Foundations of Science

, Volume 20, Issue 1, pp 59–76

Hyper-Slingshot. Is Fact-Arithmetic Possible?

Article

Abstract

The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model (world) of arithmetical theories. It may be interpreted as showing that it is impossible to construct fact-arithmetic. The importance of this conclusion arises in the context of cognitive science. In the paper, a new type of slingshot argument is presented, which is called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that the former are formulated in the semantic meta-language of mathematical theories without the use of the iota-operator or the lambda-operator as the abstractor, whereas the latter require for their expression at least one of these non-standard term-operators. Hyper-slingshots implement simpler language tools in comparison with those used in conventional slingshots.

Keywords

Slingshot Non-Fregean logic Fact-arithmetic arithmetic truth-makers Extensionality Compositionality  Facts 

References

  1. Anscombe, G. E. M. (1969). Causality and extensionality. The Journal of Philosophy, 66, 152–159.CrossRefGoogle Scholar
  2. Barwise, J., & Perry, J. (1981). Semantic innocence and uncompromising situations. Midwest Studies in Philosophy, 6, 387–403.CrossRefGoogle Scholar
  3. Baumgartner, M. (2010a). Causal slingshots. Erkenntnis, 72, 11–133.CrossRefGoogle Scholar
  4. Baumgartner, M. (2010b). Shallow analysis and the slingshot argument. Journal of Philosophical Logic, 39, 531–556.CrossRefGoogle Scholar
  5. Biłat, A. (2009). Filozoficzne podstawy logiki sytuacji’ (‘Philosophical Foundations of logic of situations’). In A. Biłat (Ed.), Aporie ontologii sytuacji (pp. 95–153). Lublin: Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej.Google Scholar
  6. Church, A. (1943). Carnap’s introduction to semantics. The Philosophical Review, 52, 298–304.CrossRefGoogle Scholar
  7. Correia, F. (2010). Grounding and truth-functions. Logique et Analyse, 53(211), 251–279.Google Scholar
  8. Cummins, R., & Gottlieb, D. (1972). On an argument for truth-functionality. American Philosophical Quarterly, 9, 265–269.Google Scholar
  9. Davidson, D. (1967). Truth and meaning. Synthese, 17, 304–323.CrossRefGoogle Scholar
  10. Davidson, D. (1969). True to the facts. The Journal of Philosophy, 66, 748–764.CrossRefGoogle Scholar
  11. Davidson, D. (1990). The structure and content of truth. The Journal of Philosophy, 87, 279–328.CrossRefGoogle Scholar
  12. Donaho, S. (1998). Are declarative sentences representations? Mind, 107, 33–57.CrossRefGoogle Scholar
  13. Drai, D. (2002). The slingshot argument: An improved version. Ratio (new series), 15, 194–204.Google Scholar
  14. Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.Google Scholar
  15. Fine, K. (2012). The pure logic of ground. The review of symbolic logic, 5, 1–25.CrossRefGoogle Scholar
  16. Føllesdal, D. (1983). Situation semantics and the “slingshot” argument. Erkenntnis, 19, 91–98.CrossRefGoogle Scholar
  17. Geach, P. T. (1976). Saying and showing in Frege and Wittgenstein. In J. Hintikka (Ed.), Essays on Wittgenstein in Honour of G.H. von Wright (pp. 54–70). Amsterdam: Acta Philosophica Fennica, 28.Google Scholar
  18. Golińska-Pilarek, J., & Huuskonen, T. (2005). Number of extensions of non-Fregean logics. Journal of Philosophical Logic, 34, 193–206.CrossRefGoogle Scholar
  19. Gödel, K. (1944). Russell’s mathematical logic. In P. A. Schlipp (Ed.), The Philosophy of Bertrand Russell (pp. 125–153). Evanston and Chicago: Northwestern University Press.Google Scholar
  20. Krysztofiak, W. (2007a). The phenomenological idealism controversy in light of possible worlds semantics. Axiomathes, 17, 75–97.CrossRefGoogle Scholar
  21. Krysztofiak, W. (2007b). Spór o ontologię sytuacji jako spór o zasadę kompozycyjności. Argument z metafory (The controversy over ontology of situations as the controversy over the compositionality principle. The argument of a metaphor). Filozofia Nauki, 15, No 4(60), pp. 51–70.Google Scholar
  22. Krysztofiak, W. (2009). Co wyraża argument slingshot? (What does the slingshot argument express?). In A. Biłat (Ed.), Aporie ontologii sytuacji (Aporias of ontology of situations) (pp. 37–73). Lublin: Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej.Google Scholar
