The Atlas of Reality
Page 45
What does it mean to say that something like a boundary is vague? Broadly, there are two theories of vagueness, epistemic and metaphysical (or semantic). According to Epistemicists, everything in fact has precise, determinate boundaries, but there are many cases in which we do not, and perhaps cannot, know exactly where those boundaries lie. Anti-Epistemicists maintain that in some cases there is no fact of the matter to be known. For Anti-Epistemicists, there are failures of the Law of Bivalence (the principle that True and False are the only truth-values a proposition can have) associated with the existence of vague boundaries. That is to say, sometimes a proposition about a boundary will be neither true nor false. Suppose Felix is a cat, and molecule M is vaguely a part of Felix. Anti-Epistemicists claim that (32) is neither true nor false:
(32) Molecule M is part of Felix.
In contrast, Epistemicists insist that (32) is either true or false, even if we cannot know which it is. Epistemicists claim that absolutely nothing in the world is vague. Every boundary of everything is precise and determinate.
12.2T Epistemicism. Every boundary is determinate. Any vagueness is merely epistemic, that is, merely a matter of our not knowing exactly where the boundary lies.
12.2A Anti-Epistemicism. Some boundaries are indeterminate, resulting in failures of the Law of Bivalence.
Epistemicism has a number of promising features. First, Epistemicists can hold to Bivalence. Bivalence provides a simple and convincing basis for the applicability of all of the principles of classical logic, including the Law of Non-Contradiction (no proposition is both true and false) and the Law of Excluded Middle (every proposition of the form p or not-p is a logical truth). Whatever we think about (32), the following propositions seem to be uncontroversially true:
(33) It is not the case that M both is and is not a part of Felix.
(34) Either M is a part of Felix or it is not.
(35) If M is a part of Felix, then M is a part of Felix.
However, if (32) is neither true nor false, it is not obvious how to explain the truth of (33)–(35).
Second, Epistemicists have no difficulty handling higher-order vagueness, vagueness about vague boundaries. For example, consider the question of whether the predicate ‘is vague’ is itself vague. For Epistemicists, this just amounts to the question of whether there are some cases in which we are ignorant about what we are ignorant about. It wouldn't be at all surprising or puzzling if we were ignorant about our ignorance.
Third, Timothy Williamson (2000) has provided an interesting and substantial theory that explains why we are ignorant about many boundaries. He does so by appealing to his margin of error principle. We can know a proposition p just in case we justifiably believe p now and we would still believe p, in any “nearby” counterfactual situation in which p remains true. So, I know that some molecule M in Felix's heart is part of Felix because I would still believe that M was part of Felix even if M were moved slightly to the left or right. This is not true of molecules at or very near the outer limits of Felix, since moving M slightly, even if it still remains part of Felix, could easily tempt me to error or at least to suspension of belief.
Finally, Epistemicists can appeal to the context-sensitivity of many vague terms and predicates (Kamp 1975). Terms are context-sensitive when what falls within the extension of a term depends on the conversational context within which the term is being used. Adjectives like ‘bald’ and ‘tall’ are paradigmatically context-sensitive. Someone might count as ‘tall’ in a conversation about college students but not in a conversation about college basketball players. These facts about contextual variation support Epistemicism in two ways. First, they make it harder for one to be sure about where the boundary lies in a particular context, since one may be unsure about some of the contextual factors. Second, they help to explain why what are actually sharp boundaries seem fuzzy or blurry: boundaries are in a constant state of flux.
Nonetheless, Epistemicism is difficult for many to swallow. It is hard to see how our reference to things and classes of things could be so much more precise than our ability to recognize their boundaries.
One way of motivating Anti-Epistemicism is by noting that it seems plausible that all vagueness is linguistic or conceptual in character. Some philosophers have asserted that vagueness exists only in language or in the mind, not “in the world”. For example, David Lewis says,
The only intelligible account of vagueness locates it in our thought and language. The reason why it's vague where The Outback begins is not that there's this thing, The Outback, with imprecise boundaries; rather there are many things, with different borders, and nobody has been fool enough to try to enforce a choice of one of them as the official referent of the word ‘outback’. Vagueness is semantic indecision. (Lewis 1986a: 212)
Taken literally, this is nonsense, since language and the mind are just as much “in the world” as anything else. It is part of the very nature of a proper name (like ‘The Outback’) that it have a single referent, just as it is part of the very nature of a concept to have a single set as its extension. Lewis's talk of “semantic indecision” obscures the fact that he's positing real, worldly vagueness in the nature of the name ‘The Outback’ (see Merricks 2001 and Salmon 2010).
The most charitable interpretation of this Linguistic Theory of Vagueness is that vagueness is simply a matter of ambiguity. Vague terms refer ambiguously to a large number of precise entities, and vague predicates ambiguously express a large number of precise properties. We can call this the ‘Multiple Meaning Theory of Vagueness’. If terms and predicates are not ambiguous in this way, and if vagueness is not merely epistemic, then these terms must refer to things that are really vague or indeterminate in their boundaries. Thus, we get two possible versions of Anti-Epistemicism.
