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HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes)

  • Hannes LeitgebEmail author
Article

Abstract

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

Keywords

Hyperintensional Intuitionistic logic Paraconsistent logic Truthmaker semantics Clutters Semantic paradoxes 

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Notes

Acknowledgements

I am grateful to Albert Anglberger, JC Beall, Johan van Benthem, Franz Berto, Walter Carnielli, Lucas Champollion, Ivano Ciardelli, J. Michael Dunn, Hartry Field, Martin Fischer, Volker Halbach, Ole Hjortland, Levin Hornischer, Leon Horsten, Johannes Korbmacher, David Makinson, Ed Mares, Joan R. Moschovakis, Yiannis Moschovakis, Julien Murzi, Carlo Nicolai, Lavinia Picollo, Andreas Pietz, Graham Priest, Greg Restall, Floris Roelofsen, Lorenzo Rossi, Benjamin Schnieder, Moritz Schulz, Stanislav Speranski, Andrew Tedder, Frank Veltman, Albert Visser, Philip Welch, Steve Yablo, and an anonymous reviewer for comments on earlier versions of this paper. This work was supported by an Alexander von Humboldt Professorship grant by the Alexander von Humboldt Foundation.

References

  1. 1.
    Almukdad, A., & Nelson, D. (1984). Constructible falsity and inexact predicates. Journal of Symbolic Logic, 49, 231–33.Google Scholar
  2. 2.
    Anderson, A.R., & Belnap, N.D. Jr. (1962). Tautological entailments. Philosophical Studies, 13, 9–24.Google Scholar
  3. 3.
    Anderson, A.R., & Belnap, N.D. Jr. (1975). Entailment: the logic of relevance and necessity Vol. 1. Princeton: Princeton University Press.Google Scholar
  4. 4.
    Anderson, A.R., Belnap, N.D. Jr., Dunn, J.M. (1992). Entailment: the logic of relevance and necessity Vol. 2. Princeton: Princeton University Press.Google Scholar
  5. 5.
    Angell, R.B. (1977). Three systems of first degree entailment. The Journal of Symbolic Logic, 42, 147.Google Scholar
  6. 6.
    Angell, R.B. (2002). A-Logic. Lanham: University Press of America.Google Scholar
  7. 7.
    Bacon, A. (2013). A new conditional for naive truth theory. Notre Dame Journal of Formal Logic, 54, 87–104.Google Scholar
  8. 8.
    Barwise, J., & Etchmendy, J. (1989). The liar: an essay in truth and circularity. Oxford: Oxford University Press.Google Scholar
  9. 9.
    Barwise, J., & Perry, J. (1983). Situations and attitudes. Cambridge: The MIT Press.Google Scholar
  10. 10.
    Beall, J. (2007). Revenge of the liar. New essays on the paradox. Oxford: Oxford University Press.Google Scholar
  11. 11.
    Beall, J. (2009). Spandrels of truth. Oxford: Oxford University Press.Google Scholar
  12. 12.
    Belnap, N.D. (1977). A useful four-valued logic. In Dunn, J.M., & Epstein, G. (Eds.) Modern uses of multiple-valued logic (pp. 8–37). Dordrecht: Reidel.Google Scholar
  13. 13.
    Berto, F. (2015). A modality called ‘negation’. Mind, 124, 761–93.Google Scholar
  14. 14.
    Bialynicki-Birula, A., & Rasiowa, H. (1957). On the representation of quasi-Boolean algebras. Bulletin Academie Polen. Sci. Cl., 3, 259–61.Google Scholar
  15. 15.
    Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–43.Google Scholar
  16. 16.
    Briggs, R. (2012). Interventionist counterfactuals. Philosophical Studies, 160, 139–66.Google Scholar
  17. 17.
    Brunner, A.B., & Carnielli, W.A. (2005). Anti-intuitionism and paraconsistency. Journal of Applied Logic, 3, 161–84.Google Scholar
  18. 18.
    Carnap, R. (1947). Meaning and necessity. Chicago: University of Chicago Press.Google Scholar
  19. 19.
    Charlwood, G.W. (1981). An axiomatic version of positive semi-lattice relevance logic. Journal of Symbolic Logic, 46, 233–39.Google Scholar
  20. 20.
    Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40, 55–94.Google Scholar
  21. 21.
    Cobreros, P., Egré, P., Ripley, D., van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41, 347–85.Google Scholar
  22. 22.