  23. Krysztofiak, W. (2011). Fakty matematyczne w świetle logiki niefregowskiej (Mathematical facts in light of non-Fregean logic). Filozofia Nauki, 19, No 4(76), 83–101.Google Scholar
  24. Krysztofiak, W. (2013). Do we need mathematical facts? History and philosophy of logic. http://www.tandfonline.com/doi/full/10.1080/01445340.2013.828517
  25. Lycan, W. D. (1974). The extensionality of cause, space and time. Mind, 83, 498–511.CrossRefGoogle Scholar
  26. Mulligan, K., Simons, P., & Smith, B. (1984). Truth-makers. Philosophy and the Phenomenological Research, 44, 287–321.CrossRefGoogle Scholar
  27. Niiniluoto, I. (2004). Tarski’s definition and truth-makers. Annals of Pure and Applied Logic, 126, 57–76.CrossRefGoogle Scholar
  28. Nowak, M. (2008). The logics of analytic equivalence. Bulletin of the Section of Logic, 37(3/4), 265–272.Google Scholar
  29. Neale, S. (1995). The philosophical significance of Gödel’s slingshot. Mind, 104, 761–825.CrossRefGoogle Scholar
  30. Omyła, M. (1986). Zarys logiki niefregowskiej (The outline of non-Fregean logic). Warszawa: PWN.Google Scholar
  31. Omyła, M. (2009), ‘Deskrypcje w logice niefregowskiej’ (‘Descriptions In non-Fregean logic’), [w:] (red. A. Biłat), Aporie ontologii sytuacji (Aporias of ontology of situations), Lublin 2009, Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej, s. 75–93.Google Scholar
  32. Oppy, G. (1997). The philosophical insignificance of Gödel’s slingshot. Mind, 106, 121–141.CrossRefGoogle Scholar
  33. Putnam, H. (1975). Do true assertion correspond to reality. In H. Putnam (Ed.), Mind, language and reality. Philosophical papers (Vol. 2, pp. 70–84). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  34. Quine, W. O. (1966). Three grades of modal involvement. In W. O. Quine (Ed.), The ways of paradox (pp. 156–174). New York: Random House.Google Scholar
  35. Quine, W. V. (1970). Philosophy of logic. Englewood: Prentice-Hall.Google Scholar
  36. Restall, G. (2004). One way to face facts. The Philosophical Quarterly, 54, 420–426.CrossRefGoogle Scholar
  37. Rodriguez-Pereyra, G. (1998). Searle’s correspondence theory of truth and the slingshot. The Philosophical Quarterly, 48(193) (1998), 513–522.Google Scholar
  38. Searle, J. (1996). The construction of social reality. Harmondsworth: Penguin.Google Scholar
  39. Shramko, Y., & Wansing, H. (2009). The slingshot argument and sentential identity. Studia Logica, 91, 429–455.CrossRefGoogle Scholar
  40. Smith, B. (1989). Logic and the sachverhalt. The Monist, 72, 52–69.CrossRefGoogle Scholar
  41. Suszko, R. (1968). Ontology in the Tractatus of L. Wittgenstein. Notre Dame Journal of Formal Logic, 9(1), 7–33.Google Scholar
  42. Suszko, R. (1998). Logika i teorie niefregowskie. In Suszko, R. Wybór pism, Polskie Towarzystwo Semiotyczne: Warszawa. [translated from: Suszko, R. (1968). Non-Fregean logic and theories. Analele Universitatii Bucuresti, Acta logica XI, 105–125].Google Scholar
  43. Widerker, D. (1983). The extensionality argument. Noûs, 17, 457–468.CrossRefGoogle Scholar
  44. Williamson, J. (1976). Facts and truth. The Philosophical Quarterly, 26, 203–216.CrossRefGoogle Scholar
  45. Wójcicki, R. (1984). Romana Suszki semantyka sytuacyjna (Roman Suszko’s situational semantics). Studia Filozoficzne, 7(224), 3–19.Google Scholar
  46. Wójtowicz, A. (2005). The slingshot argument and non-Fregean logic. In A. Brożek, J. J. Jadacki, & W. Strawiński (Eds.), Logic, methodology and philosophy of science in Warsaw (Vol. 2, pp. 185–192). Warszawa: Wydawnictwo Naukowe Semper.Google Scholar
  47. Wójtowicz, A. (2007) Znaczenie nazw a znaczenie zdań. W obronie ontologii sytuacji (The meaning of names and the meaning of sentences. In defense of ontology of situations). Warszawa: Wydawnictwo Naukowe Semper.Google Scholar
  48. Yourgrau, P. (1987). Frege on truth and reference. Notre Dame Journal of Formal Logic, 28, 132–138.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute of PhilosophyUniversity of SzczecinSzczecinPoland

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