12.2A.1T Multiple Meaning Theory of Vagueness. Some boundaries are indeterminate, but all such indeterminacy is merely a matter of ambiguous reference to fully determinate entities.
12.2A.1A Real Ontological Vagueness. Some entities lack determinate boundaries, independently of our knowledge or how we refer to them.
12.2.1 The multiple meaning theory of vagueness.
Multiple Meaning Theorists argue that vague things are really non-empty classes of precise things. Terms that seem to be singular, like the proper name ‘Felix’, really refer to a class of things, a class containing a large number of cat-like or nearly-cattish material bodies in a certain neighborhood. One way to work this out is to assume that vague terms are ambiguous, in something like the way the word ‘bank’ has at least two different meanings (the side of a river or a lending institution). Multiple Meaning Theorists should suppose that vague names like ‘Felix’ are massively ambiguous, with billions or trillions of distinct meanings and referents. (see Unger 1980 and Lewis 1993). (32) is indeterminate because molecule M belongs to some of these referents and fails to belong to others. The indeterminacy of (32) would be similar to the indeterminacy of (36):
(36) John and Mary went to a bank.
(36) could come out as true on one interpretation of ‘bank’ and false on the other, if John and Mary went to a side of a river, but not to a lending institution.
Such an ambiguity theory can provide an explanation for the truth of instances of the theorems of classical logic, including (33) through (35). The explanation takes the form of the device of supervaluations (van Fraassen 1969, Fine 1975, Kamp 1975). An ambiguous statement counts as super-true if it comes out true under every permissible interpretation or precisification of its vague/ambiguous terms. Sentences (33) through (35) are super-true, since no matter what precise referent we assign to the name ‘Felix’ (regardless of whether the referent contains or doesn't contain M), the statements come out true. However, although supervaluational semantics agrees with all of the theorems of classical logic, it does not endorse as valid all of the methods of classical logic, such as reductio ad absurdum (indirect proof) or conditional proof (Williamson 1994: 151–153).
Mu
ltiple Meaning Theory faces several serious problems. First, there is a legitimate question as to whether Multiple Meaning Theorists have pushed the concepts of meaning and ambiguity beyond their breaking points. We understand what it is for a word like ‘bank’ to be ambiguous: there are two semantical rules associated with the phonetical and grammatical symbol. However, in the case of a name like ‘Felix’, we have to imagine billions or trillions of distinct semantic interpretations, all of which are somehow encompassed as permissible by our linguistic practices.
In response, Multiple Meaning Theorists could point out that on Lewis's account there is a difference between lexical ambiguity and vagueness. Vagueness is the result of semantic incompleteness or “indecision”, as Lewis puts it. The incomplete meaning is compatible with multiple, equally good completions. Lexical ambiguity, on the other hand, is the result of our decisions, rather than indecision. The community of English-speakers, for example, have chosen to use ‘bank’ to pick out two very different types of thing.
Second, there is the problem of higher-order vagueness. Just as it is can be unclear whether molecule M is or is not part of Felix, if ‘Felix’ is ambiguous and molecule M lies in the borderline region contained by some but not all the referents of ‘Felix’, so can it be unclear whether or not some molecule M is a borderline case. The Multiple Meaning Theorists seem to be committed to saying that there is a precise answer as to which molecules belong to some referent of ‘Felix’ and which belong to none. But this seems just as indeterminate as the first-order question of which molecules belong to Felix.
Finally, Multiple Meaning Theory, when combined with the view that human persons are vague material objects, destroys the unity of each person. RCK is surely aware of his own unity as a thinking being, and yet Multiple Meaning Theory would entail that ‘I’ in RCK's mind or mouth in fact refers to a vast multitude of overlapping material thinkers. This vast multiplication of human persons would be deeply inconsistent with many of our most central ethical and political beliefs and practices. If Smith is guilty of a crime or responsible for some act of heroism, which of the trillions of Smiths should be punished or rewarded, and how do we ensure that we interact with the right ones?
12.2.2 Real ontological vagueness
Let's turn finally to Real Ontological Vagueness, the view that there is real vagueness or indeterminacy in the world. If vague objects are non-fundamental or grounded, their grounding in precisely-bounded objects is a case of what we called extra-conceptual grounding (in Section 3.4). Such grounding does not enable us to deny that vague objects really exist.
How can we make sense of this view? One way would be to give up Bivalence. Suppose that Fred is borderline bald. We might say that is neither true nor false (speaking precisely) that Fred is bald. Or, suppose molecule M is a borderline case of a proper part of Felix. We could say that it is neither true nor false that Felix includes M. This is the three-valued proposal for ontological vagueness.
There are three major drawbacks to the three-valued proposal. First, giving up Bivalence gives us good reason to give up classical logic more generally, including the Law of Excluded Middle:
Law of Excluded Middle. For every proposition p, the proposition of the form ‘p or not-p’ is a logical truth.
If some propositions are neither true nor false, then it seems that there should be cases in which neither p nor not-p are true. Given our standard understanding of thetruth-conditions for ‘or’, these cases should give rise to exceptions to Excluded Middle, as well as many other principles of classical logic.
