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A Unification of Two Approaches to Vagueness: The Boolean Many-Valued Approach and the Modal-Precisificational Approach

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Abstract

The Boolean many-valued approach to vagueness is similar to the infinite-valued approach embraced by fuzzy logic in the respect in which both approaches seek to solve the problems of vagueness by assigning to the relevant sentences many values between falsity (or 0) and truth (or 1), but while the fuzzy-logic approach postulates linearly-ordered values (or ‘degrees’) between 0 and 1, the Boolean approach assigns to sentences values in a many-element (including infinite-element) complete Boolean algebra. On the modal-precisificational approach represented by Kit Fine, if a (vague) sentence is indeterminate in truth value in some world (possibly the actual world), it is taken to be true in one precisified world accessible from that world and false in another. This paper points to a way to unify these two approaches to vagueness by showing that Fine’s version of the modal-precisificational approach can be combined with the Boolean many-valued approach instead of supervaluationism, one of the most popular approaches to vagueness.

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Notes

  1. One might think that the precisificational/supervaluational approaches in general and the present approach in particular are a species of – or at least more harmonious with – semanticism. That may be true if precisifications are taken as purely mathematical entities, as they are in this paper. However, as Akiba ([1] and [2]) has shown, it is generally not difficult to convert the precisificational/supervaluational approaches into ontic versions (so-called ‘ontic supervaluationism’) by employing a more realistic notion of precisifications. Or we can easily think of not just vague propositions but also vague properties such as baldness and tallness to have the Boolean values we discuss. Thus, it is not difficult to modify the present approach to accommodate onticism. We suspect that a similar maneuver can be adopted for epistemicism.

  2. The treatment of identity and quantifiers in the Scott-Solovey Boolean-valued models approach (see, e.g., Bell [6], Jech [18], and Kunen [22]) suggests how quantification should be incorporated. For a little more on this, also see Akiba [3].

  3. See, e.g., Goguen [16], Lakoff [23], Machina [25], and Smith [31] for philosophical defense of fuzzy logic and set theory.

  4. Generally, penumbral connections are distinguishable into logical connections (e.g., the connection between ‘bald’ and ‘bald and fat’) and conceptual connections (e.g., the connection between ‘red’ and ‘pink’). In this paper we are concerned only with the former. We believe that the latter can be dealt with by an introduction of meaning postulates (with degrees) that connect the relevant concepts in an appropriate manner.

  5. For experimental bases of this point, see, e.g., Bonini et al. [8], Alxatib and Pelletier [4], and Serchuk et al. [30].

  6. Other problematic features of fuzzy logic we won’t get into here should include its complicated notion of logical consequence (see Hajek [17]).

  7. The completeness is actually necessary only when there are universal and existential quantifiers in the language; so it is not necessary in the present context since we are dealing only with propositional logic in this paper. We put this condition in, however, so that we can expand the language and logic easily.

  8. This is what differentiates this valuation from the four-valued valuation for first-degree entailment (see, e.g., Priest [27], p. 146); in the negation table for FDE, U on the second row remains U, and −U on the third row remains −U. The other two tables are the same.

  9. The resulting many-valued logics are investigated in Beall and van Fraassen [5], Sect. 11.3.

  10. Fine himself uses ‘ p≤q’ to mean ‘q precisifies p’, but, as we shall see, our notation will prove more convenient. The ‘less means more’ convention adopted here is often employed in the forcing literature. Take \(\phantom {\dot {i}\!}\trianglelefteq \) as an arrow pointing to the direction of precisification.

  11. Strangely, Fine discusses disjunctions but does not explicitly spell out the truth (and falsity) conditions for disjunction. Disjunction Rule here is recovered from his discussion and the truth and falsity conditions for conjunction and negation.

  12. Note that conditions (a), (d), (e) plus this condition, given that \(\phantom {\dot {i}\!}\trianglelefteq \) is a partial order, would constitute the Kripke semantics of intuitionistic logic.

