Knowledge and Vagueness
KNOWLEDGE AND VAGUENESS
When anthropologists painstakingly identified the taxon of the skeleton that later became known as "Lucy's child,"
There was no eureka. There was no grand turning point. The evidence kept dribbling in, and through hard labor and some dogged thinking we did solve the puzzle, not through revelation but through a sort of absorption, just below the level of explicit consciousness. It was as if the truth had slowly seeped through our pores, until we had come know it without knowing that we did. So when the final, indisputable confirmation came, we hardly noticed the event. What had once been a mystery had become—in hindsight, mind you—obvious from the start (Johanson and James Shreeve 1989, p. 203).
Instead of there being a clear point at which the anthropologists knew that the specimen was Homo habilis, there was stratification: The researchers began from obvious ignorance, inched up to being borderline knowers, and eventually emerged as clear knowers.
The vagueness of knowledge has substantial implications. When skeptics took over Plato's Academy, they tried to prove that there can be no knowledge. Such a proof would ensure that everything is a clear negative case of "knowledge." Knowledge would be a perfectly precise term; a skeptic should think twice before complaining about the vagueness of knowledge! Typically, borderline cases are flanked by clear cases (Figure 1), so
the vagueness of "know" positively invites the inference that there is at least some knowledge.
The vagueness of knowledge also affects principles of epistemic logic such as the "KK thesis": If you know, then you know you know. If the KK thesis were true, the anthropologists would have known that they knew from the moment they knew the taxon of Lucy's child.
Given a naturalistic perspective on knowers, the vagueness of "know" should be expected. Human perceptual capacities and memory trail off in the patterns made famous by evolutionary iconography (Figure 2).
1. The Sorites Paradox
Only a vague term (e.g., human ) can serve as the inductive predicate of a sorites argument:
Base Step: There are now humans.
Induction Step: If there were humans n years ago, then there were humans n – 1 years ago.
Conclusion: There were humans five billion years ago.
Because the earth is only 4.6 billion years old, the conclusion is false. The base step is clearly true and the argument is classically valid. Therefore, people naturally suspect the induction step. However, they are unable to specify a value for n at which the generalization is false.
If vagueness is merely a kind of ignorance, there is no need to find a counterexample to the induction step. One can know a generalization is false even if one cannot pinpoint where it breaks down. Consider an anthropologist who doubts that all of the skeletal fragments in a bag belong to a single individual but cannot identify any pair of fragments as belonging to distinct specimens. When the anthropologist weighs the bag and learns there are more than enough fragments to constitute one skeleton, that is all that is needed to refute the generalization that all of the fragments come from a single individual.
In common usage, a borderline case is often simply one that cannot be settled at a given stage of inquiry. When an archeologist sorts stones, a few are obviously tools and most others are clearly just rocks. There will be another
group of stones whose status cannot be determined by unaided observation. These borderline cases are put under a field microscope. The three-way sorting begins afresh. Borderline cases that survive this second stage of inquiry may eventually wind up under an electron microscope.
Philosophers focus on the minority of borderline cases in which there is no prospect of resolution. How many years did the Middle Ages last? Is Israel a new state or an ancient state? Philosophers are at sea with these questions, and because people are unsure what would count as correctly answering these questions, their ignorance cannot be relativized to a set of resources.
Epistemicists insist there remains a crucial resemblance between these absolute borderline cases and relative borderline cases; they take all vagueness to be a form of ignorance. Epistemicists solve the sorites paradox by claiming that there is a hidden counterexample to the induction step. After all, they know the base step is true and the conclusion is false; classical logic then licenses the deduction that the induction step is false.
In classical logic, denying the induction step of the above sorites argument is equivalent to asserting there is a number n such that n years ago there was at least one human being but the year before that there were no human beings. So belief that there is a counterexample to the induction step is equivalent to the belief that there was a first human!
Incredulous anthropologists counter that nature does not draw a sharp line between humans and nonhumans. Speakers have not made up for the absence of sharp boundary by supplying an artificial one. Consequently, anyone who searches for the exact year humans appear on the evolutionary timeline is conceptually confused.
2. Infinite Regresses
David Sanford (1975a) points out that if finite sequences do not need beginnings or endings, there are neglected solutions to infinite regress problems. Consider the infinite regress of justification: A belief can only be justified by another justified belief. Justification cannot be achieved by reasoning in a circle. Nor can chains of justification be infinitely long. The skeptic concludes that no beliefs are justified. The foundationalist responds by conferring axiomatic status on some beliefs; axioms justify other beliefs without needing justification from other beliefs. The vagueness of "justified" suggests another solution to this infinite regress: Admit that the chain of justification is finite but deny it must terminate in an axiomatic belief.
Compare justification to motherhood. Each woman must have a mother. Her family tree cannot go back infinitely and cannot circle back on itself. Is one to conclude that some woman lacks a mother? Sanford instead appeals to the vagueness of "mother." As one moves down her ancestral line, what counts as a mother eventually becomes less and less clear. After passing through a stretch of borderline cases, one arrives at ancestors who clearly lack a gender and therefore are clear nonmothers. Sanford says that an insistence that finite sequences have terminal points is an incarnation of the sorites paradox.
