Notre Dame Journal of Formal Logic

The Logical Strength of Compositional Principles

Richard G. Heck Jr.

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Abstract

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement.

Article information

Source
Notre Dame J. Formal Logic Volume 59, Number 1 (2018), 1-33.

Dates
Received: 4 September 2014
Accepted: 28 February 2015
First available in Project Euclid: 5 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1499241609

Digital Object Identifier
doi:10.1215/00294527-2017-0011

Subjects
Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
truth compositionality Tarski deflationism

Citation

Heck Jr., Richard G. The Logical Strength of Compositional Principles. Notre Dame J. Formal Logic 59 (2018), no. 1, 1--33. doi:10.1215/00294527-2017-0011. https://projecteuclid.org/euclid.ndjfl/1499241609


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References

  • [1] Beklemishev, L. D., “Reflection schemes and provability algebras in formal arithmetic,” Russian Mathematical Surveys, vol. 60 (2005), pp. 197–268.
  • [2] Burgess, J. P., Fixing Frege, Princeton University Press, Princeton, 2005.
  • [3] Corcoran, J., W. Frank, and M. Maloney, “String theory,” Journal of Symbolic Logic, vol. 39 (1974), pp. 625–37.
  • [4] Craig, W., and R. L. Vaught, “Finite axiomatizability using additional predicates,” Journal of Symbolic Logic, vol. 23 (1958), pp. 289–308.
  • [5] Enayat, A., and A. Visser, “Full satisfaction classes in a general setting (Part I),” preprint, https://pdfs.semanticscholar.org/730d/be402772e16926179b92bfa1416f636ce340.pdf (accessed 21 May 2017).
  • [6] Enayat, A., and A. Visser, “New constructions of satisfaction classes,” pp. 321–35 in Unifying the Philosophy of Truth, edited by T. Achourioti, H. Galinon, J. M. Fernández, and K. Fujimoto, Springer, New York, 2015. DOI 10.1007/978-94-017-9673-6.
  • [7] Feferman, S., “Arithmetization of metamathematics in a general setting,” Fundamenta Mathematicae, vol. 49 (1960/1961), pp. 35–92.
  • [8] Field, H., “Deflationist views of meaning and content,” Mind, vol. 103 (1994), pp. 249–85.
  • [9] Field, H., “Deflating the conservativeness requirement,” Journal of Philosophy, vol. 96 (1999), pp. 533–40.
  • [10] Field, H., “Compositional principles vs. schematic reasoning,” The Monist, vol. 89 (2006), pp. 9–27.
  • [11] Grzegorczyk, A., “Undecidability without arithmetization,” Studia Logica, vol. 79 (2005), pp. 163–230.
  • [12] Gupta, A., “A critique of deflationism,” Philosophical Topics, vol. 21 (1993), pp. 57–81.
  • [13] Hájek, P., and P. Pudlák, Metamathematics of First-order Arithmetic, Springer, New York, 1993.
  • [14] Halbach, V., “Disquotationalism and infinite conjunction,” Mind, vol. 108 (1999), pp. 1–22.
  • [15] Halbach, V., “Disquotational truth and analyticity,” Journal of Symbolic Logic, vol. 66 (2001), pp. 1959–73.
  • [16] Halbach, V., “How innocent is deflationism?,” Synthese, vol. 126 (2001), pp. 167–94.
  • [17] Heck, R. G., Jr., “Truth and disquotation,” Synthese, vol. 142 (2004), pp. 317–52.
  • [18] Heck, R. G., Jr., “Consistency and the theory of truth,” Review of Symbolic Logic, vol. 8 (2015), pp. 424–66.
  • [19] Heck, R. G., Jr., “The strength of truth-theories,” preprint, http://rgheck.frege.org/pdf/unpublished/StrengthOfTruthTheories.pdf (accessed 21 May 2017).
  • [20] Heck, R. G., “Disquotationalism and the compositional principles,” preprint, http://rgheck.frege.org/pdf/unpublished/CompositionalPrinciples.pdf, 2013(accessed 21 May 2017).
  • [21] Horwich, P., Truth, Blackwell, Oxford, 1990.
  • [22] Ketland, J., “Deflationism and Tarski’s paradise,” Mind, vol. 108 (1999), pp. 69–94.
  • [23] Leigh, G. E., and C. Nicolai, “Axiomatic truth, syntax and metatheoretic reasoning,” Review of Symbolic Logic, vol. 6 (2013), pp. 613–36.
  • [24] Mostowski, A., “On models of axiomatic systems,” Fundamenta Mathematicae, vol. 39 (1952), pp. 133–58.
  • [25] Nelson, E., Predicative Arithmetic, vol. 32 of Mathematical Notes, Princeton University Press, Princeton, 1986.
  • [26] Parsons, C., “Sets and classes,” Noûs, vol. 8 (1974), pp. 1–12.
  • [27] Pudlák, P., “Cuts, consistency statements and interpretations,” Journal of Symbolic Logic, vol. 50 (1985), pp. 423–41.
  • [28] Quine, W. V. O., “Concatenation as a basis for arithmetic,” Journal of Symbolic Logic, vol. 11 (1946), pp. 105–14.
  • [29] Shapiro, S., “Proof and truth: Through thick and thin,” Journal of Philosophy, vol. 95 (1998), pp. 493–521.
  • [30] Tarski, A., “A general method in proofs of undecidability,” pp. 1–35 in Undecidable Theories, edited by A. Tarski, A. Mostowski, and A. Robinson, vol. 13 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1953.
  • [31] Tarski, A., “The concept of truth in formalized languages,” pp. 152–278 in Logic, Semantics, and Metamathematics, edited by J. Corcoran, Hackett, Indianapolis, 1958.
  • [32] Tarski, A., A. Mostowski, and A. Robinson, Undecidable Theories, vol. 13 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1953.
  • [33] Visser, A., “Growing commas: A study of sequentiality and concatenation,” Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 61–85.
  • [34] Wang, H., “Truth definitions and consistency proofs,” Transactions of the American Mathematical Society, vol. 73 (1952), pp. 243–75.
  • [35] Wilkie, A. J., and J. B. Paris, “On the scheme of induction for bounded arithmetic formulas,” Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261–302.