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The Problem of Artificial Precision in Theories of Vagueness: A Note on the Rôle of Maximal Consistency

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Abstract

The problem of artificial precision is a major objection to any theory of vagueness based on real numbers as degrees of truth. Suppose you are willing to admit that, under sufficiently specified circumstances, a predication of “is red” receives a unique, exact number from the real unit interval [0, 1]. You should then be committed to explain what is it that determines that value, settling for instance that my coat is red to degree 0.322 rather than 0.321. In this note I revisit the problem in the important case of Łukasiewicz infinite-valued propositional logic that brings to the foreground the rôle of maximally consistent theories. I argue that the problem of artificial precision, as commonly conceived of in the literature, actually conflates two distinct problems of a very different nature.

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Notes

  1. The related but distinct problem of higher-order vagueness (see e.g. Keefe 2000, pp. 31–36) will not be considered in this note.

  2. A set of conventions for omitting parentheses in formulæ is usually adopted (\(\neg\) is more binding than →), and later extended to derived connectives. I do not spell the details here, as the conventions are analogous to the ones in classical logic, and are unlikely to cause confusion.

  3. In (Cignoli et al. 2000, Chapter 4) the language has no logical constants, and consequently (A0) does not appear as an axiom. I prefer to explicitly have \(\bot\) in the language, and thus I add Ex falso quodlibet to the standard axiomatisation.

  4. However, \(\fancyscript{L}\) fails strong completeness (i.e. completeness for theories): there is a set \(S\subseteq {\textsc {Form}}\) and a formula \(\alpha \in {\textsc {Form}}\) such that S ⊧ α, but \(S \, \not\vdash \, \alpha\); see (Cignoli et al. 2000, 4.6).

  5. Not all vague predicates are alike. The predicate Tall (\(\cdot\)) comes in tandem with its opposite, Short (\(\cdot\)), over the domain of all individuals, say; but Red (\(\cdot\)) does not: there is no colour term for non-Red (\(\cdot\)) in the visible spectrum. Although I will not argue the point in this note, I believe that \(\fancyscript{L}\) cannot be an appropriate formal model of vague predicates such as Red (\(\cdot\)). Hence the shift from redness to tallness.

  6. Assuming that all propositions under consideration are built from the single atomic one “VM is tall”, and that the logic is truth-functional. Since, formally, the discussion applies to \(\fancyscript{L}_{1}\), these assumptions are satisfied.

  7. Compare Haack’s claim that “[In fuzzy logic] one is obliged to require that a predicate definitely applies to such-and-such, rather than to such-and-such other, degree” (Haack 1979, loc. cit.). It was just shown that the claim, if taken at face value, is unwarranted.

  8. There is just one such admissible evaluation in \(\fancyscript{L}_{1}\), of course. I am using the plural form in preparation for the forthcoming extension (S T ).

  9. As mentioned, \(\Uptheta_r\) is deductively closed for any \(r \in [0,1]\). Given \(\alpha \in {\textsc {Form}}_1\), suppose \(\Uptheta_{r} \vdash \alpha\). If \(w_r\colon {\textsc {Form}}_1 \to [0,1]\) is the unique evaluation such that w r (X 1) = r, then \(w_r(\Uptheta_{r})=\{1\}\) by (); since \(\Uptheta_{r} \vdash \alpha\), then w r (α) = 1 by the generalised validity theorem for \(\fancyscript{L}_1;\) hence \(\alpha \in \Uptheta_r\), again by (). Moreover, \(\Uptheta_r\) is consistent: since \(w_r(\bot)=0\) by the semantics of \(\bot, \) we have \(\bot \not \in \Uptheta_r\) in light of (). It is harder to prove that \(\Uptheta_r\) is maximally consistent, and that all maximally consistent theories are of this form. However, this is a standard result (essentially Cignoli et al. 2000, 4.6.3 and 3.5.1).

  10. In which case Proposition 1 and question (Q1) would be concerned with maximally consistent theories in \(\fancyscript{L}_{n}\) or \(\fancyscript{L}\).

  11. In which case Proposition 1 and question (Q1) would be concerned with semisimple theories, i.e. those theories for which completeness holds; see (Cignoli et al. 2000, 4.6 and 3.6).

