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Consequence Mining

Constants Versus Consequence Relations

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Abstract

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation \(\Rightarrow_X\). We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse to—more precisely, forms a Galois connection with—the Bolzano–Tarski function.

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References

  1. Abrusán, M. (2011). Presuppositional and negative islands: A semantic account. Natural Language Semantics, 19(3), 257–321.

    Article  Google Scholar 

  2. Aczel, P. (1990). Replacement systems and the axiomatization of situation theory. In R. Cooper, K. Mukai, & J. Perry (Eds.), Situation theory and its applications (Vol. 1, pp. 3–33). Stanford: CLSI Publications.

    Google Scholar 

  3. Bar-Hillel, Y. (1950). Bolzano’s definition of analytic propositions. Theoria, 16(2), 91–117.

    Article  Google Scholar 

  4. Bolzano, B. (1837). Theory of science. Edited by J. Berg. D. Reidel: Dordrecht (1973)

  5. Bonevac, D. (1985). Quantity and quantification. Noûs, 19, 229–247.

    Article  Google Scholar 

  6. Bonnay, D. (2008). Logicality and invariance. Bulletin of Symbolic Logic, 14(1), 29–68.

    Article  Google Scholar 

  7. Bonnay, D., & Westerståhl, D. (2010). Logical consequence inside out. In M. Aloni & K. Schulz (Eds.), Amsterdam colloquium 2009. LNAI (Vol. 6042, pp. 193–202). Heidelberg: Springer.

    Google Scholar 

  8. Carnap, R. (1937). The logical syntax of language. London: Kegan, Paul, Trench Trubner & Cie. Rev. ed. translation of Logische Syntax der Sprache, Wien: Springer (1934).

  9. Dunn, M., & Belnap, N. (1968). The substitution interpretation of the quantifiers. Noûs, 4, 177–185.

    Article  Google Scholar 

  10. Feferman, S. (2010). Set-theoretical invariance criteria for logicality. Notre Dame Journal of Formal Logic, 51, 3–20.

    Article  Google Scholar 

  11. Fox, D., & Hackl, M. (2006). The universal density of measurement. Linguistics and Philosophy, 59(5), 537–586.

    Google Scholar 

  12. Gajewski, J. (2002). L-analyticity and natural language. Manuscript.

  13. Gentzen, G. (1932). Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen. Mathematische Annalen, 107, 329–350. English translation. In M. E. Szabo (Ed.), The collected papers of Gerhard Gentzen. Amsterdam: North-Holland (1969).

  14. Gómez-Torrente, M. (1996). Tarski on logical consequence. Notre Dame Journal of Formal Logic, 37(1), 125–151.

    Article  Google Scholar 

  15. Hertz, G. (1923). Über Axiomensysteme für beliebige Satzsysteme. Mathematische Annalen, 87, 246–269.

    Article  Google Scholar 

  16. Lewis, C. I., & Langford, C. H. (1932). Symbolig logic. New York: Dover.

    Google Scholar 

  17. MacFarlane, J. (2009). Logical constants. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2009 ed.). http://plato.stanford.edu/archives/fall2009/entries/logicalconstants/.

  18. Peters, S., & Westerståhl, D. (2006). Quantifiers in language and logic. Oxford: Oxford University Press.

    Google Scholar 

  19. Quine, W. (1976). Algebraic logic and predicate functors. In The ways of paradox (pp. 283–307). Cambridge: Harvard University Press.

    Google Scholar 

  20. Shoesmith, D. J., & Smiley, T. J. (1978). Multiple-conclusion logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  21. Tarski, A. (1930a). Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I’. Monatshefte für Mathematik und Physik, 37, 361–404. English translation in [24].

  22. Tarski, A. (1930b). Über einige fundamentale Begriffe der Metamathematik. Comptes rendus des séances de la Societeé des Sciences et des Lettres de Varsovie, 23, 22–29. English translation in Tarski [24].

  23. Tarski, A. (1936). On the concept of logical consequence. In [24] (pp. 409–420).

  24. Tarski, A. (1956). Logic, semantics, metamathematics. Oxford: Clarendon Press. Republished 1983 by Hackett Publishing, Indianapolis.

  25. Tarski, A. (1986). What are logical notions?. History and Philosophy of Logic, 7, 145–154.

    Article  Google Scholar 

  26. van Benthem, J. (2003). Is there still logic in Bolzano’s key?. In E. Morscher (Ed.), Bernard Bolzanos Leistungen in Logik, Mathematik und Physik Bd. 16 (pp. 11–34). Sankt Augustin: Academia.

  27. Westerståhl, D. (2011). From constants to consequence, and back. Synthese (online first). doi:10.1007/s11229-011-9902-z.

    Google Scholar 

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Correspondence to Denis Bonnay.

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We thank Johan van Benthem, Stephen Read, and Göran Sundholm for helpful remarks, and in particular Lloyd Humberstone, Dave Ripley, and an anonymous referee for very useful comments on an earlier version of this paper. We also thank the audiences at several seminars and workshops where we presented this material, in various stages, for inspiring discussion.

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Bonnay, D., Westerståhl, D. Consequence Mining. J Philos Logic 41, 671–709 (2012). https://doi.org/10.1007/s10992-012-9234-6

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  • DOI: https://doi.org/10.1007/s10992-012-9234-6

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