomenologische Forschung VIII (pp. 439–809). Max Niemeyer Verlag.

    Google Scholar 

  • Becker, O. (1930). Zur Logik der Modalitäten. In E. Husserl (Ed.), Jahrbuch für Philosophie und phänomenologische Forschung XI (pp. 497–548). Max Niemeyer Verlag (English translation in Centrone and Minari, 2022b).

    Google Scholar 

  • Becker, O. (1952). Untersuchungen über den Modalkalkül. Westkulturverlag Anton Hain.

    Google Scholar 

  • Centrone, S., & Minari, P. (2022a). Becker’s rule is not Becker’s rule. In A. Klev & S. Rahman (Eds.), Festschrift for Göran Sundholm. Springer, Cham.

    Google Scholar 

  • Centrone, S., & Minari, P. (2022b). Oskar Becker, on the logic of modalities (1930): Translation, commentary and analysis. Springer, Cham.

    Google Scholar 

  • Chagrov, A., & Zakharyashchev, M. (1992). Modal companions of intermediate propositional logics. Studia Logica, 51, 49–82.

    Google Scholar 

  • Churchman, C. W. (1938). On finite and infinite modal systems. The Journal of Symbolic Logic, 3, 77–82.

    Article  Google Scholar 

  • Došen, K. (1992). The first axiomatization of relevant logic. Journal of Philosophical Logic, 21, 339–356.

    Google Scholar 

  • Feferman, S., Kleene, S., Moore, G., Solovay, R., & van Heijenoort, J. (Eds.). (1986). Kurt Gödel collected works. Volume I: Publications 1929–1936. Oxford University Press.

    Google Scholar 

  • Feys, R. (1937). Les logiques nouvelles des modalités. Revue néo-scolastique de philosophie, 56, 517–553.

    Article  Google Scholar 

  • Feys, R. (1938). Les logiques nouvelles des modalités (suite et fin). Revue néo-scolastique de philosophie, 58, 217–252.

    Article  Google Scholar 

  • Feys, R. (1950). Les systèmes formalisés des modalités aristotéliciennes. Revue philosophique de Louvain, 48, 478–509.

    Article  Google Scholar 

  • Gentzen, G. (1935). Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39, 176–210, 405–431.

    Article  Google Scholar 

  • Glivenko, V. (1929). Sur quelques points de la logique de M. Brouwer. Acad. Roy. Belg. Bull. Cl. Sci., Sér. 5, 15, 183–188.

    Google Scholar 

  • Gödel, K. (1931). Besprechung von Becker 1930: Zur Logik der Modalitäten. Monatshefte für Mathematik und Physik (Literaturberichte), 38, 5–6. (Reprinted and translated in [Feferman et al., 1986], 216–217).

    Google Scholar 

  • Gödel, K. (1933). Eine interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematischen Kolloquiums, 4, 39–40. (Reprinted and translated in [Feferman et al., 1986], 300–301).

    Google Scholar 

  • Goodman, N. D. (1984). Epistemic arithmetic is a conservative extension of intuitionistic arithmetic. The Journal of Symbolic Logic, 49, 192–203.

    Article  Google Scholar 

  • Hacking, I. (1963). What is strict implication? The Journal of Symbolic Logic, 28, 51–71.

    Article  Google Scholar 

  • Heyting, A. (1930). Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse (pp. 42–56, 57–71, 158–169).

    Google Scholar 

  • Kripke, S. (1963). Semantical analysis of modal logic, I. Normal modal propositional calculi. Zeitschr. f. Math. Logik und Grundlagen d. Math., 9, 67–96.

    Article  Google Scholar 

  • Kripke, S. (1965a). Semantical analysis of intuitionistic logic. In J. Crossley & M. Dummett (Eds.), Formal systems and recursive functions (pp. 92–130). North-Holland Publishing Company.

    Chapter  Google Scholar 

  • Kripke, S. (1965b). Semantical analysis of modal logic, II. Non-normal modal propositional calculi. In J. W. Addison, L. Henkin, & A. Tarski (Eds.), The theory of models. Proceedings of the 1963 international symposium at Berkeley (pp. 206–220). North-Holland Publishing Company.

    Google Scholar 

  • Lemmon, E. J. (1957). New foundations for Lewis modal systems. The Journal of Symbolic Logic, 22, 176–186.

    Article  Google Scholar 

  • Lewis, C. I. (1918). A survey of symbolic logic. University of California Press.

    Google Scholar 

  • Lewis, C. I. (1920). Strict implication—An emendation. Journal of Philosophy, Psychology and Scientific Methods, 17, 300–302.

    Article  Google Scholar 

  • Lewis, C. I., & Langford, C. H. (1932). Symbolic logic. The Century Co. (Dover Publications, New York \(\,^2\)1959).

    Google Scholar 

  • Martin, G. (1969). Oskar Beckers Untersuchungen über den Modalkalkül. Kant-Studien, 60, 312–318.

    Google Scholar 

  • McKinsey, J. C. C., & Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. The Journal of Symbolic Logic, 13, 1–15.

    Article  Google Scholar 

  • Mints, G. (1992). On Novikov’s hypothesis (1978, Russian). In G. Mints (Ed.), Selected papers in proof theory (pp. 147–151). Bibliopolis, Naples and North-Holland Publishing Company.

    Google Scholar 

  • Oakes, C. (1999). Interpretations of intuitionistic logic in non-normal modal logics. Journal of Philosophical Logic, 28, 47–60.

    Article  Google Scholar 

  • Orlov, I. (1928). The calculus of compatibility of propositions (in Russian). Mathematics of the USSR, Sbornik, 35, 263–286.

    Google Scholar 

  • Parry, W. T. (1939). Modalities in the survey system of strict implication. The Journal of Symbolic Logic, 4, 137–154.

    Article  Google Scholar 

  • Prior, A. (1955). Formal logic. Oxford University Press.

    Google Scholar 

  • Shapiro, S. (Ed.). (1985). Intensional mathematics. North-Holland Publishing Company.

    Google Scholar 

  • Zimny, L. (1969). Oskar Becker—Bibliographie. Kant-Studien, 60, 319–330.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Philosophie, FernUniversität in Hagen, Hagen, Germany

    Stefania Centrone

  2. Dipartimento di Lettere e Filosofia, Università degli Studi di Firenze, Firenze, Italy

    Pierluigi Minari

Authors
  1. Stefania Centrone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierluigi Minari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierluigi Minari .

Editor information

Editors and Affiliations

  1. Dipartimento di Lettere e Filosofia, Università di Firenze, Florence, Italy

    Francesco Ademollo

  2. Department of Humanities, Social Sciences, and Cultural Industries, Università di Parma, Parma, Italy

    Fabrizio Amerini

  3. Laboratoire SPHère, CNRS/Université Paris Cité, Paris, France

    Vincenzo De Risi

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Centrone, S., Minari, P. (2022). Oskar Becker and the Modal Translation of Intuitionistic Logic. In: Ademollo, F., Amerini, F., De Risi, V. (eds) Thinking and Calculating. Logic, Epistemology, and the Unity of Science, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-030-97303-2_18

Download citation

Publish with us

Policies and ethics

Search

Navigation