Lewis and Vagueness (guest post, by Daniel Nolan, University of Notre Dame)

This month’s letter is an eleven-page letter David Lewis sent to Timothy Williamson in May 1999. The letter divides into three main sections. In the first, Lewis sets out his own views about the connection between evidence (or one thing we could call ‘evidence’) and knowledge (or things ‘know’ could mean in various contexts). The second section is a sustained discussion of Williamson’s own arguments about evidence, knowledge and luminosity. (E.g. the question of whether you always know what your immediate sensory evidence is.)  The third section, from p 8 onwards is a defence Lewis offers of his own “supervaluationist” views about vagueness from influential criticisms that appeared in Williamson’s 1994 book Vagueness.

This third section (pp 8-11), together with some things Lewis says in a slightly later letter to Williamson in June 1999, have the status of an underground legend among some philosophers working on vagueness. I first heard rumours of this letter in 2001 or 2002. Before I heard about this letter, I had assumed Lewis was a “supervaluationist” about vagueness in the sense of holding a view like that expressed in Fine 1975 or Keefe 2000, and the view criticised under that name by Williamson 1994. Lewis’s remarks about the “linguistic theory of vagueness” in On The Plurality of Worlds (Lewis 1986) as involving semantic indecision, for example, are often read as expressing standard supervaluationist sentiments. It becomes clear in this letter that Lewis’s view on vagueness by the late 1990s was not that view. Lewis labels his view supervaluationism, but it is at least a very unorthodox supervaluationism. In some moods I would be inclined to not classify the view he defends here as a kind of supervaluationism at all, though I won’t make a fuss about the label.

Lewis defends the view that sentences containing vague expressions are never true or false. Nor is our aim in assertion to only assert the true ones. Instead, supertruth replaces truth in our account of the aim of assertion. Assertion is to be guided by producing sentences that are supertrue, or near enough supertrue. (This marks a change even from the view along these lines discussed in Lewis 1993.) An approach like this has a range of unusual features: it seems revisionary indeed about which claims are true, and presumably any presentation of theory itself is not intended as true, since any presentation will doubtless employ vague language. (Lewis does keep an apparatus of precise propositions that are literally true in the background, though, so at least propositional truth might be more straightforward.) Lewis’s view expressed here has some close affinities with the view defended in Braun and Sider 2007.

In this blog post I want to give a bit of an “executive summary” of what we can learn about Lewis’s view of vagueness, and related views that come out about truth, validity, assertion, and the like. The sections of Lewis’s letter on epistemic matters, like evidence and luminosity, are very interesting as well, and would repay careful reading by people working on those topics: but since the material on vagueness strikes me as the most important part for current thinking, that’s what I’ll focus on here. I have written up a more detailed discussion of the view of vagueness we get from Lewis here, and Lewis’s responses to Williamson’s arguments against supervaluationism here. There I’ll go into some things in more detail, and also include a section on Lewis’s remarks on evidence (section 1 of the letter).

First, a few paragraphs on “orthodox” supervaluationism to set the scene:

An orthodox supervaluationist view about vagueness, or indeterminacy generally, treats vagueness as a matter of expressions in our language having a meaning that is somehow unsettled between multiple candidate precise meanings. Sentences with vague expressions in them can be associated, one-to-many, with “precisifications”: assignments of precise semantic values to a sentence. (And likewise, parts of sentences like names or predicates can also be associated with “precisifications” that assign them precise values.) The usual semantic value associated by each precisification with an entire sentence will be a proposition with an entirely determinate truth-value. Some of these precisifications are “admissible” precisifications: assigning that precise semantic value to an expression would not get the wrong truth-value for any of the sentences that the expression appears in which are determinate in truth value. (There is not complete agreement about what makes a precisification an admissible one: you might think that an admissible assignment respects “all the rules of use for the expressions in the sentence”, for example, but only if there is nothing about the rules of use for the expression that guarantees that sometimes it fails to apply completely precisely!) I will talk about a sentence’s truth value on a precisification, to mean the truth-value that the proposition associated with a given precisification has.

Sentences with vague vocabulary in them are supertrue if and only if all of their admissible precisifications are true; superfalse if and only if all of their admissible precisifications are false. A sentence in a context is plain true if and only if it is supertrue; plain false if and only if it is superfalse, and neither true nor false otherwise. Finally, supertruth plays the role in assertion that you would normally take truth to play: plausibly (but controversially), a claim is assertable only if it is supertrue.

Outside borderline areas, ordinary sentences behave as if they were precise: “Michael Jordan is tall” is supertrue, and so plain true. Inside borderline areas, some sentences will be neither true nor false: if Bob is a borderline case of “tall”, “Bob is tall” will be neither true on all admissible precisifications, nor false on all admissible precisifications, but rather true on some and false on others. But complex sentences built up from these simple sentences may still be supertrue: “Bill is tall or Bill is not tall” will be true on all precisifications, and so supertrue (/true). With the right rules for admissibility, we can ensure other claims are true on all precisifications: “If Jim is tall, and Bill is taller than Jim, then Bill is tall” should be true on all precisifications, even if both Jim and Bill have heights that make them borderline for “tall”.

The way sentences interact with a sentential “determinacy” operator (“it is determinate that…”) for orthodox supervaluationists is as follows: a sentence is determinately true on an admissible precisification provided all of its admissible precisifications are true. Or to put it directly in terms of the operator, DΦ is true on a given admissible precisification when Φ is true on all admissible precisifications. An inference is deductively valid when it is necessarily supertruth-preserving: the supertruth of all of the premises always guarantees the supertruth of the conclusion. (This is, near enough, “global” validity in the sense of Williamson 1994 p 147-148).