    Conforti, M., Cornuéjols, G., Zambelli, G. (2014). Integer programming. Heidelberg: Springer.Google Scholar
  23. 23.
    Cordovil, R., Fukuda, K., Moreira, M.L. (1991). Clutters and matroids. Discrete Mathematics, 89, 161–71.Google Scholar
  24. 24.
    Correia, F. (2016). On the logic of factual equivalence. The Review of Symbolic Logic, 9, 103–22.Google Scholar
  25. 25.
    Cresswell, M.J. (1975). Hyperintensional logic. Studia Logica, 34, 25–38.Google Scholar
  26. 26.
    Dunn, J.M. (1966). The algebra of intensional logics. PhD thesis, University of Pittsburgh.Google Scholar
  27. 27.
    Dunn, J.M., & Belnap, N.D. Jr. (1975). Intensional algebras. In Anderson, A.R. (Ed.) Entailment (pp. 180–206). Princeton: Princeton University Press.Google Scholar
  28. 28.
    Dunn, J.M. (1976). Intuitive semantics for first-degree entailments and coupled trees. Philosophical Studies, 29, 149–68.Google Scholar
  29. 29.
    Dunn, J.M. (1993). Star and perp: two treatments of negation. Philosophical Perspectives, 7, 331–357.Google Scholar
  30. 30.
    Dunn, J.M. (1996). Generalized ortho negation. In Wansing, H. (Ed.) Negation: a concept in focus (pp. 3–26): de Gruyter.Google Scholar
  31. 31.
    Dunn, J.M. (1999). A comparative study of various semantical treatments of negation: a history of formal negation. In Gabbay, D., & Wansing, H. (Eds.) What is negation? (pp. 23–51). Dordrecht: Kluwer Academic Publishers.Google Scholar
  32. 32.
    Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 66, 5–40.Google Scholar
  33. 33.
    Dunn, J.M., & Restall 2002, G. (2002). Relevance logic. In Guenthner, F., & Gabbay, D. (Eds.) Handbook of philosophical logic (Vol. 6, pp. 1–128). Dordrecht: Kluwer.Google Scholar
  34. 34.
    Duzi, M., Jespersen, B., Materna, P. (2010). Procedural semantics for hyperintensional logic. foundations and applications of transparent intensional logic, logic, epistemology, and the unity of science Vol. 17. Dordrecht: Springer.Google Scholar
  35. 35.
    Edmonds, J., & Fulkerson, D.R. (1970). Bottleneck extrema. Journal of Combinatorial Theory, 8, 299–306.Google Scholar
  36. 36.
    Fefermann, S. (1984). Toward useful type-free theories I. Journal of Symbolic Logic, 49, 75–111.Google Scholar
  37. 37.
    Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.Google Scholar
  38. 38.
    Field, H. (2014). Naive truth and restricted quantification: saving truth a whole lot better. Review of Symbolic Logic, 7, 1–45.Google Scholar
  39. 39.
    Field, H. (forthcoming). Indicative conditionals, restricted quantifiers, and naive truth, forthcoming in Review of Symbolic Logic.Google Scholar
  40. 40.
    Fine, K. (2012). A guide to ground. In Correia, F., & Schnieder, B. (Eds.) Metaphysical grounding (pp. 37–80). Cambridge: Cambridge University Press.Google Scholar
  41. 41.
    Fine, K. (2012). Counterfactuals without possible worlds. Journal of Philosophy, 109, 221–46.Google Scholar
  42. 42.
    Fine, K. (2012). The pure logic of ground. Review of Symbolic Logic, 25, 1–25.Google Scholar
  43. 43.
    Fine, K. (2014). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43, 549–577.Google Scholar
  44. 44.
    Fine, K. (2014). Permission and possible worlds. Dialectica, 68, 317–36.Google Scholar
  45. 45.
    Fine, K. (2016). Angellic content. Journal of Philosophical Logic, 45, 199–226.Google Scholar
  46. 46.
    Fine, K. (2017). Truthmaker semantics. In Hale, B., Wright, C., Miller, A. (Eds.) A companion to the philosophy of language. 2nd edn. (pp. 556–77). Chichester: Wiley.Google Scholar
  47. 47.
    Fine, K. (forthcoming (a)). A theory of truth-conditional content I: conjunction, disjunction and negation, forthcoming in the Journal of Philosophical Logic.Google Scholar
  48. 48.
    Fine, K. (forthcoming (b)). A theory of truth-conditional content II: subject-matter, common content, remainder and ground, forthcoming in the Journal of Philosophical Logic.Google Scholar
  49. 49.