3. The Logical Predicament
Because the sorites argument is classically valid, David Sanford must espouse a deviant logic. Supplemental logics (modal logic, deontic logic, etc.) merely add theorems to the standard stock; they cannot subtract the sorites from the list of valid arguments. So Sanford must target classical logic, weakening it just enough to stop its validation of the sorites—without causing too much collateral damage. In standard fuzzy logic, almost all classical theorems are rejected—except for the special case in which the truth-values equal full truth or full falsehood (Machina 1976). Sanford (1975b) accepts degrees of truth but prefers to keep all classical theorems by rejecting the truth-functionality of the logical connectives. Other deviant logicians reject some classical inference rules. For instance, intuitionists closely associate proof with truth and so try to derail the sorites paradox by rejecting the validity of double-negation (Putnam 1983). Supervaluationists either reject inference rules such as contraposition and reductio ad absurdum or reject core semantic principles such as Tarski's convention T (McGee and McLaughlin 1995).
These changes occur at the center of the human web of belief and so reverberate widely. Because knowledge implies truth, new questions are raised by the fuzzy logician's talk of degrees of truth. For instance, can one know a proposition that has a degree of truth less than one? The fuzzy logician wants to explain human ignorance of typical borderline cases and so is committed to saying that people are ignorant of propositions that are as close to being false as to being true. But what about propositions that are nearly true? Fuzzy logicians say that many propositions that appear to be clear truths merely have a high degree of truth. So if knowledge implies full truth, people know less than they seem to know.
4. The Credibility Gap
Knowledge does seem to imply full truth because knowledge implies belief and one can only believe what one considers to be fully true. "It is not fully true that the Black Skull is an australopithecine but I believe it is an australopithecine" is as hard on the ear as G. E. Moore's paradoxical sentence "It is raining but I do not believe it" (Moore 1942, 543).
This credibility gap hinders efforts to moderate epistemicism. Intuitively, people's wishy-washy attitude toward borderline cases seems like a reaction to the vagueness of these cases. But a subjectivist may reverse the relationship and say that the wishy-washy attitudes are what make propositions vague. If indeterminacy is a projection of human ambivalence, then people may hope to avoid the metaphysical burden of epistemicism. The epistemicist would be right in basing vagueness in the subject's limitations but wrong in postulating sharp thresholds.
Crispin Wright (2001) says that x is a borderline case of F-ness if two parties can disagree about whether x is F without either party being guilty of a cognitive shortcoming. Each party knows all the relevant facts, each is a competent speaker, and each has reasoned well. Wright compares this faultless stalemate with the cultural variation that makes relativism popular among ethnographers.
Critics of Wright object that anyone who takes a position on a borderline statement is guilty of a cognitive shortcoming; they ought to be agnostic. If one thinks that same-sex civil unions are borderline cases of marriages, then one cannot believe that they are marriages.
Stephen Schiffer (1998) has suggested that people have a special attitude toward cases that they take to be borderline. "Vague partial belief" differs from the belief humans extend to precise propositions. It also differs from the degrees of belief that people associate with probability theory. The probability calculus instructs people to assign a higher probability to a disjunction than either of its contingent disjuncts. But when the disjuncts are borderline cases, Schiffer only assigns the disjunction as much vague partial belief as he assigns the strongest disjunct.
This result (which echoes the fuzzy logician's rule for calculating disjunctions) grates against the observation that hedging a claim can make it more assertible. One can know that Blaise Pascal died at thirty-nine but not be sure whether this counts as dying as a young man. However, one can confidently say that either Pascal died as a young man or as a man in middle age.
Supervaluationists have a simple explanation of why people do not believe borderline statements: they lack truth-values. Belief aims at truth, thus people cannot believe a statement that they believe to be borderline. However, this explanation overgeneralizes, for it does seem possible to have weak propositional attitudes (guessing, doubting, and suspecting) toward statements that one acknowledges to be borderline.
Supervaluationists also have trouble explaining why one can make a statement more credible by adding an epistemic hedge. If one believes linguistic indecision prevents "ten is a small number" from having a truth-value, then one cannot believe it may be true. Yet if ten clearly is a borderline case of a small number, then it is appropriate to shrug one's shoulders and conclude "ten might be a small number and ten might not be a small number." Indeed, prefixing any statement that is clearly borderline with "maybe" seems to make it clearly true.
Supervaluationists use truth-value gaps and the principle that knowledge implies truth to explain why humans are absolutely ignorant of borderline statements. God cannot know when a fetus becomes a human being because there is nothing to know.
Supervaluationists pride themselves on the modesty of their revision of classical logic. The workhorse of their adjustment is the notion of super-truth: A statement is super-true if and only if it comes out true under all admissible precisifications of the statement. For instance, "Either the specimen is a Homo erectus or an archaic Homo Sapien " is super-true because it comes out true regardless of how one precisifies Homo erectus and archaic Homo sapien.
Any statement that has the form of a classical tautology will be super-true even if it contains vague terms. So supervaluationists claim to preserve all the theorems of classical logic.
