I wrote this months ago and have basically completely disavowed it. But I spent a lot of work on it, so it would be a shame if it never saw the light of day. If someone wants it, I can dig up the bibliography.
Vagueness is a troublesome and mysterious feature of natural languages. It is troublesome because it can be exploited to nefarious ends in an argument called “the sorites paradox.” It is mysterious because it is not altogether clear why language is vague at all, especially to the degree that it is. In using the term “vague,” I mean to distinguish it from other phenomena such as ambiguity, generality, context-dependence and uninformativeness. Ambiguity is the result of a confusion between multiple linguistic tokens, like “bat” (mammal of order Chiroptera) and “bat” (a variety of baseball equipment.) A predicate is general when it doesn’t make distinctions that more specific predicates might. An example of a general predicate is “child” (which does not specify gender.) Context-dependent predicates have different truth-conditions under different background conditions, like who is speaking and what the reference class is. “Tall” denotes different ranges of heights in humans and buildings. An uninformative predicate is one that supplies less information than is appropriate in a context, like “child” in this hypothetical exchange:
Speaker 1: Does Shmuel have any daughters?
Speaker 2: Shmuel has one child.
I will follow Mark Sainsbury in saying that vague concepts are “concepts without boundaries” (Sainsbury 1996.) That is, vague predicates classify objects without sorting them into well defined sets. This is not a theory-neutral definition of vagueness: it rules out many of the most popular competitors. However, I think that the absence of boundaries is such a basic feature of vague concepts that we can safely reject any theory that posits them.
The Sorites Paradox
The sorites is a very simple paradox that most people are roughly familiar with without knowing that it has a name or that it is even a paradox. It is an ancient puzzle, but there is something mysterious about it that has made most people (even some philosophers) dismiss it without even trying to put forth a solution. There are many variations, but here is a simple one suited for our purposes:
P1: 1,000,000,000 grains of sand in a pile constitute a heap
P2: If 1,000,000,000 grains of sand in a pile constitute a heap, then 999,999,999 grains of sand in a pile constitute a heap
P3: If 999,999,999 grains of sand in a pile constitute a heap, then 999,999,998 grains of sand in a pile constitute a heap
P1000000001: If 2 grains of sand in a pile constitute a heap, then 1 grain of sand in a pile constitutes a heap
C: Therefore, one grain of sand in a pile constitutes a heap
This is a paradox because while each of the premises seems indisputable, together they ought to entail an unacceptable conclusion. Any solution to the paradox must hold that in every context, one of the premises isn’t true or the argument is invalid. (Or that the conclusion is true!)
Solving the Paradox
The philosopher’s job is to figure out how this is so. Approaches that deny one of P2-1000000001 can be called sharpist, using Bryan Frances’s terminology (Frances 1.) Most sharpists will hold that vague predicates do have boundaries, but that these boundaries are unknowable (epistemicism,) indeterminate (supervaluationism,) extremely variable (contextualism,) or in possession of some other status that supposedly makes them less problematic. I am mentioning these theories only for the purpose of completeness: I consider them misguided for reasons I indicated in the introduction.
An approach that rejects P1 could be called nihilist, and an approach that accepts the conclusion could be called trivialist. I do not know of any defenses of the trivialist view in the literature, so I will say no more about it except for that it seems to share important similarities with the nihilist option. The nihilist view is also rare, but not unheard of. The clearest case of a true nihilist is Peter Unger, who takes the sorites paradox to be evidence for the nonexistence of ordinary material objects like heaps, chairs, and mountains. (Unger 1979.) Other nihilists take their views to be harmless and void of any serious metaphysical consequences. (Braun, Sider 2007) establishes a novel theory of ignoring vagueness within a nihilist (they call themselves “semantic nihilists”) framework. They are in agreement with the broad supervaluationist picture that vagueness is to be seen as the existence of multiple acceptable precisifications of a predicate, but they disagree with the identification of truth-under-every-acceptable-precification (what supervaluationists call supertruth and Braun and Sider call approximate truth) with truth simpliciter. Braun and Sider hold that “There is typically a cloud of propositions in the neighborhood of a sentence uttered by a vague speaker. Vagueness prevents the speaker from singling out one of these propositions uniquely, but does not banish the cloud” (Braun, Sider 4.) Braun and Sider see themselves as vindicating “[the] old and attractive view [that] vagueness is to be eliminated before semantic notions (truth, implication, and so on) may be applied” (Braun, Sider 1.)
