Abstract
Both the phenomenon of presupposition and that of vagueness have motivated the use of one form or another of trivalent logic, in which a declarative sentence can not only receive the standard values true (1) and false (0), but also a third, non-standard truth-value which is usually understood as ‘undefined’ (#). The goal of this paper is to propose a multivalent framework which can deal simultaneously with presupposition and vagueness, and, more specifically, capture their projection properties as well as their different roles in language. Now, there is a prima facie simple way of doing this, which simply consists in assimilating the two phenomena, and using an appropriate type of trivalent logic. On this view, we just need a compositional system that deals with the ‘undefined’ truth-value, and does not care about whether the source of undefinedness is ‘presuppositional’ or related to vagueness. I will argue that such a simple solution cannot succeed, and point out a number of desiderata that any successful approach must meet. I will then present and discuss two seven-valued semantics, inspired, respectively, by the Strong Kleene semantics and by supervaluationism, which meet these desiderata.
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Notes
There is a recent debate as to whether the predicted asymmetry is real. Schlenker (2008) and Chemla and Schlenker (2012) argue for a kind of ambiguity, whereby the predictions made by the Middle Kleene truth-tables correspond to one of two possible readings (though these proposals do not adopt a trivalent approach to presupposition projection), while the other reading corresponds to what would be predicted by the symmetric Strong Kleene truth-tables (see below). Rothschild (2011), on the other hand, claims that there is no real asymmetry in presupposition projection.
There is of course a close relationship between Strong Kleene and Middle Kleene. Middle Kleene can be viewed as an asymmetric variant of Strong Kleene. In Middle Kleene, for every binary connective, whenever the first argument received a standard truth-value (0 or 1), the corresponding line is identical to its counterpart in the Strong Kleene truth-table. When the first argument receives the value \(\#\), then the sentence as a whole receives the value \(\#\). George 2008 offers a way of deriving in a systematic way Middle Kleene truth-tables on the basis of classical truth-tables, simply by using a different definition of a ‘repair’ from the one used below in (6). George furthermore extends this account to predicate logic and beyond.
Priest (1984), followed by Ripley (2013), offers a hierarchy of levels, where the semantics at level \(n+1\) is derived from the one at level \(n\) by means of a generalization of the rule given in (8). The 1st-level semantics in Ripley (2013) is just classical, bivalent semantics. The 2nd-level truth-values consist of the non-empty subsets of \(\{0,1\}\), i.e. \(\{0\}\), \(\{1\}\), and \(\{0, 1\}\), and the associated semantics is then Strong Kleene, with \(\{0,1\}\) playing the role of the third truth-value. The 3rd level in Ripley’s system thus corresponds to my 2nd level, and its semantics is derived exactly as stated in (8), but not on the basis of Middle Kleene—so the resulting semantics is different from the one presented here.
As Paul Egré pointed out to me (p.c.), this might actually be reasonable, because the threshold for tallness might be different for boys and girls, or for young children and adult children. I am assuming here that the standard for tallness is uniform for all of John’s children. One way of making this natural is to consider a case where John has no daughter, and only adult sons, who happen to be borderline-tall and to have blue eyes.
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Acknowledgments
I would like to gratefully acknowledge the important role that my numerous conversations with Jérémy Zehr and Paul Egré played in stimulating my thoughts and thereby shaping the ideas that I present here. I also thank Paul Egré for his careful reading of a first draft of this paper. The research leading to these results has received support from the Agence Nationale de la Recherche (grants ANR-10-LABX-0087 IEC, ANR-10-IDEX-0001-02 PSL, & ANR-14-CE30-0010-01 TriLogMean).
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Spector, B. Multivalent Semantics for Vagueness and Presupposition. Topoi 35, 45–55 (2016). https://doi.org/10.1007/s11245-014-9292-1
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DOI: https://doi.org/10.1007/s11245-014-9292-1