But can one believe a statement by virtue of its super-truth? Truth under all disambiguations is not enough. Suppose a person says "bachelors are mammals" and it is not clear whether that person is referring to unmarried men or to college graduates or to just any young male mammal. One knows the statement expresses a truth but does not know which truth it expresses. Ambiguous statements are not objects of knowledge.
But vague statements are objects of knowledge. People know "the number of men is either an even number or an odd number" even though the vagueness of "man" makes it impossible to count the number of men. Supervaluationists have trouble accepting asymmetries between vagueness and ambiguity. They characterize vagueness in semantic terms rather than epistemic terms, so supervaluationism looks more like a logic of ambiguity (Lewis 1982).
5. Higher Order Vagueness
In Purity and Danger Mary Douglas conjectures that the bearers of taboos are borderline cases (moles, eels, twilight, and so on). She interprets rituals of purification as attempts to reclassify doubtful cases (as when hermaphrodites are declared men through a rite of passage). Assessment of Douglas's hypothesis is hindered by the vagueness of "borderline case."
Borderline cases of "borderline case" are normal with vague terms. In addition to there being borderline cases of "human," there are borderline cases of "borderline human." So in addition to first order vagueness there is second order vagueness, third order vagueness, and so on, apparently ad infinitum.
Higher order vagueness is a problem for deviant logicians because they employ classical logic and set theory in the metalanguages they use to describe vague terms. This classical medium forces them to represent the transition from clear to borderline cases as a sharp threshold. For instance, supervaluationist semantics implies that there is a first point at which "x is a human" is true. So instead of having the epistemicist's sharp threshold between truth and falsehood, the supervaluationist has a sharp threshold between truth and absence of truth. Similarly, the fuzzy logician has sharp thresholds between each degree of truth, and can only approximate vagueness by using a large quantity of discrete microtransitions. The fuzzy logician's representation of vagueness is like a dot matrix printer's representation of gray—a black and white affair when examined close up.
What originally bothered philosophers were sharp thresholds, not sharp thresholds between truth and falsehood. Thus epistemicists advertise themselves as just self-consciously biting a bullet that others gnaw absentmindedly.
6. Explaining the Ignorance
Recent epistemicists are careful to endorse the principle that inquiry into borderline cases is futile. That is why they stress that borderline statements are unknowable. But if these statements have truth-values, why can't they be known? One response is to challenge the presumption in favor of knowability—to portray ignorance as a natural state in need of no explanation.
However, Timothy Williamson (1994) directly answers the question of why borderline statements cannot be known. He traces the unknowability of borderline statements to the knower's need for a margin for error. When at a stadium, one can know there are about ten thousand people. But one cannot know there are exactly ten thousand, for a person cannot reliably discriminate between there being ten thousand and there being ten thousand and one. Given that "human" has the sort of precise threshold epistemicists allege, anyone who happened to correctly believe that humans originated n years ago, would have to be right by luck. For all this person knows, the origin could have been a year earlier.
Williamson believes that thresholds for vague predicates are determined by the psychology, social conditions, and environment of the speech community. These conditions are too complicated to allow humans to ascertain the threshold for vague terms.
The margin for error principle yields different limits for different kinds of knowers. For much of the history of Homo sapiens there were other hominids who had different cognitive capacities. Williamson's theory does not preclude these hominids from knowing the threshold of some vague terms. Some of what is chaotic to humans may be predictable to these homonids. Williamson is committed to the relativity of all borderline cases. Supervaluationists claim an advantage over Williamson insofar as they neatly model absolute borderline cases.
Roy Sorensen (2001) has speculated that an epistemicist can match the neatness of the supervaluationists by using truth-maker gaps instead of truth-value gaps. A truth-maker is what makes a proposition true. For instance, "Humans and chimpanzees had a common ancestor seven million years ago" is made true by a Miocene primate who had as descendants both Noam Chomsky and Nim Chimpsky. One learns the truth-value of propositions only by becoming appropriately related to their truth-makers. Propositions that lack truth-makers have truth-values that are not anchored to any piece of reality. This objective indeterminacy makes the propositions absolutely unknowable.
7. Vagueness and Epistemic Logic
If the relationship between knowledge and borderline cases is orderly, epistemicists can offer a logic of vagueness as a branch of epistemic logic. For instance, Timothy Williamson elaborates his "logic of clarity" in a way that makes it isomorphic to supervaluationism. The basic idea is that a statement is definitely true if it comes out true "under all sharp interpretations of the language indiscriminable from the right one" (Williamson 1999, p. 128). This mirrors the supervaluationist's principle that a statement is definitely true if it comes out true under all admissible completions of the language.
Epistemicists are divided on how closely vagueness is bound up with borderline cases. Everybody agrees that a vague term need not have actual borderline cases. Possible borderline cases are sufficient. But what about borderline cases that are merely epistemically possible? Perhaps the mere threat of an objective borderline case can be enough to make a predicate vague (Sorensen 2001). After all, if the threat cannot be exposed as false, then there will be irremediable linguistic ignorance without borderline cases. One would be able to embed the predicate in a sorites argument and bedevil people with doubts about termination points.
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Roy Sorensen (2005)