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From (Paraconsistent) Topos Logic to Universal (Topos) Logic

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The Road to Universal Logic

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

In this chapter, I describe how complement toposes, with their paraconsistent internal logic, lead to a more abstract theory of topos logic. Béziau’s work in Universal Logic – including his ideas on logical structures, axiomatic emptiness and on logical many-valuedness – is central in this shift and therefore it is with great pleasure that I wrote this chapter for the present commemorative volume.

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Notes

  1. 1.

    A good starting point are Chapters 1, 2 and 4 of [26].

  2. 2.

    This elucidation of toposes in logical terms follows closely [1].

  3. 3.

    As Awodey has noted, this is Russell’s notion of propositional function, for example in The Principles of Mathematics § 22 or Principia Mathematica, pp. 14 and 161.

  4. 4.

    Note by the way that, unlike many authors, I prefer the equalizers presentation of logic, not the pullbacks one.

  5. 5.

    Let \(f\!:\!X\!\longrightarrow\!Y\) and \(g\!:\!X\!\longrightarrow\!Y\) be morphisms in a category C. An equalizer in C for f and g is given by an object W and a morphism \(i\!:\!W\!\longrightarrow\!X\) in C with the following two properties: (1) \(f\circ i=g\circ i\) and (2) for any morphism \(h\!:\!Z\!\longrightarrow\!X\) in C, if \(f\circ h=g\circ h\), then there is exactly one morphism in C \(k\!:\!Z\!\longrightarrow W\) such that \(h=i\circ k\).

  6. 6.

    Nice introductions to this category can be found in [38] and [22].

  7. 7.

    I have made a little abuse of notation, for I used ‘\({}_{S}p\)’ in both \(\models_{{}_{S}\mathcal{E}}\) and \(\models_{I}\). In rigour, \({}_{S}p\) is a morphism which corresponds to a formula \((_{S}p)^{\ast}\) in a possibly different language, but there is no harm if one identifies them. A proof can be found in [15, see \(\mathsection 8.3\) for the soundness part and \(\mathsection 10.6\) for the completeness part].

  8. 8.

    I use the word ‘slogan’ here pretty much in the sense of van Inwagen: ‘a vague phrase of ordinary English whose use is by no means dictated by the mathematically formulated speculations it is supposed to summarize’ [36, p. 163], ‘but that looks as if it was’, I would add.

  9. 9.

    And maybe also of (S4), due mostly to the appearance of Hegelian terminology (‘objective’), very frequent in Lawvere but not in other topos-theorists. Omitting that, one can add [2] and [15] as supporters of this slogan.

  10. 10.

    It is important to set their individual contributions. Of the ten diagrams in [28, Ch. 11], Mortensen drew the first one and the final five, while Lavers drew the remaining four. The diagram for the dual-conditional never was explicitly drawn, but it was discussed in [28, p. 109]. The full story, as told by Mortensen in personal communication is as follows. Mortensen gave a talk at the Australian National University (Canberra) in late 1986, on paraconsistent topos logic, arguing the topological motivation for closed set logic. He defined a complement topos, drew the first three diagrams from Inconsistent Mathematics, chapter 11, that is including the complement versions of \({}_{S}\mathit{true}\) and paraconsistent negation, and criticized Goodman’s views on the conditional. But it was not seen clearly at that stage how the logic would turn out. Peter Lavers was present (also Richard Routley, Robert K. Meyer, Michael A. McRobbie, Chris Brink and others). For a couple of days in Canberra, Mortensen and Lavers tried without success to thrash it out. Mortensen returned home to Adelaide and two weeks later Lavers’ letter arrived in Adelaide, in which he stressed that inverting the order is the key insight to understanding the problem, drew the diagrams for conjunction and disjunction, and pointed out that subtraction is the right topological dual for the conditional. Mortensen then responded with the four diagrams for the \(S5\) conditional, and one for quantification (last five diagrams in Inconsistent Mathematics, chapter 11). A few months later (1987), Mortensen wrote the first paper, with Lavers as co-author, and sent it to Saunders Mac Lane and William Lawvere (also Routley, Meyer, Priest). Mac Lane replied but Lawvere did not. A later version of that paper became the eleventh chapter of Inconsistent Mathematics.

  11. 11.

    Mortensen and Lavers use the names complement-classifier and complement topos, which are now the names set in the literature (cf. [11, 28, 29, 37]). I have decided not to use the name ‘dual topos’ because the adjective ‘dual’ applied to categories has another well-entrenched meaning in category theory.

  12. 12.

    I have attempted such a categorial description of this kind of duality in [10].

  13. 13.