    Fox, C., & Lappin, S. (2001). A framework for the hyperintensional semantics of natural language with two implementations, LACL (logical aspects of computational linguistics), lecture notes in computer science, (pp. 175–92). Dordrecht: Springer.Google Scholar
  50. 50.
    van Fraassen, B. (1969). Facts and tautological entailments. Journal of Philosophy, 66, 477–87.Google Scholar
  51. 51.
    van Fraassen, B. (1980). The scientific image. Oxford: Oxford University Press.Google Scholar
  52. 52.
    Gabbay, D., & Maksimova, L. (2005). Interpolation and definability. Oxford: Clarendon Press.Google Scholar
  53. 53.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H. (2007). Residuated lattices: an algebraic glimpse at substructural logics, studies in logic and the foundations of mathematics Vol. 151. Amsterdam: Elsevier.Google Scholar
  54. 54.
    Gärdenfors, P. (2000). Conceptual spaces: the geometry of thought. Cambridge: The MIT Press.Google Scholar
  55. 55.
    Gärdenfors, P. (2014). The geometry of meaning: semantics based on conceptual spaces. Cambridge: The MIT Press.Google Scholar
  56. 56.
    Gödel, K. (1933). Zum intuitionistischen Aussagekalkül. Ergebnisse eines mathematischen Kolloquiums, 4, 40.Google Scholar
  57. 57.
    Goldblatt, R.I. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 19–35.Google Scholar
  58. 58.
    Görnemann, S. (1971). A logic stronger than intuitionism. The Journal of Symbolic Logic, 36(/2), 249–261.Google Scholar
  59. 59.
    Groenendijk, J., & Stokhof, M. (1984). Studies on the semantics of questions and the pragmatics of answers. PhD Thesis, University of Amsterdam.Google Scholar
  60. 60.
    Gupta, A., & Standefer, S. (2017). Conditionals in theories of truth. Journal of Philosophical Logic, 46, 27–63.Google Scholar
  61. 61.
    Gurevich, Y. (1977). Intuitionistic logic with strong negation. Studia Logica, 36, 49–59.Google Scholar
  62. 62.
    Halbach, V. (2011). Axiomatic theories of truth. Cambridge: Cambridge University Press.Google Scholar
  63. 63.
    Halbach, V., & Horsten, L. (2006). Axiomatizing Kripkes theory of truth. Journal of Symbolic Logic, 71, 677–712.Google Scholar
  64. 64.
    Halpern, J.Y. (2003). Reasoning about uncertainty. Cambridge: The MIT Press.Google Scholar
  65. 65.
    Hansen, J., Pigozzi, G., van der Torre, L. (2007). Ten philosophical problems in deontic logic. In Boella, G. et al. (Eds.) Normative multi-agent systems. Dagstuhl: Dagstuhl Seminar Proceedings. http://drops.dagstuhl.de/opus/volltexte/2007/941.
  66. 66.
    Hermes, H. (1967). Einführung in die Verbandstheorie. Berlin: Springer.Google Scholar
  67. 67.
    Holliday, W., Hoshi, T., Icard, T. (2012). Uniform logic of information dynamics. In Bolander, T. et al. (Eds.) Advances in modal logic (Vol. 9, pp. 348–367). London: College Publications.Google Scholar
  68. 68.
    Hornischer, L. (2017). Hyperintensionality and synonymy. A logical, philosophical, and cognitive investigation, MSc Thesis, Institute for Logic. Language and Computation, University of Amsterdam.Google Scholar
  69. 69.
    Horsten, L. (2011). The Tarskian turn. Deflationism and axiomatic truth. Cambridge: MIT Press.Google Scholar
  70. 70.
    Jago, M. (2014). The impossible. An essay on hyperintensionality. Oxford: Oxford University Press.Google Scholar
  71. 71.
    Jennings, R.E., & Chen, Y. (2013). FDE: a logic of clutters. In Tanaka, K., Berto, F., Mares, E., Paoli, F. (Eds.) Paraconsistency: logic and applications (pp. 163–72). Dordrecht: Springer.Google Scholar
  72. 72.
    Kalman, J.A. (1958). Lattices with involution. Transactions of the American Mathematical Society, 87, 485–91.Google Scholar
  73. 73.
    Kamp, H. (1973). Free choice permission. Proceedings of the Aristotelian Society, 74, 57–74.Google Scholar
  74. 74.
    Kapsner, A. (2014). Logics and falsifications. A new perspective on constructivist semantics, trends in logic Vol. 40. Heidelberg: Springer.Google Scholar
  75. 75.
    Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press.Google Scholar
