Notre Dame Journal of Formal Logic

Inferentialism and Quantification

Owen Griffiths

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Logical inferentialists contend that the meanings of the logical constants are given by their inference rules. Not just any rules are acceptable, however: inferentialists should demand that inference rules must reflect reasoning in natural language. By this standard, I argue, the inferentialist treatment of quantification fails. In particular, the inference rules for the universal quantifier contain free variables, which find no answer in natural language. I consider the most plausible natural language correlate to free variables—the use of variables in the language of informal mathematics—and argue that it lends inferentialism no support.

Article information

Notre Dame J. Formal Logic, Volume 58, Number 1 (2017), 107-113.

Received: 23 September 2013
Accepted: 24 May 2014
First available in Project Euclid: 25 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B65: Logic of natural languages [See also 68T50, 91F20] 03B10: Classical first-order logic 03B35: Mechanization of proofs and logical operations [See also 68T15]

inferentialism quantifiers logical constants logical consequence proof theory


Griffiths, Owen. Inferentialism and Quantification. Notre Dame J. Formal Logic 58 (2017), no. 1, 107--113. doi:10.1215/00294527-3768059.

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  • [1] Brandom, R., Making it Explicit, Harvard University Press, Cambridge, Mass., 1994.
  • [2] Dummett, M., Frege: Philosophy of Mathematics, Harvard University Press, Cambridge, Mass., 1991.
  • [3] Kleene, S. C., Introduction to Metamathematics, North-Holland, Amsterdam, 1952.
  • [4] Negri, S., and J. von Plato, Structural Proof Theory, with Appendix C by A. Ranta, Cambridge University Press, Cambridge, 2001.
  • [5] Prawitz, D., “On the idea of a general proof theory,” Synthese, vol. 27 (1974), pp. 63–77.
  • [6] Read, S., “Harmony and autonomy in classical logic,” Journal of Philosophical Logic, vol. 29 (2000), pp. 123–54.
  • [7] Rosser, J. B., Logic for Mathematicians, McGraw-Hill, New York, 1953.
  • [8] Steinberger, F., “Why conclusions should remain single,” Journal of Philosophical Logic, vol. 40 (2011), pp. 333–55.