    Again, I have made a little abuse of notation, for I used ‘\({}_{D}p\)’ in both \(\models_{{}_{D}\mathcal{E}}\) and \(\models_{I}\). In rigour, \({}_{D}p\) is a morphism which corresponds to a formula \((_{D}p)^{\ast}\) in a possibly different language.

  14. 14.

    By abuse of notation but to simplify reading, I will not indicate that the order here is dual to that in standard toposes, unless there is risk of confusion.

  15. 15.

    Inconsistency-tolerant categorial structures are studied further in [28, chapter 12, written by William James] and in [16].

  16. 16.

    Thus, as Vasyukov ([37] p. 292) points out: ‘(…) in Set we always have paraconsistency because of the presence of both types of subobject classifiers (…)’ just as we always have in it (at least) intuitionistic logic. The presence of paraconsistency within classical logic is not news. See, for example [8], where some paraconsistent negations in S5 and classical first-order logic are defined.

  17. 17.

    It is easy to verify that after making all the necessary changes, i.e. changing \({}_{S}\mathit{true}_{\textbf{${S}^{\downarrow\downarrow}$}}\) for \({}_{D}\mathit{false}_{\textbf{${S}^{\downarrow\downarrow}$}}\), etc. the names are ordered in the same way as they are in \({}_{S}\) \({S}^{\downarrow\downarrow}\).

  18. 18.

    Someone could argue that these are not notions of logical consequence at all, since logical consequence has to satisfy the Tarskian conditions. I guess (and hope) that readers of this Festschrift do not have this kind of doubts. For a defense of the logicality of non-Tarskian relations of logical consequence, see for example [12].

References

  1. Awodey, S.: Structure in mathematics and logic: A categorical perspective. Philos. Math. 4(3), 209–237 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Awodey, S.: Continuity and logical completeness: An application of sheaf theory and topoi. In: van Benthem, M.R.J., Heinzmann, G., Visser, H. (eds.) The Age of Alternative Logics. Assesing Philosophy of Logic and Mathematics Today, pp. 139–149. Springer, Dordrecht (2006)

    Chapter  Google Scholar 

  3. Bell, J.L.: Lectures on the Foundations of Mathematics. Available at publish.uwo.ca/~jbell/foundations%20of%20mathematics.pdf

    Google Scholar 

  4. Bell, J.L.: From absolute to local mathematics. Synthese 69, 409–426 (1986)

    Article  MathSciNet  Google Scholar 

  5. Bell, J. L.: Toposes and Local Set Theories: An Introduction. Clarendon Press, Oxford (1988)

    MATH  Google Scholar 

  6. Béziau, J.-Y.: Universal logic. In: Childers, T., Majer, O. (eds.) Logica ’94 – Proceedings of the 8th International Symposium, pp. 73–93. Czech Academy of Sciences, Prague (1994)

    Google Scholar 

  7. Béziau, J.-Y.: From paraconsistent logic to universal logic. Sorites 12, 5–32 (2001)

    Google Scholar 

  8. Béziau, J.-Y.: S5 is a paraconsistent logic and so is first-order classical logic. Log. Investig. 9, 301–309 (2002)

    Google Scholar 

  9. de Queiroz, G.d.S.: On the Duality between Intuitionism and Paraconsistency, in Portuguese. PhD thesis, Universidade Estadual de Campinas (1998)

    Google Scholar 

  10. Estrada-González, L.: Quaternality in a topos-theoretical setting, unpublished typescript (2011)

    Google Scholar 

  11. Estrada-González, L.: Complement-topoi and dual intuitionistic logic. Australasian J. Log. 9 26–44 (2010)

    Google Scholar 

  12. Estrada-González, L.: Fifty (more or less) shades of logical consequence. In: Arazim, P., Dančák, M. (eds.) LOGICA Yearbook 2014. College Publications, London (2015)

    Google Scholar 

  13. Frankowski, S.: Formalization of a plausible inference. Bull. Sect. Log. 33, 41–52 (2004)

    MATH  MathSciNet  Google Scholar 

  14. Frankowski, S.: P-consequence versus q-consequence operations. Bull. Sect. Log. 33, 197–207 (2004)

    MATH  MathSciNet  Google Scholar 

  15. Goldblatt, R.: Topoi: The Categorial Analysis of Logic. Studies in Logic and the Foundations of Mathematics, vol. 98, revised edition. North Holland Publishing, Amsterdam (1984)

    Google Scholar 

  16. James, W., Mortensen, C.: Categories, sheaves, and paraconsistent logic. Unpublished typescript (1997)

    Google Scholar 

  17. Kotas, J., da Costa, N.C.A.: Some problems on logical matrices and valorizations. In: Arruda, A., da Costa, N.C.A., Sette, A.M. (eds.) Proceedings of the Third Brazilian Conference on Mathematical Logic, pp. 131–145. Sociedade Brasileira de Lógica, São Paulo (1980)

