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Somehow Things Do Not Relate: On the Interpretation of Polyadic Second-Order Logic

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Abstract

Boolos has suggested a plural interpretation of second-order logic for two purposes: (i) to escape Quine’s allegation that second-order logic is set theory in disguise, and (ii) to avoid the paradoxes arising if the second-order variables are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Rayo and Yablo suggest an new interpretation for polyadic second-order logic in a Boolosian spirit. The present paper argues that Rayo and Yablo’s interpretation does not achieve the goal.

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Notes

  1. In essence, a model-theoretic semantics that assigns sets as values to predicates and predicate variables does just that.

  2. This informal characterization of the the proposal should suffice for our purpose. Boolos [1] defines a precise recursive translation procedure. Pioneering work in plural logic building on Boolos’s approach has been done, e.g., by Agustín Rayo and Gabriel Uzquiano [19] and Rayo [18].

  3. The insistence on the correct semantic (or thematic) role in the interpretation of predicates has been formulated by Fred Landman [9], §5.4; see also Thomas McKay [11], ch. 2.

  4. This point has previously been made by Crispin Wright [24]; see also [25].

  5. [15], chapter 2, esp. p. 35–38.

  6. Rayo and Yablo [20], p. 84. Regarding the legitimacy of the anaphorical ‘so’ in clause (e), Rayo and Yablo ([20], pp. 82–83) refer to Dorothy Grover [8].

  7. For simplicity’s sake, (16) is formulated using parameters, ‘a’ and ‘b’, rather than quantified variables, which would strictly speaking be required for Rayo and Yablo’s translation. Nothing hangs on this.

  8. I am indebted to Agustín Rayo, R. K. B. Vaas, Crispin Wright, and an anonymous referee for helpful comments.

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Authors and Affiliations

  1. Philosophy Department , University of Connecticut, 344 Mansfield Rd., Storrs, CT, 06269-1054, USA

    Marcus Rossberg

  2. Northern Institute of Philosophy, University Of Aberdeen, Aberdeen, Scotland

    Marcus Rossberg

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  1. Marcus Rossberg
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Correspondence to Marcus Rossberg.

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Rossberg, M. Somehow Things Do Not Relate: On the Interpretation of Polyadic Second-Order Logic. J Philos Logic 44, 341–350 (2015). https://doi.org/10.1007/s10992-014-9326-6

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