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Journal of Logic, Language and Information

, Volume 25, Issue 2, pp 215–268 | Cite as

Subatomic Natural Deduction for a Naturalistic First-Order Language with Non-Primitive Identity

  • Bartosz WięckowskiEmail author
Article

Abstract

A first-order language with a defined identity predicate is proposed whose apparatus for atomic predication is sensitive to grammatical categories of natural language (e.g., common nouns, verbs, adjectives, adverbs, modifiers). Subatomic natural deduction systems are defined for this naturalistic first-order language. These systems contain subatomic systems which govern the inferential relations which obtain between naturalistic atomic sentences and between their possibly composite components. As a main result it is shown that normal derivations in the defined systems enjoy the subexpression property which subsumes the subformula property with respect to atomic and identity formulae as a special case. The systems admit a proof-theoretic semantics which does not only apply to logically compound but also to atomic and identity formulae—as well as to their components. The potential of the defined systems for a meticulous first-order analysis of natural inferences whose validity crucially depends on expressions of some of the aforementioned categories is demonstrated.

Keywords

Identity Natural deduction Natural logic Normalization  Proof-theoretic semantics Simple rules 

Notes

Acknowledgments

I would like to thank Nissim Francez for discussions on proof-theoretic semantics and an anonymous referee for her/his feedback. This work was supported by the DFG (Grant WI 3456/2-1).

