Rumfitt on truth-grounds, negation, and vagueness

Abstract

In The Boundary Stones of Thought (2015), Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.

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Notes

  1. 1.

    That is, a one-place predicate symbol is assigned a function from D to \(\{T, F\}\), a two-place predicate a function from \(D^2\), etc.

  2. 2.

    Sambin uses \(\mathscr {F}\) for both the pre-topology and its closure operator; I’ll use Rumfitt’s notation C for both.

  3. 3.

    “Saturated” in Sambin’s terminology.

  4. 4.

    Sambin defines \(U \rightarrow _C V = \{x : x \bullet y \in C(V)\,{\text{for}}\, {\text{all}}y\in U\}\), and in our context we are only concerned with V where \(V = C(V)\).

  5. 5.

    Rumfitt uses Brouwer’s original example of the occurrence of 0123456789 in the decimal extension of \(\pi\). It is now known that it does occur (Borwein 1998), so assume s is a sequence which so far has not been found among the decimal expansion of \(\pi\).

  6. 6.

    The ancestral \(R^*\) of a relation R is the relation which holds between x and y iff there is a finite sequence \(z_i\) such that \(xRz_1Rz_2 \dots z_nRy\).

  7. 7.

    He merely states: “[W]hich color predicates an object satisfies depends on which poles are maximally close in color to it. Hence, ...the members of \(X^\perp\) will be those objects whose color status is incompatible with being A” (p. 244).

  8. 8.

    Rumfitt’s proposal for a semantics underwriting Wright’s approach to the sorites using intuitionistic logic in Section 8.3 seems to also follow this pattern, as the appeal to Tarski’s completeness result suggests that an interpretation maps statements to open sets in a topology. However, in the ensuing discussion, it seems that the open sets are, like in the semantics for polar predicates, extensions of the predicates and not possible values of statements. It is not obvious to me that we can this easily switch between a semantics of statements and a semantics of predicates and assume that the resulting logic is the same.

References

  1. Arai, T. (2010). Intuitionistic fixed point theories over Heyting arithmetic. In S. Feferman & W. Sieg (Eds.), Proofs, Categories and Computations: Essays in Honor of Grigori Mints (pp. 1–14). London: College Publications.

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Correspondence to Richard Zach.

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Zach, R. Rumfitt on truth-grounds, negation, and vagueness. Philos Stud 175, 2079–2089 (2018). https://doi.org/10.1007/s11098-018-1114-7

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Keywords

  • Negation
  • Vagueness
  • Intuitionism
  • Logical pluralism
  • Non-classical logic