  13. This name is taken from Jech [18], p. 204.

  14. In this paper the symbol − is used in two ways: −X, where X is a Boolean value, is a Boolean complement (see Section 2); X−Y, where X and Y are sets, is a set subtraction. (Note, however, that a Boolean value can at the same time be a set).

  15. Furthermore, if P is finite, we can also easily introduce the degree [ [ϕ] ] P in [0,1] as the ratio between the number of ps such that \(\phantom {\dot {i}\!}p \ {{\Vdash }_{P}} \ \phi \) and the cardinality of the power set of P. Kamp [20] introduces degrees into supervaluationism in such a fashion.

  16. In the Kripke semantics of S4 modal logic, the following conditions hold:

    1. 1.

      \(\phantom {\dot {i}\!}p \ {{\Vdash }_{P}} \ \square \phi \ \,\Leftrightarrow \ \,\forall q\trianglelefteq p.\ q \ {{\Vdash }_{P}} \ \phi \text {;}\)

    2. 2.

      \(p \ {{\Vdash }_{P}} \ \square \neg \phi \ \,\Leftrightarrow \ \,\forall q\trianglelefteq p.\ q \ {{{\not {\Vdash } }}_{P}} \ \phi ;\)

    3. 3.

      \(\phantom {\dot {i}\!}p \ {{\Vdash }_{P}} \ \phi \wedge \psi \ \,\Leftrightarrow \ \,p \ {{\Vdash }_{P}} \ \phi \text { and}\;p \ {{\Vdash }_{P}} \ \psi \text {;}\)

    4. 4.

      \(\phantom {\dot {i}\!}p \ {{\Vdash }_{P}} \ \square \diamondsuit (\phi \vee \psi )\ \,\Leftrightarrow \ \,\forall q\trianglelefteq p\ \exists r\trianglelefteq q\ (r \ {{\Vdash }_{P}} \ \phi \text { or}\;r \ {{\Vdash }_{P}} \ \psi ),\)

    where \(\phantom {\dot {i}\!}\trianglelefteq \) may be a partial order, and \(\phantom {\dot {i}\!}\square \) and ♢ are genuine modal operators ‘necessarily’ and ‘possibly’. Compare these with Eq. 8, 8, 8, and 8 as well as Eq. 33. Another way to get the results we’ve just got, suggested by the similarity revealed here, is to translate the sentences we have been considering into S4 modal formulas, inserting the modal operators \(\phantom {\dot {i}\!}\square \) and ♢ in the appropriate places, and to show that the modal formulas As such that \(\phantom {\dot {i}\!}A \leftrightarrow \square \diamondsuit A\) (the modal counterparts of regular open sets) constitute a Boolean algebra. Note that \(\phantom {\dot {i}\!}\square \diamondsuit A \leftrightarrow \square \diamondsuit \square \diamondsuit A\) for every A in S4. Smullyan and Fitting [33] exploit such a translation to connect Cohen’s forcing with the Scott-Solovay Boolean-valued models.

  17. There is another popular way to introduce a ‘definitely’ operator of sort, i.e., as a full-fledged counterpart of the ‘necessity’ operator in (S4) modal logic. (Recall that Fine’s ‘definitely’ operator is closer to the ‘actually’ operator than to the necessity operator). Just read \(\phantom {\dot {i}\!}\square \) in note 16 above as D (‘definitely’). However, as the aforementioned correspondence between the clauses in note 16 and those in Eq. 8 indicates, such an operator will be totally redundant in the present setting.

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Acknowledgments

Our utmost gratitude goes to the referee for Journal of Philosophical Logic, whose invaluable comments on the two previous drafts of this paper have helped us to improve the paper tremendously.

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  1. Department of Philosophy, Virginia Commonwealth University, Richmond, VA, 23284-2025, USA

    Ken Akiba

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Akiba, K. A Unification of Two Approaches to Vagueness: The Boolean Many-Valued Approach and the Modal-Precisificational Approach. J Philos Logic 46, 419–441 (2017). https://doi.org/10.1007/s10992-016-9405-y

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