Download the PDF version here. Originally published in Prometheus on 12/31/2009.
In this paper, I examine David Lewisʼ treatment of vagueness as a problem of “semantic indecision” and conclude that this position on vagueness is inconsistent with the metaphysics of his theory of modal realism, with specific regard for counterparthood and the counterpart relation. To reach this conclusion, I employ a thought experiment in which an exact counterpart of Lewis is subjected to a series of possible worlds treatments designed to satisfy Lewisʼ criteria for counterparthood, as well as to test the limits of semantic treatments of higher-order vagueness. In doing this, I find that Lewisʼ suggestions for dealing with vagueness fails to pick out counterparts at several points in this series, even when given a satisfactorily precisified set of criteria for the qua relation. Rather than a clear candidate for counterparthood, one encounters instead a problem of infinite regression that could destabilize the whole project of Lewisian de re modal realism. I conclude by noting that accepting metaphysical vagueness into the Lewisian theory of modal realism changes nothing in the overall theory, and may in fact be the only way to save the theory from its problem of infinite regression.
The sentence “All bachelors are unmarried.” and the sentence “The author of this paper is typing on a computer.” are both true. However, these two sentences are not true in the same way. To understand how they are different, let us examine how each sentence can be considered true. In the first example, truth appears to be a function of the meaning of the word “bachelor” in relation to the rest of the sentence. This is because the first example states a specific and important property of the concept of being a bachelor. It is almost a kind of definition, and its truth is derived from the relationship between its form (what kind of sentence it is) and its semantic content (what the sentence means). The first example sentence demonstrates that the relationship between the form and the content of a sentence is one way to measure the truth of a sentence. The second example presents an exception to this rule. The form of the second example sentence does not follow the pattern of the first, yet (much to the authorʼs chagrin as I watch the clock) it is still true.
So what does it mean to say that the two sentences are true in different ways? In this case, it means that they arrive at being true by taking two different routes. The first example must always be true; the word “bachelor” carries somewhere in its meaning the idea of being unmarried, and the property of being unmarried is a definitive property of the word “bachelor”. The second example need not be, but by all respects still is, true. Truths that must be true, like the first example, are called necessary truths. This means that there is some quality of the words of the sentence or the things being discussed by the words in that sentence which requires the sentence to be true. The second example illustrates what is called a contingent truth. Contingent truths are not true in virtue of structure or meaning per se, but rather are true with regard to a given situation. Another way to put this is to say that there is no quality about me that would require, as a matter of necessity, that I be sitting at my desk writing this paper. I could just as easily be sleeping, or going for a walk, or touring the country with my world famous band instead. ! Hypothetical statements like the one above, which gives some examples of ways in which my life might have been different, are called “counterfactuals”. Counterfactuals are just what they sound like: statements that consider cases in which the facts are contrary to the way things actually are. Often stated in the form of conditionals (“If X then Y”), counterfactuals are aimed at examining the ways in which the world could (and could not) be different by positing alternate situations for conceptual analyses. Take the statement “If I werenʼt writing this paper, I would be asleep right now” as an example. The purpose is to assert that under specific circumstances that are contrary to the actual ones (“If I werenʼt writing this paper,”), a different set of statements about the world would turn out to be true, and I would be in bed.
What counterfactuals illustrate is called “modality”, or the measure of necessary or contingency. It would seem that in order to be able to evaluate modality correctly, and in doing so evaluate the truth-value of the sentence in question, there must be some properties, or kinds of properties, of the thing being discussed that remain true about that thing across all counterfactual statements. Here, I must bring up another distinction, one concerning interpretations of things about which the modalilty is in question. With regard to modality, there are two ways to gloss a given sentence. One such way is called a de re reading (from the Latin for ʻof the thingʼ). The other reading is a de dicto reading (meaning ʻof the wordʼ). De re readings of sentences are concerned with the modality of the actual physical thing(s) referred to by the terms of a sentence. De dicto readings are concerned with the modality of the words of the examined sentence itself. This differentiation is relevant to the ways by which the truth-value of a given sentence can be evaluated as necessary or contingent. To illustrate the difference between a de re and a de dicto reading, examine the sentence “The President of the United States could be a woman.” Under a de dicto reading, this sentence means that it is possible for a woman to become the President of the United States. Under a de re reading, this sentence means that the legal status of Barack Obamaʼs marriage is in serious jeopardy under current law.
In contemporary analytic philosophy, the truth-values regarding the modal properties of a sentence are often evaluated in terms of “possible worlds”. Though there is significant disagreement about the nature of these worlds, such as when it is appropriate to assign de re or de dicto readings to statements made about them, it must be the case that possible worlds are at minimum conceptual spaces in which one can run thought experiments to determine the necessity of a truth-value of a sentence. The classic examples used to illustrate this are “Aristotle is Aristotle” and “Aristotle is the teacher of Alexander”. Under a de re reading, it is simple to see that the first statement must always be true, because if there is an “Aristotle” about whom this statement can be made, then that Aristotle must be self-identical. However, there is no quality about such an Aristotle that necessitates that he be “the teacher of Alexander”. Rather, this is something that happened to be the case only as a matter of course and not as a matter of the things involved those circumstances.
