Abstract
This paper examines the acceptance of so-called borderline contradictions involving vague adjectives. A close look at the available data from previous studies points toward a preference for “gappy” descriptions of the form “x is neither P nor not P” over “glutty” descriptions of the form “x is P and not P”. We present the results of an experiment in which we tested for that difference systematically, using relative gradable adjectives. Our findings confirm that both kinds of descriptions are accepted, but indeed that “neither”-descriptions are to a large extent preferred to “and”-descriptions. We examine several possible explanations for that preference. Our account relies on the distinction proposed by Cobreros et al. (J Philos Logic, 1–39, 2012) between strict and tolerant meaning for vague adjectives, as well as on a specific implementation of the strongest meaning hypothesis endorsed by Cobreros et al. as well as Alxatib and Pelletier (Mind Lang 26(3): 287–326 2011a). Our approach, however, argues in favor of local pragmatic strengthening instead of global strengthening in order to derive that preference.
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- 1.
Sentence (4)-b may be judged outright false of course, since \(\sqrt{2}\) is not a prime number. It seems to us acceptable in a context in which a teacher, let us say, would want to cut short a dispute between two pupils, one of them arguing that \(\sqrt{2}\) is a prime integer, and the other arguing that \(\sqrt{2}\) is not a prime integer, both mistakenly thinking it is an integer. Both pupils would wrongly presuppose that \(\sqrt{2}\) is an integer, and the point of (4)-b would be to reject that presupposition. See Sect. 4.4.2 below for more on the analysis of such presuppositional sentences.
- 2.
See Ruytenbeek et al. (2017) for a published follow-up to that work.
- 3.
In this paper, we take this non-extreme property to be definitory of borderline cases. By contrast, one might consider that an extremely insane person in a perfect physical shape could be a borderline case for healthy if judgments for the sentence “this person is healthy” can be unclear and ambivalent. However, we excluded multi-dimensional adjectives from our sample as far as possible (see the discussion of evaluativity below), in order to rule out borderline cases arising from the relative weight of competing dimensions.
- 4.
The maximum standard to consider a glass “full” is still context-dependent: for instance, McNally (2011) discusses how wine need not reach the top of the glass for it to be considered full. A glass half-filled with wine can be called “full of wine” in some contexts, thereby threatening the generalization in (9). We don’t think that undermines the point we are making here, however, as further tests can be used to corroborate the classification of “full” as absolute. However, the reader is invited to replace full by empty in our examples, as “empty” appears to show less context-sensitivity (in relation to glasses at least).
- 5.
Some authors consider that the status of what we call borderline cases for absolute adjectives is due to a different phenomenon from the one at play with relative adjectives. For instance Kennedy (2007) claims that calling an almost-full glass “full” is a manifestation of imprecision, but that there is in fact a sense in which such a glass is uncontroversially not full. By contrast, for relative adjectives, there would be no non-arbitrary way to settle the question for borderline cases for they are a manifestation of vagueness proper.
- 6.
- 7.
In this case, exclusion of participants would lead the regression models that we ran to not converge, because of an insufficient variability on the controls.
- 8.
The results were descriptively similar in all conditions. A graph presenting the results for each condition is included in Appendix 3.
- 9.
See Appendix 3 for a list of the models.
- 10.
Alxatib and Pelletier talk of sub- and super-interpretation. As pointed out by Cobreros et al. (2012), we can talk of sub- and super-interpretations, but provided we do not mistake the resulting logic of vague predicates for the subvaluationist and supervaluationist logics respectively, which are not truth-functional, unlike the strict-tolerant logic used in Cobreros et al. (2012). See Ripley (2013) and Alxatib et al. (2013) for discussion and comparison. The approach, while akin to the contextualist strategy outlined at the end of the previous section, involves no mechanism of indexing. See Ripley (2011b) for more on this and the varieties of contextualism.
- 11.
See also Égré et al. (2013) for discussion, where a related phenomenon is discussed under the name “Hump Effect”. The term “conjunction effect” now strikes us as more general and adequate.
- 12.
- 13.
Since the algorithm we propose operates on high-level linguistic representations, it equally derives the expected interpretation regardless of the form of the logical translation of the neither descriptions.
- 14.
