Abstract
In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness—largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature—are logically nonclassical and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of the comparative form of gradable adjectives and of various types of gradable precise adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. This paper argues that the fuzzy kind of theory can match the benefits of the classical kind and hence that the burden is on the latter to match the advantages of the former. In particular, we develop a new version of the fuzzy logic approach that—unlike existing fuzzy theories—yields a compelling analysis of the comparative as well as an adequate account of gradable precise predicates, while still retaining the advantage of genuinely solving the sorites paradox.
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Notes
\(x_1,\ldots ,x_n\) denote the objects in the series from first (\(x_1\)) through to last (\(x_n\)). x ranges over all objects in the sorites series except the last object and \(x'\) is the object immediately after x in the series.
See e.g. Fara (2000).
As mentioned in Sect. 1, the argument that fuzzy theories of vagueness handle the sorites better than other kinds of theory is presented elsewhere (Smith , forthcoming) and in this paper we do not repeat the argument in full.
The fundamental idea is that truth comes in degrees. The representation of these degrees of truth by real numbers between 0 and 1 inclusive is convenient but not essential to the very idea of degrees of truth, which could potentially be modelled using other structures. We shall discuss this issue further below.
Cf. Forbes (1983, pp. 243–244, 1985, pp. 171–172), Williamson (1994, pp. 123–124), and Paoli (2003, p. 365, forthcoming).
On terminology: Suppose that some speakers’ implicit grasp of certain semantic facts leads them to use language the way they do—and suppose that a certain semantic theory T is a correct codification of those semantic facts. In such a situation, we say that those speakers ‘internalise’ that semantic theory. This is purely for ease of expression and should not be taken to suggest (for example) that the semantic theory comes first and then speakers learn it. On the contrary, the speakers’ usage comes first and the semantic theory seeks to codify those semantic facts, implicit grasp of which leads speakers to use language as they do.
It is relatively straightforward to allow also function symbols in the language and/or consider a special symbol for equality, but we decide to omit them here for simplicity.
By \(v[x{\mapsto }d]\) we denote the \({\textbf{M}}\)-valuation, resulting from v by setting \(v[x{\mapsto }d](x) = d\) and keeping the values of the remaining variables unchanged.
Observe that the notion of consequence we define can be characterized as a sentential truth-preserving notion of consequence, i.e., one which is concerned with whether the property of “having value 1 for all valuations” is preserved. Other alternatives, not considered in the paper, include the non-sentential truth-preserving consequence, which is concerned with whether the property of “having value 1” is preserved for each valuation, or the degree-preserving consequence, which is concerned with whether for all valuations the value of conclusion is not less than the value of each of the premises; see Bou et al. (2009).
It is also possible to define a second disjunction, by means of the formula \(\lnot \varphi \rightarrow \psi \), linked with the conjunction \( \mathbin { \& }\) by De Morgan laws. However, since this disjunction connective will not have a general counterpart in the framework introduced later, we will skip it here.
Actually, it can be shown that n can be taken as the minimum number of times one needs to use the premise \(\varphi \) in a formal proof of \(\psi \) from \(\Gamma \cup \{\varphi \}\) (see Chvalovský & Cintula, 2012).
Alternatively, it can also be obtained by adding the axiom of excluded middle, \(\varphi \vee \lnot \varphi \), or the axiom of contraction: \((\varphi \rightarrow (\varphi \rightarrow \psi ))\rightarrow (\varphi \rightarrow \psi )\); see Hájek (1998).
Observe that while \(*_{\textrm{G}}\) and \(*_\Pi \) are actually continuous (like
), \(*_{\textrm{NM}}\) is only left-continuous.Note that this pattern also fails for some precise gradable predicates, namely, those whose antonym pairs are contraries (rather than contradictories)—i.e. antonyms A and B such that A is not equivalent to not-B. For instance, it fails for ‘empty’/‘full’, despite these being precise (in some contexts).
These approaches are named after the characters in the following dialogue from the 1984 motion picture This Is Spinal Tap: Nigel Tufnel: This is a top to a, you know, what we use on stage, but it’s very, very special because, if you can see Marty DiBergi: Yeah NT: the numbers all go to eleven. Look, right across the board, eleven, eleven, eleven and MD: Oh, I see. And most amps go up to ten? NT: Exactly. MD: Does that mean it’s louder? Is it any louder? NT: Well, it’s one louder, isn’t it? It’s not ten. You see, most, most blokes, you know, be playing at ten, you’re on ten here, all the way up, all the way up, all the way up, you’re on ten on your guitar, where can you go from there? Where? MD: I don’t know. NT: Nowhere. Exactly. What we do is, if we need that extra push over the cliff, you know what we do? MD: Put it up to eleven. NT: Eleven. Exactly. One louder. MD: Why don’t you just make ten louder and make ten be the top number and make that a little louder? NT: [long pause] These go to eleven.
