Abstract
From Fine and Kamp in the 70’s—through Osherson and Smith in the 80’s, Williamson, Kamp and Partee in the 90’s and Keefe in the 00’s—up to Sauerland in the present decade, the objection continues to be run that fuzzy logic based theories of vagueness are incompatible with ordinary usage of compound propositions in the presence of borderline cases. These arguments against fuzzy theories have been rebutted several times but evidently not put to rest. I attempt to do so in this paper.
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Notes
As we shall see later, however, the resources of fuzzy logic far outrun those introduced in this section.
See Smith (2008, Sect. 2.2.1) for further details.
See Fine (1975, pp. 269–270).
See Osherson and Smith (1981, pp. 45–46). They present their argument in terms of degrees of membership of objects in sets rather than degrees of truth of statements.
See Kamp (1975, p. 546).
See Fine (1975, p. 269).
See Osherson and Smith (1981, pp. 43–45). Essentially the same objection (using the example of pussy willow and willow) was made earlier by Kay (1975, p. 153). For subsequent empirical work on this kind of case see Storms et al. (1998). Osherson and Smith (1981, pp. 46–48) also present a dual objection concerning alleged disjunctions that are apparently less true than either disjunct.
Note that some authors are explicit about the intuitive basis of their claims while others simply assert what they take to be obvious without explicitly noting that intuition is the sole support for their assertions.
I have omitted a superscript and a subscript from Kamp’s notation because they add complexity that is irrelevant in the present context. Cf. Fine (1975, p. 270): “Surely \( P\ \& -P\) is false even though \(P\) is indefinite.”
Bonini et al. (1999, pp. 389–390) express a similar intuition—and notably (given that their paper includes empirical work) do not subject this intuition to experimental testing.
Cf. also Lakoff (1987, p. 141), Fuhrmann (1988, pp. 323–324], Keefe (2000, p. 164) (who unlike fellow supervaluationists Fine and Kamp & Partee does not take \(Fa\vee \lnot Fa\) to be assertible when \(a\) is a borderline case of \(F\)), Belohlavek et al. (2002, p. 578), Belohlavek et al. (2009, p. 31), Smith (2008, p. 86) and Ripley (2013, p. 341). Some of the objectors have at least noted the existence of differing intuitions—and yet this appears to take no wind from their sails. See e.g. Osherson and Smith (1982, p. 313), Williamson (1994, p. 293 n. 47), Kamp and Partee (1995, pp. 149 n. 13, 179 n. 33), Osherson and Smith (1997, p. 201), and Bonini et al. (1999, p. 390).
Furthermore, moving from the empirical to the theoretical literature, Smith (2008) presents a sustained argument that the correct account of vagueness must involve degrees of truth.
Serchuk et al. (2011, p. 561) subsequently raise the worry themselves that “any confusion between (4) and (5) was neutralized by their apparent juxtaposition on the survey instrument: each participant was asked about both ‘\(\phi \)’ and ‘definitely \(\phi \)’.” Their response is that “This worry can be set aside by considering Experiment #2, where participants were asked for the boundaries for either ‘\(\phi \)’ or ‘definitely \(\phi \)’.” But this does not answer the worry about their experiment: it in effect concedes that the experiment is fundamentally flawed and hence directs our attention to a different experiment. (Also, we are told that the first experiment involved 350 undergraduates at the University of Calgary and was conducted in 2005, and that the second experiment involved 164 undergraduates at the University of Calgary and was conducted in 2005. We are not told whether the group of 164 was a subset of or overlapped with the group of 350. If there was overlap, then the second experiment does face a version of the problem raised above for the first experiment.) Furthermore, the second experiment still faces the second worry raised in the text above: that we should not be asking ordinary speakers for their reactions to ordinary sentences incorporating the word ‘definitely’.
This is not to say that the data have no explanation at all: just that it is not the job of fuzzy theories of vagueness to provide the explanation.
The point that degrees of membership and truth on the one hand and degrees of typicality on the other hand need to be carefully distinguished has been made by numerous authors including Zadeh (1982, p. 293), Smith and Osherson (1988, pp. 51–52), Kalish (1995), Kamp and Partee (1995, pp. 131, 133), Osherson and Smith (1997, p. 191), Belohlavek et al. (2002, p. 578) and Belohlavek and Klir (2011b, pp. 132–133). Hampton (2007, Sect. 2) agrees that membership and typicality are distinct functions but argues that both are determined by a single underlying psychological process of measuring the resemblance between an object and the prototypes for a concept.
Sentences (A) and (C) are omitted from the quotation because they are irrelevant to the present discussion.
Setting aside the problem noted in the text above, a further issue with this experiment is that subjects are forced to break ties: this would seem to build a bias against views according to which (B) and (D) are equivalent into the very design of the experiment.
Oden (1977, p. 568) asks subjects to indicate their judgements “by placing a pin in a 200-mm cork-topped board so that the position of the pin corresponded to the judged truthfulness, with the right end labeled “absolutely true” and the left end “absolutely false.” The position of the pin was measured using a ruler attached to the back of the board.”
Cf. Smith (2011, p. 61).
The fuzzy toolbox is vast and ever growing and what follows is certainly not a complete presentation of its contents.
The residuum exists iff the t-norm is left-continuous.
Contrast the methodology here with that of supervaluationists such as Kamp and Partee (1995), who focus on a relatively small number of isolated data points—e.g. \([\alpha \vee \lnot \alpha ]=1\), \([\alpha \wedge \lnot \alpha ]=0\) and \([\alpha \wedge \alpha ]=[\alpha ]\)—and then try to hit them. In t-norm fuzzy logics, by contrast, a broad system of constraints that anything worthy of the name ‘conjunction’ should satisfy is outlined (i.e. the t-norm conditions) and operations satisfying these constraints are then investigated—together with other connectives defined so that they all fit together in ways that are important in logic.