10 participants did accept the “and”-description while rejecting the “neither”-description at least once (5 participants did so on one trial, 5 participants did so on two trials). This could be noise produced by inattention on these trials, even though our participants were highly accurate. It is also conceivable, in principle, that on one or two trials those 10 participants exceptionally went through the whole procedure described above for the “and”-description but stuck with a classical, logically contradictory interpretation for the “neither”-description.
- 15.
Note that in (17), strictly would again appear under the first not and typically trigger a short break in the prosodic contour. Relatedly, when we paraphrased the meaning of (17), we used “not tall tall” to mean “not strictly tall”. The repetition of the adjective seems to be another focus-related strategy to embed the strictly operator under negation (see (ii) above).
- 16.
(19-a) may be acceptable if John is a borderline case of someone who stopped smoking. But set aside the vagueness of “stop” and “smoke” (assume they are fully crisp predicates).
- 17.
Spector writes the operator in question W, for “weak truth”, and writes the dual B. Note that we developed our account independently of Spector’s, that is from the vantage of the strict-tolerant account of vagueness, without thinking about presupposition accommodation. Spector was motivated primarily by the phenomenon of local accommodation, and with no heed to the specific asymmetry between “neither” and “and”-sentences we discuss in the vagueness case.
- 18.
And indeed, while the account of strict and tolerant we adopted here is cashed out in trivalent terms, the original account of Cobreros et al. (2012) defines the tolerant meaning of an adjective in terms of an existential quantification and the strict meaning in terms of a universal quantification, not directly over acceptable standards, but in ways that are intertranslatable with that approach.
- 19.
Thanks to G. Sassoon for making that suggestions.
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Acknowledgements
We thank the Agence Nationale de la Recherche under program TrilLogMean ANR-14-CE30-0010-01 for support, as well as grants ANR-10-LABX-0087 IEC and ANR-10-IDEX-0001-02 PSL* for research carried out at the Departement of Cognitive Studies of ENS. PE thanks the Swedish Collegium for Advanced Study who hosted him as a visiting fellow during the time this paper was written (Feb-June 2016). JZ thanks the National Science Foundation grant BCS-1349009 attributed to Florian Schwarz, and Florian Schwarz himself, for their support. We are particularly grateful to Galit Sassoon, to Benjamin Spector, and to three anonymous referees for detailed comments and suggestions. We also thank Sam Alxatib, Elitzur Bar-Asher Siegal, Pablo Cobreros, Chris Cummins, Matti Eklund, Vladimir Goranko, Yael Greenberg, Yossi Grodzinsky, Nat Hansen, Hans Kamp, Louise McNally, Marie-Christine Meyer, Peter Pagin, Roumy Pancheva, David Ripley, Robert van Rooij, Hans Rott, Uli Sauerland, Stephanie Solt, Steven Verheyen, Åsa Wikforss for various discussions. We also thank the participants of the “Vagueness and Three-valued Logics” course given by PE at the LLCC at the University of Jerusalem in 2015, as well as audiences in Madrid, Stockholm, Uppsala, and at the ESSLLI 2016 summer school in Bolzano (Course on “Trivalent Logics and Natural Language Meaning”). We warmly thank the editors, Elena Castroviejo Miró, Louise McNally, and Galit Sassoon for their editorial work and feedback.
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Appendices
Appendix 1: Scenarios
1.1 Scenarios for Human-Oriented Adjectives
1.1.1 Rich
A survey on wealth has been conducted in your country. In the population there are people with a very high degree of wealth, and people with a very low degree of wealth. Then there are people who lie in the middle between these two areas.
Imagine that Sam is one of the people in the middle range. Comparing Sam to other people in the population, is it true to say the following?
1.1.2 Tall
A survey on heights has been conducted in your country. In the population there are people of a very high height, and people of a very low height. Then there are people who lie in the middle between these two areas.
Imagine that Sam is one of the people in the middle range. Comparing Sam to other people in the population, is it true to say the following?
1.1.3 Old
A survey on age has been conducted in your country. In the population there are people whose age is very high, and people whose age is very low. Then there are people who lie in the middle between these two areas.
Imagine that Sam is one of the people in the middle range. Comparing Sam to other people in the population, is it true to say the following?