  76. 76.
    Kraus, S., Lehmann, D., Magidor, M. (1990). Non-monotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.Google Scholar
  77. 77.
    Kripke, S.A. (1965). Semantical analysis of intuitionistic logic. In Crossley, J., & Dummett, M.A.E. (Eds.) Formal systems and recursive functions (pp. 92–130). Amsterdam: North-Holland.Google Scholar
  78. 78.
    Kripke, S.A. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.Google Scholar
  79. 79.
    von Kutschera, F. (1969). Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle. Archiv für Mathematische Logik und Grundlagenforschung, 12, 104–18.Google Scholar
  80. 80.
    Leitgeb, H. (1999). Truth and the liar in De Morgan-valued models. Notre Dame Journal of Formal Logic, 40, 496–514.Google Scholar
  81. 81.
    Leitgeb, H. (2005). What truth depends on. Journal of Philosophical Logic, 34, 155–92.Google Scholar
  82. 82.
    Leitgeb, H. (2007). On the metatheory of field’s ‘solving the paradoxes, escaping revenge’. Beall, 2007, 159–83.Google Scholar
  83. 83.
    Leitgeb, H. (2010). On the Ramsey test without triviality. Notre Dame Journal of Formal Logic, 51, 21–54.Google Scholar
  84. 84.
    Leitgeb, H. (2017). The stability of belief. How rational belief coheres with probability. Oxford: Oxford University Press.Google Scholar
  85. 85.
    Lewis, D. (1973). Counterfactuals. Oxford: Blackwell.Google Scholar
  86. 86.
    Lewis, D. (1988). Statements partly about observation. Philosophical Papers, 17, 1–31.Google Scholar
  87. 87.
    Makinson, D. (1993). Five faces of minimality. Studia Logica, 52, 339–79.Google Scholar
  88. 88.
    Mares, E.D. (1997). Relevant logic and the theory of information. Synthese, 109, 345–360.Google Scholar
  89. 89.
    Mares, E.D. (2004). Relevant logic: a philosophical interpretation. Cambridge: Cambridge University Press.Google Scholar
  90. 90.
    Mares, E.D. (2013). Information, negation, and paraconsistency. In Tanaka, K., Berto, F., Mares, E., & Paoli, F. (Eds.) Paraconsistency: logic and applications (pp. 43–55). Dordrecht: Springer.Google Scholar
  91. 91.
    Martin, R.L., & Woodruff, P.W. (1975). On representing ‘True-in-L’ in L. Philosophia, 5, 213–7.Google Scholar
  92. 92.
    Moltmann, F. (2007). Events, tropes and truthmaking. Philosophical Studies, 134, 363–403.Google Scholar
  93. 93.
    Moschovakis, J. (2005). Notes on the foundations of constructive mathematics, unpublished manuscript, available at: http://www.math.ucla.edu/~joan/newnotes.pdf.
  94. 94.
    Moschovakis, Y.N. (2006). A logical calculus of meaning and synonymy. Linguistics and Philosophy, 29, 27–89.Google Scholar
  95. 95.
    Muskens, R. (1995). Meaning and partiality. CSLI Publications: Stanford.Google Scholar
  96. 96.
    Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14, 16–26.Google Scholar
  97. 97.
    Nolan, D. (2014). Hyperintensional metaphysics. Philosophical Studies, 171, 149–160.Google Scholar
  98. 98.
    Odintsov, S.P., Speranski, S.O., Shevchenko, I.Yu. (2018). Hintikka’s independence-friendly logic meets Nelson’s realizability. Studia Logica, 106, 637–70.Google Scholar
  99. 99.
    Perry, J. (1989). Possible worlds and subject matter. In Perry, J. (Ed.) The problem of the essential indexical and other essays (pp. 145–60). CSLI Publications: Stanford.Google Scholar
  100. 100.
    Pietz, A., & Rivieccio, U. (2013). Nothing but the truth. Journal of Philosophical Logic, 42, 125–35.Google Scholar
  101. 101.
    Pollard, C. (2008). Hyperintensions. Journal of Logic and Computation, 18, 257–82.Google Scholar
  102. 102.
    Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–41.Google Scholar
  103. 103.
    Priest, G. (1987). In contradiction. A study of the transconsistent. Dordrecht: Martinus Nijhoff.Google Scholar
  104. 104.
    Priest, G. (2002). Paraconsistent logic. In Gabbay, D., & Guenther, F. (Eds.) Handbook of philosophical logic. 2nd Edn. (Vol. 6, pp. 287–393). Dordrecht: Kluwer.Google Scholar