References

  1. Davidson, D. (2001). The logical form of action sentences. In D. Davidson, Essays on actions and events (pp. 105–122). Oxford: Clarendon Press. (Originally published in N. Rescher (ed.), The Logic of Decision and Action, Pittsburg, University of Pittsburg Press, 1967.).Google Scholar
  2. Davies, R., & Pfenning, F. (2001). A modal analysis of staged computation. Journal of the ACM, 48(3), 555–604.CrossRefGoogle Scholar
  3. Dummett, M. (1991). The logical basis of metaphysics. Cambridge, MA: Harvard University Press.Google Scholar
  4. Fitch, F. B. (1973). Natural deduction rules for English. Philosophical Studies, 24(2), 89–104.CrossRefGoogle Scholar
  5. Forbes, G. (2013). Intensional transitive verbs. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2013 Edition). http://plato.stanford.edu/archives/fall2013/entries/intensional-trans-verbs/.
  6. Francez, N. (2015a). Proof-theoretic semantics. London: College Publications.Google Scholar
  7. Francez, N. (2015b). On the notion of canonical derivations from open assumptions and its role in proof-theoretic semantics. The Review of Symbolic Logic, 8(2), 296–305.CrossRefGoogle Scholar
  8. Francez, N., & Ben-Avi, G. (2015). Proof-theoretic reconstruction of generalized quantifiers. Journal of Semantics, 32(3), 313–371.CrossRefGoogle Scholar
  9. Francez, N., & Dyckhoff, R. (2010). Proof-theoretic semantics for a natural language fragment. Linguistics and Philosophy, 33(6), 447–477.CrossRefGoogle Scholar
  10. Francez, N., & Dyckhoff, R. (2012). A note on harmony. The Journal of Philosophical Logic, 41(3), 613–628.CrossRefGoogle Scholar
  11. Francez, N., Dyckhoff, R., & Ben-Avi, G. (2010). Proof-theoretic semantics for subsentential phrases. Studia Logica, 94(3), 381–401.CrossRefGoogle Scholar
  12. Francez, N., & Więckowski, B. (forthcoming). A proof theory for first-order logic with definiteness. IfCoLog Journal of Logics and their Applications.Google Scholar
  13. Frege, G. (1879). Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle an der Saale: Verlag von Louis Nebert. (Reprint: I. Angelelli (ed.), Hildesheim, Georg Olms Verlag, 1998).Google Scholar
  14. Frege, G. (1893/1903). Grundgesetze der Arithmetik I/II. Jena: Verlag von Hermann Pohle. (Reprint: C. Thiel (ed.), Hildesheim, Georg Olms Verlag, 2009).Google Scholar
  15. Gamut, L. T. F. (1991). Logic, language, and meaning: Intensional logic and logical grammar (Vol.2). Chicago: The University of Chicago Press.Google Scholar
  16. Gentzen, G. (1934). Untersuchungen über das logische Schließen I, II. Mathematische Zeitschrift, 39, 176–210, 405–432Google Scholar
  17. Griffiths, O. (2014). Harmonious rules for identity. The Review of Symbolic Logic, 7(3), 499–510.CrossRefGoogle Scholar
  18. Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1, 5–32.Google Scholar
  19. Maienborn, C. (2011). Event semantics. In C. Maienborn, K. von Heusinger, & P. Portner (Eds.), Semantics: An international handbook of natural language meaning (Vol. 1, pp. 802–829). Berlin: de Gruyter Mouton.CrossRefGoogle Scholar
  20. Montague, R. (1974). Formal philosophy. Selected papers of Richard Montague, R. H. Thomason (Ed.). New Haven: Yale University Press.Google Scholar
  21. Pfenning, F., & Davies, R. (2001). A judgmental reconstruction of modal logic. Mathematical Structures in Computer Science, 11, 511–540.CrossRefGoogle Scholar
  22. Piecha, T., de Campos Sanz, W., & Schroeder-Heister, P. (2015). Failure of completeness in proof-theoretic semantics. Journal of Philosophical Logic, 44(3), 321–335.CrossRefGoogle Scholar
  23. Prawitz, D. (1965). Natural deduction. A proof-theoretical study. Stockholm: Almqvist & Wiksell. (Reprint: Mineola/NY, Dover Publications, 2006.).Google Scholar
  24. Prawitz, D. (1971). Ideas and results in proof theory. In J. E. Fenstad (Ed.), Proceedings of the second Scandinavian Logic symposium (Oslo 1970) (pp. 235–309). Amsterdam: North-Holland.CrossRefGoogle Scholar
  25. Prawitz, D. (1973). Towards a foundation of a general proof theory. In P. Suppes (Ed.), Logic, methodology and philosophy of science IV (pp. 225–250). Amsterdam: North-Holland.Google Scholar
  26. Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148(3), 507–524. Special issue on Proof-Theoretic Semantics edited by R. Kahle and P. Schroeder-Heister.Google Scholar
  27. Quine, W. V. (1969). Existence and quantification. In W. V. Quine (Ed.), Ontological relativity and other essays (pp. 94–96). New York: Columbia University Press.Google Scholar
  28. Read, S. (2004). Identity and harmony. Analysis, 64, 113–119. (Revised version: Identity and harmony revisited, 2014).Google Scholar
  29. Read, S. (2015). General elimination harmony and higher-level rules. In H. Wansing (Ed.), Dag Prawitz on proofs and meaning (pp. 293–312). Berlin: Springer.Google Scholar
  30. Sandqvist, T. (2015). Hypothesis-discharging rules in atomic bases. In H. Wansing (Ed.), Dag Prawitz on proofs and meaning (pp. 313–328). Berlin: Springer.Google Scholar
  31. Schroeder-Heister, P. (2013). Proof-theoretic semantics. In Zalta, E. N. (Ed.), The Stanford encyclopedia of philosophy (Spring 2013 Edition). http://plato.stanford.edu/archives/spr2013/entries/proof-theoretic-semantics/.
  32. Schroeder-Heister, P. (2015). Harmony in proof-theoretic semantics: A reductive analysis. In H. Wansing (Ed.), Dag Prawitz on proofs and meaning (pp. 329–358). Berlin: Springer.Google Scholar
  33. Steinberger, F. (2013). On the equivalence conjecture for proof-theoretic harmony. Notre Dame Journal of Formal Logic, 54(1), 79–86.CrossRefGoogle Scholar
  34. Tarski, A. (1995). Pisma Logiczno-Filozoficzne: Tom 1. Prawda, J. Zygmunt (Ed.), Warszawa: Wydawnictwo Naukowe PWN.Google Scholar
  35. Tranchini, L. (2015). Harmonising harmony. The Review of Symbolic Logic, 8(3), 411–423.CrossRefGoogle Scholar
  36. Troelstra, A. S., & Schwichtenberg, H. (2000). Basic proof theory (2nd ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  37. van Dalen, D. (2004). Logic and structure (4th ed.). Berlin: Springer.CrossRefGoogle Scholar
  38. Wansing, H. (2001). The idea of a proof-theoretic semantics and the meaning of the logical operations. Studia Logica, 64(1), 3–20.CrossRefGoogle Scholar
  39. Wansing, H. (2015). Dag Prawitz on proofs and meaning: Outstanding contributions to logic (Vol. 7). Berlin: Springer.Google Scholar
  40. Więckowski, B. (2010). Associative substitutional semantics and quantified modal logic. Studia Logica, 94(1), 105–138.CrossRefGoogle Scholar
  41. Więckowski, B. (2011). Rules for subatomic derivation. The Review of Symbolic Logic, 4(2), 219–236.CrossRefGoogle Scholar
  42. Więckowski, B. (2012). A constructive type-theoretical formalism for the interpretation of subatomically sensitive natural language constructions. Studia Logica, 100(4), 815–853. Special issue on Logic and Natural Language edited by N. Francez and I. Pratt-Hartmann.Google Scholar
  43. Więckowski, B. (2015). Constructive belief reports. Synthese, 192(3), 603–633. Special section on Hyperintensionality edited by M. Duží and B. Jespersen.Google Scholar
  44. Więckowski, B. (forthcoming). Refinements of subatomic natural deduction, Journal of Logic and Computation, (first published online: August 20, 2014).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institut für PhilosophieGoethe-Universität Frankfurt am MainFrankfurt am MainGermany

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