  12. The main issue in generalizing (Q1) to other systems is that Proposition 1 most often fails, so that it is not enough to consider maximally consistent theories only. For example, in the important Gödel-Dummett logic (Hájek 1998, Chapter 4), \(\fancyscript{M}\) turns out to be exactly the collection of all prime theories in the one-variable fragment, where a theory \(\Uptheta\) is prime if it proves either αβ or βα for any two formulæ α and β.

  13. If \(\fancyscript{S}\) is such that (Q1) has negative answer, then the formal semantics of \(\fancyscript{S}\) violates Leibniz’s Identity of Indiscernibles: in deference to which, we ought to identify real numbers (= possible worlds) \(r,s \in [0,1]\) whenever they satisfy \(\Uptheta_{r}=\Uptheta_{s}\), provided all we are concerned with are those properties of r and s that are expressible within \(\fancyscript{S}. \) If we insist on not identifying r and s, then there must be distinguishing properties of these two possible worlds, not expressible with the linguistic resources of \(\fancyscript{S}\), that we nonetheless wish our formal semantics to record. A formal semantics strictly richer than the available language is of course a perfectly reasonable construct, but it had better result from an explicit choice—not from overlooking a negative answer to (Q1).

  14. Algebraically, Proposition 2 amounts to the representation theorem for 1-generated simple MV-algebras, see (Cignoli et al. 2000, Chapter 3). Via Mundici’s categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a strong order unit (Cignoli et al. 2000, 7.1), this is equivalent to Hölder’s theorem, for which the interested reader may consult (Bigard et al. 1977, § 2.6).

  15. It can be proved that, given \(r \in [0,1]\), \(\Uptheta_{r}\) as in () is finitely axiomatisable in \(\fancyscript{L}_{1}\) if, and only if, r is rational. Moreover, it is possible to exhibit an algorithm (for definiteness, a Turing machine) that, given any rational number \(r \in [0,1]\) as input, outputs a formula α r (X 1) satisfying \(\Uptheta_{r}=\{\alpha_{r}(X_{1})\}^{\vdash}\). Everything hinges on the theory of continued fractions and Schauder hats; see (Cignoli et al. 2000, Chapter 3).

  16. Since Machina (1976), Łukasiewicz logic has been widely discussed in the philosophical literature as a candidate for a logic of vagueness. More often than not, it has been rejected; cf. e.g. Williamson (1994), Keefe (2000). To the best of my knowledge, though, the quite specific question (Q2) has not been addressed.

  17. Cf. Footnote 5.

  18. Caution: no circularity is involved in this passage. The objection of artificial precision can only be raised against theories that (i) have already committed to degrees of truth, and (ii) have embraced [0, 1], or some other precisely specified structure, as a mathematical model for degrees of truth and their relationships. The charge that we are here using a comparative notion of truth to explain artificial precision, without justifying the assumption that truth does come in degrees, has therefore no force. Similary, the problem of justifying why degrees of truth are modelled by the real numbers rather than, say, the octonions, may well be a problem—there is no paucity of objections to (i–ii) in the literature—but it is a different one.

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Acknowledgments

A preliminary version of parts of this paper was presented at the meeting Epistemic Aspects of Many-Valued Logics, held in Prague at the Institute of Philosophy of the Academy of Sciences of the Czech Republic, from the 13th to the 16th of September 2010. I am grateful to the organisers, Timothy Childers, Christian Fermüller, and Ondrej Majer, for having given me a chance to present some of these ideas before an audience that included several philosophers who have thought deeply about vagueness. I am indebted to many participants for questions, discussions, and criticism that have been helpful in improving my initial ideas on the subject matter of this paper. In this connection, I should particularly like to thank Christian Fermüller, Colin Howson, Nicholas J. J. Smith, and Timothy Williamson. Finally, I am indebted to the two reviewers of this paper for suggestions that led to improvements of the presentation.

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Marra, V. The Problem of Artificial Precision in Theories of Vagueness: A Note on the Rôle of Maximal Consistency. Erkenn 79, 1015–1026 (2014). https://doi.org/10.1007/s10670-013-9587-4

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