    Google Scholar 

  18. Lawvere, F.W.: Continuously variable sets: Algebraic geometry = geometric logic. In: Rose, H.E., Shepherdson, J.C. (eds.) Logic Colloquium ‘73 (Bristol, 1973), pp. 135–156. North Holland, Amsterdam (1975)

    Google Scholar 

  19. Lawvere, F.W.: Variable quantities and variable structures in topoi. In: Heller, A., Tierney, M. (eds.) Algebra, Topology, and Category Theory. A Collection of Papers in Honor of Samuel Eilenberg, pp. 101–131. Academic Press, New York (1976)

    Chapter  Google Scholar 

  20. Lawvere, F.W.: Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds.) Proceedings of the 1990 Meeting on Category Theory held in Como, Italy, Lecture Notes in Mathematics, vol. 1488, pp. 279–281. Springer, Berlin (1991)

    Google Scholar 

  21. Lawvere, F.W., Rosebrugh, R.: Sets for Mathematics. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  22. Lawvere, F.W., Schanuel, S.H.: Conceptual Mathematics. A First Introduction to Categories, second reprint. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  23. Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, New York (1992)

    Book  MATH  Google Scholar 

  24. G. Malinowski. Q-consequence operation. Rep. Math. Log. 24, 49–59 (1990)

    MATH  MathSciNet  Google Scholar 

  25. Malinowski, G.: Towards the concept of logical many-valuedness. Folia Philos. 7, 7–103 (1990)

    Google Scholar 

  26. McLarty, C.: Elementary Categories, Elementary Toposes. Oxford Clarendon Press, Toronto (1995)

    MATH  Google Scholar 

  27. McLarty, C.: Two constructivist aspects of category theory. Philos. Sci. 27 (Cahier spécial 6), 95–114 (2006)

    Google Scholar 

  28. Mortensen, C.: Inconsistent Mathematics. Kluwer Mathematics and Its Applications Series. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  29. Mortensen, C.: Closed set logic. In: Brady, R.T. (ed.) Relevant Logics and Their Rivals, vol. II, pp. 254–262. Ashgate Publishing, Aldershot (2003)

    Google Scholar 

  30. Reyes, G., Zawadowski, M.: Formal systems for modal operators on locales. Stud. Log. 52(4), 595–613 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Reyes, G., Zolfaghari, H.: Bi-Heyting algebras, toposes and modalities. J. Philos. Log. 25(1), 25–43 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Routley, R., Meyer, R.K.: Every sentential logic has a two-valued worlds semantics. Log. Anal. 19(74–76), 345–365 (1976)

    MATH  MathSciNet  Google Scholar 

  33. Scott, D.: Background to formalization. In: Leblanc, H. (ed.) Truth, Syntax and Modality, pp. 244–273. North Holland, Amsterdam (1973)

    Chapter  Google Scholar 

  34. Scott, D.: Completeness and axiomatizability in many-valued logics. In: Henkin, L., et al. (eds.) Proceedings of the Tarski Symposium, pp. 411–436. American Mathematical Society, USA (1974)

    Chapter  Google Scholar 

  35. Tsuji, M.: Many-valued logics and Suszko’s thesis revisited. Stud. Log. 60(2), 299–309 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. van Inwagen, P.: Metaphysics, 3rd edn. Westview Press, Boulder, Colorado (2008)

    Google Scholar 

  37. Vasyukov, V.: Structuring the universe of universal logic. Log. Univers. 1(2), 277–294 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Vigna, S.: A guided tour in the topos of graphs. Technical Report 199–7, Università di Milano, Dipartimento di Scienze dell’Informazione (1997). Available at http://vigna.dsi.unimi.it/papers.ToposGraphs.pdf

  39. Wansing, H., Shramko, Y.: Suszko’s thesis, inferential many-valuedness and the notion of logical system. Stud. Log. 88(1), 405–429 (2008). See also the erratum in vol. 89, p. 147 (2008)

    Google Scholar 

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Acknowledgement

I want to thank for the support from the CONACyT project CCB 2011 166502 “Aspectos filosóficos de la modalidad”. Diagrams were drawn using Paul Taylor’s diagrams package v. 3.94.

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Authors and Affiliations

  1. Instituto de Investigaciones Filosóficas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Luis Estrada-González

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  1. Luis Estrada-González
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Correspondence to Luis Estrada-González .

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  1. City University of New York, New York, New York, USA

    Arnold Koslow

  2. Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil

    Arthur Buchsbaum

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To Jean-Yves Béziau for his 50th birthday, and also to Christian Edward Mortensen for his 70th birthday.

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Estrada-González, L. (2015). From (Paraconsistent) Topos Logic to Universal (Topos) Logic. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15368-1_12

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