1.1.4 Heavy
A survey on weight has been conducted in your country. In the population, there are people of a very high weight, and people of a very low weight. Then there are people who lie in the middle between these two areas.
Imagine that Sam is one of the people in the middle range. Comparing Sam to other people in the population, is it true to say the following?
1.2 Scenarios for Object-Oriented Adjectives
1.2.1 Fast
A survey on people’s cars has been conducted in your country. In the population, there are people who own very high speed cars, and people who own very low speed cars. Then there are people who own cars that lie in the middle between these two areas.
Imagine that Sam is one of the people owning a car in the middle range. Comparing Sam’s car to the cars of other people in the population, is it true to say the following?
1.2.2 Large
A survey on people’s houses has been conducted in your country. In the population, there are people who own houses with a lot of space, and people who own houses with very little space. Then there are people who own houses that lie in the middle between these two areas.
Imagine that Sam is one of the people owning a house in the middle range. Comparing Sam’s house to the houses of other people in the population, is it true to say the following?
1.2.3 Loud
A survey on people’s voice has been conducted in your country. In the population, there are people whose voice has a very high intensity, and people whose voice has a very low intensity. Then there are people whose voice lie in the middle between these two areas.
Imagine that Sam is one of the people whose voice lie in the middle range. Comparing Sam’s voice to the voices of other people in the population, is it true to say the following?
1.2.4 Wide
A survey on people’s feet has been conducted in your country. In the population, there are people with a very high foot breadth, and people with a very low foot breadth. Then there are people whose foot breadth lie in the middle between these two areas.
Imagine that Sam is one of the people with a foot breadth in the middle range. Comparing Sam’s feet to the feet of other people in the population, is it true to say the following?
Appendix 2: Questionnaires
1.1 Pre-questionnaire
1.2 Post-questionnaire
The last two questions and their input fields would appear only if the participants reported a difference between the descriptions in the first and in the second halves of the experiment.
Appendix 3: Results Per Group and Regression Models
We ran several models, differing with respect to the complexity of their random effect structures. They all included a random intercept, but no random slope for Participant (\(\mathrm{N}=148\)). As explained in the Design subsection of Sect. 3, each participant was assigned to one of four groups, determined by two factors. Block Order indicates whether the participant responded before or after judging another set of descriptions with antonyms, and Adjective Type indicates whether the participant responded to descriptions directly referring to a human being as their subject or to an object or a property of a human being.
We included a random intercept for Adjective (\(\mathrm{N}=8\)) in all our models. We were also able to fit models including a random slope for Adjective. One set of such models added a random intercept plus a random slope for Block Order, and another set of such models added a random intercept plus a random slope for Adjective Type. For all these models, we ran three versions: one with Neither as baseline for Description, one with True as a baseline and one with False as a baseline. The outputs of the models for the former structure (with random intercepts and slopes for Block Order) are presented in Table 4, the outputs of the models for the latter structure (with random intercepts and slopes for Adjective Type) are presented in Table 5. By removing the random slope for Adjective, we were also able to fit models including a random intercept and slope both for Block Order and Adjective Type with Neither as a baseline. The contrast with Control True remains significant (Table 6).
Mean acceptance by description type and by group
To further investigate the source of the significant contrasts between Neither and Control True, and with regard to the apparently mixed descriptive results in Fig. 6 above, we ran additional models on subsetted data. We first considered four subsets: the responses of all the participants in the human-oriented adjective groups, those of all the participants in the non-human-oriented adjective groups, those of all the participants who responded in the first block and those of all the participants who responded in the second block. All these models had a random intercept for Participant, and a random intercept plus slope for Adjective, Block Order and Adjective Type. None of them yielded a significant difference between Neither and True (\(0.2< p < 0.25\)). We then subsetted the data to each minimal group of participants. The models failed to converge for the responses from the first block for the non-human oriented descriptions, and for the responses from the second block for the human-oriented descriptions. The other two models indicated a significant contrast between Neither and Control True (\(p < 0.01\)).
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Égré, P., Zehr, J. (2018). Are Gaps Preferred to Gluts? A Closer Look at Borderline Contradictions. In: Castroviejo, E., McNally, L., Weidman Sassoon, G. (eds) The Semantics of Gradability, Vagueness, and Scale Structure. Language, Cognition, and Mind, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-77791-7_2
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