The two limit points give a largest and a least truth-value whose presence is necessary for our purposes of covering a wide class of graded predicates. Of course, these extremal truth values are not actually used in the modelization of all kinds of graded predicates (see Sect. 6).
Let us note that we could work instead with the optically more manageable interval \([-0.1, 1.1]\) (yet with the same cardinality), which is perhaps more consonant with our up-to-eleven spirit. Or we could, as well, use the unit internal [0, 1] to stress the similarity with t-norms. Any suitable monotone bijection of the extended reals with any chosen closed interval of reals would yield an isomorphic semantics (more details later); however, we would pay the price of losing the natural and simple arithmetic interpretations of conjunction and implication that we have on the extended reals. The choice of a particular closed interval of real numbers as the set of truth values for our algebra is inessential from the point of view of the proposed modeling of the sorites paradox in the next section, so we can pick one just out of aesthetic considerations like those just mentioned.
This correspondence obeys a precise technical sense, which we will not explain here; see right after the definition and the end of the section for some hints.
However, note that this is not the only suitable assignment. For the sake of simplicity, we shall not make this explicit in our formulations, but we advocate a view—called ‘fuzzy plurivaluationism’—which leaves room for the acceptance of multiple equally-suitable models. On this view, instead of each vague discourse being associated with a unique intended fuzzy model, it is associated with multiple acceptable fuzzy models. The acceptable models are all those that our usage and usage dispositions do not rule out as being incorrect interpretations of our language. These models must be adequate, in the sense that they must respect the meaning-determining facts (i.e. what speakers say, and the circumstances in which they say these things; what speakers are disposed to say in possible circumstances; etc.). For example, they must all assign the adjective ‘rich’ a model such that any individual who all speakers agree is paradigmatically rich is assigned a designated value. Moreover, for any two people such that one is clearly richer than the other, all adequate models must assign ‘rich’ a model that maps the former to a higher degree than the latter. The idea is that, even though the meaning-determining facts do not suffice to pin down a single adequate fuzzy model, this does not necessarily lead to abandoning the fuzzy route. We just take all adequate models to be on a par. For more details, see Smith (2008).
Although this may seem very obvious in this setting, other proposals are not able to offer such a simple analysis. For instance, Kennedy (2007) needs to pose a null morpheme transforming so-called ‘bare adjectives’, which denote measure functions, into proper predicates, understood as classical crisp predicates.
This particular choice of analysis forbids comparatives involving different predicates (e.g. ‘Alex is taller than he is smart’), but we could easily modify it so as to leave room for these (by analysing the comparative as a function from a pair of properties to a binary relation). Nevertheless, since this topic falls outside the scope of this article, we will stick to the analysis above.
For more details on this distinction, see Burnett (2016, Chap. 3).
This is one of the ways in which context interacts with the semantic analysis of gradable predicates, but it is not the only one. For instance, another point where context must be taken into account is to determine the exact form of the extension of (a use of) a predicate (and, possibly, anti-extension, depending on the kind of analysis). This is the sense in which ‘tall’ has different extensions depending on whether we are talking, say, about basketball players or about 10 year-old children. The latter collections of objects are usually referred to as ‘the class of comparison’ (see (Kennedy , 2007; Burnett , 2016)) and are usually provided by the context.
In all these cases, we are focusing on the precise use of the predicates in question.
These distributions are described in Kennedy (2007).
We could have chosen any other designated value. For present purposes, it does not need to be the top of the chain. However, we choose the top for simplicity and to capture the intuition that bigraded predicates are a limiting case of any other type of gradable precise predicates.
Once again, we could have chosen any other anti-designated value.
Formulas of the form \( MF (c,d)\) correspond to our old \({\textbf{f}}(c)>{\textbf{f}}(d)\) (i.e. the result of multiple \(\beta \)-reductions on \(\lambda {\textbf{g}}\lambda {\textbf{f}}.\lambda x.{\textbf{g}}(x)>\lambda y.{\textbf{f}}(y)\)). These two representations belong to two different levels of linguistic analysis. The lambda calculus expressions make explicit the compositional steps by which we get to the semantic analysis of complex statements, in this case, comparatives. By contrast, the first-order notation employed here simply expresses the resulting propositional content of those statements. In a sense, we can see the lambda expressions as providing instructions to arrive at the semantics of a binary predicate \(M\!F\), given a corresponding monadic predicate F.
Nevertheless, the question of the axiomatizability of the logic of \({\varvec{C}}\) is left for future research.
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Acknowledgements
We are grateful to the anonymous referees for helpful comments. The research leading to these results was supported by the Grant GA18-00113S of the Czech Science Foundation. Finally, we acknowledge support by the European Union’s Marie Sklodowska–Curie grant no. 101007627 (MOSAIC project).
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Cintula, P., Grimau, B., Noguera, C. et al. These degrees go to eleven: fuzzy logics and gradable predicates. Synthese 200, 445 (2022). https://doi.org/10.1007/s11229-022-03909-2
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DOI: https://doi.org/10.1007/s11229-022-03909-2