This is a heavy price to pay for a dubious metaphysical gain. First, we should surely want to say things like ‘If Fred is bald, then Fred is bald’, or ‘If Fred is bald, then someone just like him with even fewer hairs is bald’ to be logically or definitionally true, but this will be hard to do without the resources of classical logic.
Second, consider the principle known as ‘Tarski's Schema’ (after Alfred Tarski):
PTruth 2 Tarski's Schema. For any sentence s, if ‘S’ is a name for s, then we should affirm the sentence of the form: S is true if and only if s.
The final clause, ‘S is true if and only if s’, is sometimes called a ‘Tarski biconditional’. On the left side of the biconditional (‘if and only if’) the name S is named or mentioned, and on the right side it is used. A classic example of a Tarski biconditional is (37):
(37) ‘Snow is white’ is true if and only if snow is white.
Not only do Tarski biconditionals seem to be logically or definitionally true, but they seem to have a further interesting characteristic: the left-hand and right-hand sides seem to be saying exactly equivalent things. These two sides should have the same semantic value. However, the three-valued proposal violates this rule. If Fred is a borderline case of baldness, then ‘Fred is bald’ is neither true nor false. If so, then (38) will be false:
(38) ‘Fred is bald’ is true.
(38) is false, since ‘Fred is bald’ is neither true nor false. Consider the relevant instance of Tarski's biconditional:
(39) ‘Fred is bald’ is true if and only if Fred is bald.
Since (38) is simply false, the left-hand of (39) is false and the right-hand side is neither true nor false, violating the semantic equivalence between the two.
Third, the three-valued proposal does not give us a workable account of higher-order vagueness. Suppose that Fred is neither definitely vaguely bald nor definitely definitely bald. That is, suppose he is a borderline case of a borderline case of baldness. What should we say about the sentence ‘Fred is bald’ in that case? Is it true, false or neither? If we say that it is true or that it is false, then we must conclude that Fred is definitely not a case of vagueness. If we say that it is neither true nor false, then we must conclude that Fred is definitely a case of vagueness. How can we get that he is a vague case of vagueness? Do we have to keep increasing the stock of non-standard truth-values, like vaguely true or vaguely false? Where, if ever, does this process end? And how do the various truth-values relate to one another? The three-valued proposal seems to be leading into a morass.
There is an alternative approach that we might consider, the Many Actual Worlds approach (see Elizabeth Barnes 2010). On this view, there might be more than one world that is actual. If Fred is a case of baldness, then both a world in which he is bald and a world in which he is not bald might be actual.
We will discuss the idea of possible worlds in more detail in Chapters 14–16, but our discussion here will require a bit more detail that we've needed to this point. For now, think of a possible world as a comprehensive state of affairs, a comprehensive way for things to be. Some states of affairs, like the Patriots' winning Super Bowl XLIX, are relatively non-comprehensive, leaving many facts undetermined. A possible world is a maximal state of affairs, one that settles all of the matters of fact in a possible scenario. More precisely, a state of affairs A is maximal if and only if there is no state of affairs A′ such that it's possible both that A obtains A′ obtains and that A obtains and A′ doesn't obtain. Alternatively, we could think of worlds as maximal compossible sets of states of affairs. A set of states of affairs is compossible if it is possible that all the members of the set obtain together. But what makes such a set maximal? Just that no further state of affairs can be added to the set without producing a non-compossible set. (These definitions are from Plantinga 1974.)
However, if two or more mutually inconsistent worlds can all be actual together, as on the Many Actual Worlds approach, then none of them will qualify as maximal. We'll have to define ‘possible world’ in a different way: a possible world is a minimal set of states of affairs that could include everything that is actual. Each possible world is comprehensive enough that it could comprise the whole of reality, but no bigger.
If more than one world can be actual, how do we define ‘truth in a world’? We can't define it in Plantinga's way (1974): namely, p is true in w if and only if p would be true if w were actual. A wo
rld w in which p is false might be actual, along with another world in which p is true. World w's being actual isn't enough to ensure that p would be false, even if p is false ‘in’ or ‘according to’ w. We will have to appeal instead to what would be the case if w were the only actual world:
p is true in w if and only if p would be true if w were the only actual world.
We can now say that a proposition p is super-true if and only if p is true in every actual world. Now there will be propositions that are neither super-true nor super-false, but all logical truths and necessary truths will be super-true, including Excluded Middle. We get to use all of classical logic, by van Fraassen's (1969) method of supervaluations, now interpreted ontologically instead of semantically.
However, there is a fly in the ointment: it will turn out to be super-true that there is only one actual world! Given our definition of truth in a world, the proposition that w is the only actual world will always be true in w, and so it will always be super-true, no matter how many worlds are actual. Our account of truth undermines the very theory we started with.
So, proponents of Many Actual Worlds need a more complicated theory of truth. The initial definition will work for all atomic propositions other than those that predicate actuality of individual worlds. Let's say that a set of worlds is a cluster of worlds if it is possible for all of them to be actual together. We can now define what it is for a proposition to be true relative to a pair