Abstract
A way of modelling the meanings of vague terms directly in terms of the uncertainties they give rise to is explored. These uncertainties are modelled with a probabilistic, Bayesian version of situation semantics on which meaning is captured as the correlations between uses of words and the types of situations those words are used to refer to. It is argued that doing so provides a framework from which vagueness arises naturally. It is also claimed that such a framework is in a position to better capture the boundarylessness of vague concepts.
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Notes
- 1.
This notion of semantic information is more rigorously developed in Sutton (2013).
- 2.
This may be to strong. There could be some priors we have, or reasoning we could engage in, that would make some heights more reasonable than others. My thanks go to Noah Goodman for helpful discussion.
- 3.
I do not mean to take, as assumed, a clear, semantically important division between CNs and adjectives. Below, I will treat words like ‘tall’ and ‘green’ as predicate modifiers and words like ‘car’ and ‘man’ as predicates. In many cases, the traditional classifications of words as CNs and adjectives will overlap with these semantic divisions. Nothing I say rests on whether they all do.
- 4.
This point is related to the commonly held idea that vague adjectives are context-sensitive or interpretable only relative to a comparison class (Cresswell 1974).
- 5.
In English at least, length is not simply a euphemism for height. Trains, passageways, halls, to name a few, can have a great length without having a great height.
- 6.
An interesting possibility is that the application of domain specific modifiers to NPs that do not carry information about that domain will generate metaphorical interpretations. Metaphor, humour and other such subjects are outside of the scope of this paper. There may also be an interesting link to be explored between the ideas put forward here and work done on scalar adjectives in de Marnee et al. (2010).
- 7.
This was pointed out by an anonymous reviewer for the BNLSP 2013 Workshop.
- 8.
That is not to say that ‘red’ cannot relate to non-stereotypical shades (such as in ‘red onion’). Elsewhere (Sutton 2013), I suggest a way for this account to model the information carried by ‘pet fish’ where goldfish are not the most stereotypical fish, nor the most stereotypical pets.
- 9.
This way of viewing infons (as types) propagates through into situation theoretic approaches with richer type systems. See, for example Cooper (2012).
- 10.
This notation for infons is essentially Devlin’s. However, for polarities, I adopt Barwise and Perry’s ‘yes’ and ‘no’ instead of Devlin’s ‘1’, ‘0’. This is to avoid potential confusion with the limit cases of probability values [0, 1].
- 11.
More recent situation theoretic approaches take types to be objects. An example of this rich type-theoretic approach is Type Theory with Records (TTR) (Cooper 2012).
- 12.
This is close to Devlin’s notation, but with Cooper’s use of ‘\(\lambda \)’ for abstractions instead of Devlin’s (\(\dot{s}|\dot{s} \vDash \sigma \)). This variation is to avoid possible confusion resulting from the use of ‘|’ in probability theory.
- 13.
This can be so even given the assumption that speakers are not being deliberately deceptive. What is being tracked here is the extent to which, in general, properties such as heights of individuals correlate with types of utterances, such as describing them as tall.
- 14.
The hedging here is included because, although I assume a domain of situations (as a set), situations are best not viewed as a set.
- 15.
A difference between this, my own, and similar proposals will be explained in Sect. 7.
- 16.
I assume for simplicity that these properties are not vague. Nothing turns on this however. Even if properties were vague in the sense that someone may genuinely be of indeterminate (exact) height, one would still need to explain how words like ‘tall’ admit of wide ranges of those properties in a graded way.
- 17.
I will from now on suppress all reference to locations and times, which are also types in situation theory.
- 18.
In a more sophisticated model, one would wish the values of priors to be set in accordance with the learning experiences of agents. One way this has been implemented will be discussed in Sect. 7.2.
- 19.
Of course, given particularly skewed learning data, anomalies in individuals semantic representations may occur. There will, nonetheless, be general patterns of use across whole language communities. What learners are assumed to be implicitly doing is approximating to the patterns of use in their learning communities as a whole.
- 20.
I ignore the contribution of the indefinite article and treat this as a simple predication. I leave the modelling of quantification in this system for future research.
- 21.
Which will be 1 over the sum of the modified distribution.
- 22.
This could also be described by adjusting values of parameters on a Gaussian function.
- 23.
I assume that 0.01 is the arbitrarily small value. \(\varPhi \) is an abbreviation for \(\lambda [\dot{s}](\dot{s} \vDash \langle \!\langle \mathtt {height\!\!=}h, \mathtt j , yes\rangle \!\rangle \mid \dot{d}\vDash \langle \!\langle \mathtt {utters}, \dot{a}, \mathrm {MAN}, \mathtt j , yes\rangle \!\rangle ) \).
- 24.
Bayes’ Rule, stated in regular notation is: \(P(C|A) = \dfrac{P(C \wedge A)}{P(A)}\).
- 25.
Where the output judgement is decided by whether an ‘apple’ or an ‘not-apple’ judgement receives a higher value.
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Acknowledgments
I would like to thank the organisers and participants of the Bayesian Natural Language Semantics and Pragmatics workshop held at ESSLLI 2013 and the anonymous reviewers for the workshop and this volume for the helpful comments and improvements that they have suggested. Thanks also to the participants of the King’s College London Language and Cognition Seminar, with special thanks to Alex Clark, Ruth Kempson, Shalom Lappin, and Wilfried Meyer-Viol for their invaluable comments on earlier versions of this paper.
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Sutton, P.R. (2015). Towards a Probabilistic Semantics for Vague Adjectives. In: Zeevat, H., Schmitz, HC. (eds) Bayesian Natural Language Semantics and Pragmatics. Language, Cognition, and Mind, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-17064-0_10
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