Another approach is the tolerance approach. Tolerantists accept, in one form or another, all instances of this principle:
where P is a vague predicate like “a number of grains of sand that suffices to make a heap of sand” and ~P is a relation of similarity in aspects relevant to whether something is P. Because a soritical chain of objects can be constructed for most vague predicates (a series of objects in which the first element falls under the extension of the predicate, the second element does not fall under the extension of the predicate, and any two adjacent members are similar in all aspects relevant to whether they fall under the extension of the predicate,) a revision of classical logic appears to be required. (Although, see (Pagin 2010.)) This revision usually takes the form of the denial of the transitivity of entailment: the principle that from A⊢B and B⊢C one can derive A⊢C. The easiest way to generate a non-transitive logic is to demand different standards for premises (the symbols on the left hand side of ⊢) and conclusions (the symbols on the right hand side of ⊢.) For example, one could say that a valid argument is one where if the premise is assigned the value 1, then the conclusion must be assigned the value 1 or the value ½.
My own perspective falls between the nihilist and tolerantist camps. I agree with the nihilists that all predicates lack extensions, but I blame this not on vagueness, but on the possibility of verbal disputes surrounding those predicates. I will discuss this issue in a separate paper. I follow tolerantists in advocating for the adoption of a non-transitive logic to deal with the sorites paradox.
Dissolving the Paradox
Most people are not philosophers, of course, and so most people do not care about solving the sorites paradox. This is not troubling at all. What is troubling is what I would call dismissivism, the idea that there isn’t a solution to be found at all. Dismissivists substantiate this by blaming the paradox on what they see as a mistaken conception of the relation between reality and language. For example, some may reject the standard package of the correspondence theory of truth (the thesis that truth consists of a special relation between a statement and the world) and truth-conditional semantics (the thesis that the meaning of a statement consists of the conditions under which it is true.) One example of this approach can be found in (Correia 2013,) which puts forth a very interesting and radical analysis of the sorites paradox in terms of so-called “signalling games,” but falls short of providing a solution, even suggesting that any theory that avoided the paradox would be a misrepresentation of natural language (Correia 15.) It is obvious to me that natural language does avoid the paradox. And so, a formalization of this intuition would be appreciated.
Resolving the Paradox?
I agree with those who seek to sweep vagueness’s paradoxes under the rug in that I hold that no substantive account of vague truth can be supplied. However, we certainly use vague predicates in our day to day lives coherently, and it would be nice for logic to respect that. Logic on the picture I am advocating is not a matter of determining what might be the case but a matter of determining what inferences are acceptable. The sorites paradox is paradoxical not because it threatens our ideas about heaps but because it threatens our ideas of what can be derived from intuitively acceptable premises.
There are a great deal many non-transitive logics to choose from. The current favorite among tolerantists appears to be the logic ST (strict/tolerant), which can be characterized similarly as the “gappy” K3 and the “glutty” LP. If a statement is assigned the value ½ in a model of K3, it is treated as neither true nor false in that model (a gap), and if a statement is assigned the value ½ in a model of LP, it is treated as both true and false (a glut). ST splits the difference: premises are evaluated as in L3 and conclusions are evaluated as in LP. Another logic, “pb” (super/sub), with analogues to supervaluationist logic and subvaluationist logic, is more at home in a classical worldview. Although each instance of P(x)∧(x~P y)→P(y) follows from the claim that we have a Sorites series, the negation of the universal claim does as well (Not for all x and y, P(x)∧(x~P y)→P(y)), which may disqualify it in the eyes of very picky philosophers. ST arguably also suffers the same problem, although to a lesser extent: both the universal form of the tolerance principle and its negation follow from the existence of a Sorites series. I have a hunch that this could be avoided by considering non-transitive analogues of weakenings of LP that don’t prove
as in (Beall 2004).
1) This conception of vagueness can be fitted to a pluralist theory of vagueness just as easily as it can be to a nihilist theory. Under the nihilist picture, when we make vague statements we aren’t saying anything at all. Under a pluralist picture, when we make vague statements we are saying many things. Braun and Sider’s theory of ignoring vagueness could easily be extended as well, perhaps treating the ignoring of the plurality of what we say when we speak vaguely as an instance of conflation. A tolerant theory of vagueness based on conflation is described in (Ripley 2016). It is not clear exactly how the pluralist would resolve the sorites paradox without resorting to a tolerant framework. The literature contains some discussion of plurivaluationism, which seems to be what I am referring to as pluralism, so there may be some clues there.
2) If entailment is a relation between sets of premises and sets of conclusions, then a valid argument is one where if all the premises are true, then some of
the conclusions are true.