  105. 105.
    Priest, G. (2008). An introduction to non-classical logic. From if to is. Cambridge: Cambridge University Press.Google Scholar
  106. 106.
    Punčochář, V. (Unpublished). Substructural inquisitive logics, unpublished manuscript.Google Scholar
  107. 107.
    Rasiowa, H., & Sikorski, R. (1963). The mathematics of metamathematics. Państwowe Wydawnictwo Naukowe: Warsaw.Google Scholar
  108. 108.
    Rauszer, C., & Sabalski, B. (1975). Notes on the Rasiowa-Sikorski lemma. Studia Logica, 34, 265–8.Google Scholar
  109. 109.
    Read, S. (1988). Relevant logic. Oxford: Blackwell.Google Scholar
  110. 110.
    Restall, G. (1995). Four-valued semantics for relevant logics (and some of their rivals. Journal of Philosophical Logic, 24, 139–60.Google Scholar
  111. 111.
    Restall, G. (1996). Information flow and relevant logics. In Seligman, J., & Westerstahl, D. (Eds.) Logic, language and computation (pp. 463–77). Stanford: CSLI.Google Scholar
  112. 112.
    Restall, G. (2000). Modelling truthmaking. Logique et Analyse, 169–170, 211–30.Google Scholar
  113. 113.
    Rossi, L. (2016). Adding a conditional to Kripke’s theory of truth. Journal of Philosophical Logic, 45, 485–529.Google Scholar
  114. 114.
    Van Rooij, R. (2017). A fine-grained global analysis of implicatures. In Pistioa-Reda, S., & Domaneschi, F. (Eds.) Linguistic and psycholinguistic approaches on implicatures and presuppositions (pp. 73–110). Cham: Palgrave Macmillan.Google Scholar
  115. 115.
    Routley, R., & Routley, V. (1972). Semantics of first-degree entailment. Nous, 3, 335–59.Google Scholar
  116. 116.
    Routley, R., Plumwood, V., Meyer, R.K., Brady, R.T. (1982). Relevant logics and their rivals, Ridgeview Press.Google Scholar
  117. 117.
    Schindler, T. (2014). Axioms for grounded truth. Review of Symbolic Logic, 7, 73–83.Google Scholar
  118. 118.
    Schnieder, B. (2011). A logic for ‘because’. Review of Symbolic Logic, 4, 445–65.Google Scholar
  119. 119.
    Seymour, P.D. (1976). The forbidden minors of binary clutters. Journal of the London Mathematical Society (2), 12, 356–60.Google Scholar
  120. 120.
    Thomason, R.H. (1969). A semantical study of constructible falsity. Zeitschrift for mathematische Logik und Grundlagen der Mathematik, 15, 247–57.Google Scholar
  121. 121.
    Urquhart, A. (1972). Semantics for relevant implication. Journal of Symbolic Logic, 37, 15–69.Google Scholar
  122. 122.
    Veltman, F. (1985). Logics for conditionals. PhD Thesis, University of Amsterdam (UvA).Google Scholar
  123. 123.
    Visser, A. (1984). Four valued semantics and the liar. Journal of Philosophical Logic, 13, 181–212.Google Scholar
  124. 124.
    Wansing, H. (1993). The logic of information structures. Berlin: Springer.Google Scholar
  125. 125.
    Williamson, T. (1994). Never say never. Topoi, 13, 135–145.Google Scholar
  126. 126.
    Woodruff, P.W. (1984). Paradox, truth and logic I. Journal of Philosophical Logic, 13, 213–232.Google Scholar
  127. 127.
    Wright, C. (2001). On being in a quandary: relativism, vagueness, logical revisionism. Mind, 110, 45–98.Google Scholar
  128. 128.
    Yablo, S. (1982). Grounding, dependence, and paradox. Journal of Philosophical Logic, 11, 117–37.Google Scholar
  129. 129.
    Yablo, S. (2014). Aboutness. Princeton: Princeton University Press.Google Scholar
  130. 130.
    Yablo, S. (2016). Ifs, ands, and buts: an incremental truthmaker-semantics for indicative conditionals. Analytic Philosophy, 57, 175–213.Google Scholar
  131. 131.
    Yolov, N. (2016). Unpublished: blocker size via matching minors, unpublished manuscript (arXiv:1606.06263).
  132. 132.
    Zalta, E.N. (1983). Abstract objects: an introduction to axiomatic metaphysics. Dordrecht: Reidel.Google Scholar

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Authors and Affiliations

  1. 1.LMU MunichMunichGermany

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