So much for orthodox supervaluationism (or at least my sketch of it for today). Lewis’s view is not this package of views. This is despite the fact that his discussion of the “linguistic theory of vagueness” in On The Plurality of Worlds (Lewis 1986) for example, is often read as expressing standard supervaluationist sentiments. (As Lewis himself notes in his letter, there are some remarks in his “Many, But Almost One” (1993) paper that also suggest a very non-standard supervaluationist attitude to vagueness.)

Lewis’s view employs admissible precisifications, and shares some features with orthodox supervaluationism: for example, that disjunctions can be supertrue even though both disjunctions fail to be supertrue. And he can employ a determinacy operator like that of the supervaluationists’, holding that for DΦ to be true on an admissible precisification Φ must be true on all admissible precisifications. But he rejects the identification of supertruth with truth. Rather, as mentioned above, we replace sentential truth with supertruth in parts of our theory of assertion: co-operative speakers will aim at asserting a sentence only if it is supertrue, or near enough to supertrue. Sentences containing vague language are not to be treated as true or false simpliciter, but only true or false relative to a precisification: in this respect, they are analogous to ambiguous sentences, which may not be true or false simpliciter but only true or false relative to a disambiguation. Lewis gets much of the effect of using vague language as an orthodox supervaluationist predicts, without allowing for the truth of sentences containing vague vocabulary even in the clearest cases.

His remarks responding to particular challenges from Williamson reveal some further unusual stances he takes. Here is a brief list of five of the positions I found most surprising for Lewis to take:

1) On validity: in his response to Williamson’s Section V.3, Lewis appears to endorse a pluralism about logical validity (at least logical validity for sentences). And he says that paraconsistent systems like Priest’s LP or Dunn’s RM are among the appropriate systems for logical validity!

2) Orthodox supervaluationism allows that there is one grain of sand that makes a difference between a heap and a non-heap, even though for each grain of sand, it fails to be true that that is the crucial grain of sand. But Lewis goes one further: since the standard of assertion is near-enough-supertruth, we can assert about each grain of sand that it is not the crucial one that makes the difference, while also claiming that there is one that is the crucial one that makes the difference. Vagueness lets us assert a jointly inconsistent set of sentences.

3) Lewis allows that some of the phenomena of vagueness might be due to the kind of context dependence Kamp 1975 and Kamp 1984 discuss, as well as the apparatus of supervaluations.

4) Lewis takes a surprising stand on determinacy in the infinite limit in his response to Williamson V.6. Just as it would be odd to have a sharp transition from clear positive cases clear negative cases (e.g. a sharp height cutoff between the shortest tall man to the tallest not-tall man), it looks implausible that there would be a sharp cutoff between clear positive cases and borderline cases (e.g. between the shortest tall man to the tallest indeterminate-whether-he-is-tall man). What about the “perfectly straightforward applications” of expressions: cases where it is determinate that Ann is tall, determinate that it is determinate that Ann is tall, determinate that it is determinate that it is determinate that Ann is tall… and so on, for any finite number of iterations of “it is determinate that”?  This status, of being determinate* that Ann is tall, is a puzzling one: it is tempting to say its boundaries are also vague, but Williamson argues that a supervaluationist will need to rely on additional structure in her model of vagueness to capture this. Lewis, on the other hand, wants to say no case is one of someone being determinate* tall, no matter what her height. (And presumably says this about anyone of any possible height). Even if you were 100′ tall, at some point you would be in the borderline for tall, or the borderline for the borderline for tall, or the borderline for the borderline for the borderline for tall, or…  Furthermore, Lewis seems to be saying that people who would not agree with him about this have made a mistaken assumption “lazily”.

5) Lewis is stuck with some challenges about how to explain ordinary attributions of truth to sentences containing vocabulary susceptible to vagueness. (His position, strictly speaking, is that none of these sentences are true: but presumably there is something all right about our ordinary sorting of sentences uttered on occasions into true and false ones, when we use “true” outside technical discussions of language.) One option that he seems to close off is that it is okay to say that a sentence S is true when it is supertrue (or near enough supertrue).

My thanks to all of those on the Age of Metaphysical Revolution Project, and particularly to Anthony Fisher for an invitation to be a guest poster on the Letter of the Month blog. Thanks also to Timothy Williamson for being willing to have me present this letter.

As mentioned above, for more discussion of the views Lewis expresses in this letter, and some critical commentary, please look at my longer discussion: Lewis on Williamson: Evidence, Knowledge and Vagueness. Lewis’s remarks in this letter about vagueness seem to me a fruitful source for ideas and problems in exploring non-standard accounts of vagueness in terms of the relationship between vague language and multiple acceptable precisifications.



Braun, D. and Sider, T. 2007. “Vague, so Untrue”. Nous 41.2: 133-146

Fine, K. 1975. “Vagueness, Logic and Truth”. Synthese 30: 265-300

Kamp, R. 1975. “Two Theories of Adjectives” in Keenan, E.L. (ed.) Formal Semantics of Natural Language. Cambridge: Cambridge University Press.

Kamp, R. 1981. “The Paradox of the Heap” in Mönnich, U. (ed). Aspects of Philosophical Logic, Some Logical Forays into Linguistics and Philosophy. Dordrecht: D. Reidel

Keefe, R. 2000. Theories of Vagueness. Cambridge: Cambridge University Press.

Lewis, D. 1986. On the Plurality of Worlds. Oxford: Blackwell.

Lewis, D. 1993. “Many, But Almost One” in Campbell, K., Bacon, J. and Reinhardt, L. (eds) Ontology, Causality and Mind: Essays on the Philosophy of D. M. Armstrong. Cambridge: Cambridge University Press, pp 23-38. Reprinted in Lewis 1999b, pp 164-182.

Williamson, T. 1994. Vagueness. London: Routledge.