So in Gödel logic, there is no difference between the strong and weak conjunction.
Where \(\wedge \) is the Łukasiewicz t-norm and \(\vee \) is its dual: \(x\vee y=1-((1-x)\wedge (1-y))\).
Where \(\wedge \) is the Łukasiewicz t-norm and \(\vee \) is its dual. See Belohlavek et al. (2009, p. 31), Belohlavek and Klir (2011b, p. 138) and Paoli (forthcoming).
For details see Belohlavek and Klir (2011a, pp. 57–60).
This point is made by Belohlavek et al. (2002, p. 578). (Note that this also works as a response to the dual disjunction objection mentioned in n. 7 above; cf. Belohlavek et al. (2002, p. 580).) There are other ways of accommodating Osherson & Smith’s data within a fuzzy framework—see e.g. Zadeh (1978) and Zadeh (1982, p. 291) (and Osherson and Smith (1982, Sect. 4.1) for criticism—and Belohlavek et al. (2009, p. 33) for counter-criticism).
For details see Smith (2008, Sect. 5.5).
In Tappenden’s example the foreman waits until you have a pile of samples, and then says “Every one of these samples is either red or orange”.
Of course this is just an example, to make clear the possibility that one might be competent with a word, and yet one’s particular performance with that word (on some occasion) might not in any way flow from that competence. For all I know it may be that one can gain useful information about the semantics of words by seeing in what colours subjects choose to write them. My point is just that for this to be so, their performance would indeed have to flow from their semantic competence—and this is a substantive assumption: it is not automatic.
See Verkuilen et al. (2011, Sect. 6.4) for further discussion of this case, including a list of the nine major symptoms.
Even if the data were highly regular (which, as we shall see, they are not) this would not show that the regularly observed behaviour flows from competence. Compare: it might be that for some reason connected with their upbringing—but not connected to their competence with the words involved—all subjects write certain sentences in green and certain other sentences in red. Not all regularly observed behaviours are results of competence: there are also other kinds of regularities (e.g. widespread systematic biases).
Note that these studies concern simple predications, not compound sentences. For further results showing persistent disagreement and inconsistency amongst responses see Parikh (1994, p. 524) (and further references there), Hampton (2011, Sect. 9.4) and Egré et al. (2013) (and further references there).
Ripley (2011a) asks subjects for their responses to the sentences ‘The circle is near the square and it isn’t near the square’ and ‘The circle both is and isn’t near the square’. He says (p. 174) that his results are similar to those reported by Alxatib and Pelletier (2011b). However this is not clear, because Ripley asks subjects for their level of agreement (he gives subjects seven possible responses, with 1 labelled ‘Disagree’ and 7 labelled ‘Agree’) whereas Alxatib and Pelletier (like Serchuk et al.) ask subjects for responses framed in terms of truth. Cf. Sect. 5.1 above.
I raised a version of this hypothesis in Smith (2011, p. 61) but my remarks there were necessarily brief. Ripley (2011b, pp. 63–64) responds, concluding: “Since the goal is to learn about borderline cases, diving in and asking participants about borderline cases is an important source of data. It cannot be dismissed as unreliable on the grounds of participants’ discomfort; that discomfort is itself part of the phenomenon to be studied. This sort of methodology has resulted in considerable success when it comes to simple categorization judgments, and there is no reason to expect it to be less reliable when it comes to compound judgments.” My point, however—as I hope is clear from my longer discussion here—is not simply that subjects are uncomfortable around borderline cases: it is that competence may well impose no requirements here.
By a ‘valuation’ I mean the part of a model that assigns values to primitive nonlogical symbols—as distinct from the part that says how values are assigned to complex expressions, given a valuation.
Here and in the following, for ease of presentation I explicitly mention only connectives when considering complex expressions—but the points made are general.
A three-valued set is a total function from the domain to a set of three truth values, say \(\{0,*,1\}\). Objects mapped to 1 are thought of as definite cases of the predicate, objects mapped to 0 as definite noncases of the predicate and the remaining objects as borderline cases. The points made here about three-valued valuations apply equally to partial valuations, in which the extensions of predicates are partial or gappy sets: partial functions from the domain to the classical truth values \(\{0,1\}\).
See Smith (2012) for an introduction to such logics.
Here ‘extend’ means that they retain all mappings to 1 and 0.
See Smith (2008, Sect. 2.4) for an introduction to these views.
A related point concerns the enormous success of fuzzy logics in applications (e.g. in engineering and computer science). (In this connection, Serchuk et al. (2011, p. 561 n. 14) state that “degree theory is the only theory of vagueness that has been put to use”.) This uptake and success is itself an empirical fact that demands explanation. From a point of view according to which, say, classical or supervaluationist logic provides the (only) correct treatment of vagueness, this fact is incomprehensible.
This point tends to be overlooked. For example, Osherson and Smith (1982, p. 303) simply assume that the “gradient theorist”—i.e. the theorist who takes concepts to have degrees of membership—will want to take a degree-functional approach to conjunctive concepts.
For helpful comments, I am grateful to Radim Belohlavek, Richard Dietz, Francesco Paoli and two anonymous referees.
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Smith, N.J.J. Undead argument: the truth-functionality objection to fuzzy theories of vagueness. Synthese 194, 3761–3787 (2017). https://doi.org/10.1007/s11229-014-0651-7
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DOI: https://doi.org/10.1007/s11229-014-0651-7