Note the inherent import here of a robust idea of identity. When employing possible worlds as a measure of modal properties, especially under a de re reading, identity is assumed as a most basic property. Generally, identity can be understood with the following two premises: (1) All things are self-identical and (2) no two things are identical to each other in all ways. Identity is a philosophical issue unto itself, however due to the limited scope of this paper it must suffice to say that identity is the metaphysical property of self-sameness. As such, an issue like metaphysical vagueness would appear to be an important area to explore when discussing de re modality, if only because in any dialogue that takes as its subject the modal properties of things must start first with an understanding of the thing about which modal properties are to be discussed. If one is to understand how a thing could have been different, as well as how it could not have been different, then one must first encounter that thingʼs identity. In this paper, I will contrast David Lewisʼ treatment of de re modality (Lewisian modal realism) against that of another philosopher, Saul Kripke. I argue that modal realism necessarily admits of metaphysical vagueness, and until that theory is modified to accept this fact, modal realism is not a feasible theory of modality.
II. De Re and Vagueness
Saul Kripke explores de re interpretations in his work Naming and Necessity. In this lecture series, Kripke abandons the skepticism of his predecessors such as Quine about de re readings. He argues that rigid designators, which are terms that pick out the same thing across all possible worlds (where that thing exists), are the appropriate means of evaluating claims de re. Examples of rigid designators are names, numbers, and natural kind terms such as “water” or “gold”. Under this treatment, possible worlds need be nothing more than hypothetical scenarios run in oneʼs mind. For Kripke, to exist is to be the extension of a term.
The Kripkean treatment of de re modality is what is called “ersatz” modality (ersatz meaning substitution) and is semantic in nature. This is to say that Kripkean possible worlds are constructs of meaning located in the mind, and are intended to exemplify counterfactual possibilities about actually extant things. Kripke does not assert that using possible worlds to evaluate modal claims has any ontological implications. As such, a Kripkean treatment de re allows for the existence of “transworld identity” which means that the singular identity of a thing being discussed can be distributed over all possible worlds. Another way to put this is to say that for Kripke, the “Aristotle” in any of the worlds in which Aristotle was not the teacher of Alexander is self- identical to the Aristotle in this world.
David Lewis accepted the premise of possible worlds re-introduced to analytic philosophy by Kripke. The function of possible worlds for Lewis is almost exactly the same as for Kripke; they measure the necessity of a claim. However, according to Lewis, possible worlds are real physical places—as real as our own world—in which real things exist. This is Lewisian modal realism. As Lewis discusses at great length in On The Plurality of Worlds, there are innumerable such real worlds, and the inhabitants of these worlds are the subjects of the counterfactual conditionals that possible world scenarios evaluate. Among other things, Lewis asserts that counterfactuals are evaluated by means of a relation of counterparts in possible worlds to things in the actual world. This means that when we say something could have been different in any way from the way it actually is (for example, to say that I might have been a concert violinist instead of a philosopher) is to say that there is some possible world in which the counterpart of that thing actually is that way (my counterpart is a concert violinist in some possible world).
In §4 of his book, Lewis argues that possible worlds are spatiotemporally isolated from each other. He also argues that spatiotemporal location is a necessary property of identity, and that the criteria of identity can only be sufficiently met by sharing exactly all the same qualities, including spatiotemporal location. From these two premises arise the need for the counterpart relation to take the place of transworld identity, because identity can only be granted to objects that are at least spatiotemporally identical, and also because in Lewisʼ theory individuals in different possible worlds are separated and thus anything in one world is spatiotemporally isolated from anything in another. What will eventually run Lewis aground here remains unproblematic for Kripke because Kripkeʼs specific theory of possible worlds has no ontological implications and all he needs to grant identity is that a termʼs extension be the same across all possible worlds. However, for Lewis it is clear that it is not possible to grant identities across possible worlds because their referents differ in spatiotemporal location, so he must create some new theory to take its place.
The counterpart relation is the theory that Lewis proposes for this purpose. It is the basis for being able to assess counterfactuals in Lewisʼ account. The counterpart relation is one of “relevant similarity”, or comparative similarity of desired properties between token-specific candidates for counterparts across possible worlds. That which makes an individual in another world a counterpart of an individual in the actual world is an overall comparison of similarity among all possible candidates in any given possible world. Counterparts are only counterparts to each other in virtue of a given “qua relation.” For example, if a counterfactual involving having a certain number of hairs on oneʼs head is being discussed, the proper way to assign counterparthood would be to say that person X in World 1 (W1) is a counterpart of person Y in W2 qua Xʼs and Yʼs number of head-hairs. Thus, one person can be a viable candidate for the counterpart of another person if and only if those two people, as Lewis states, “closely resemble [each other] in important aspects.”
There is no room in Kripkeʼs ersatz treatment of possible worlds for vagueness to arise as an issue. In virtue of the fact that there are no ontological assertions made by ersatz possible worlds, there can be no ontological vagueness. Furthermore, because rigid designators grant identity across possible worlds and the fact that to have an extension, as far as Kripke is concerned, is to exist, there can only be semantic vagueness in the evaluations of modal claims.
This is not the case for Lewisʼ theory of modal realism, though. Modal realism is a large and complex theory that leaves much room in which the problems of vagueness might take hold. Notably, however, Lewis famously decries the whole project of metaphysical vagueness:
"The only intelligible account of vagueness locates it in our thought and language. The reason it’s vague where the outback begins is not that there’s this thing, the outback, with imprecise borders; 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 1986, 212).
Many such quotes fill the pages of On The Plurality of Worlds. The idea is always the same: the world is not vague, but rather it is our representations of the world, our words, that are. That David Lewis, the famous metaphysician, is so vehemently opposed to metaphysical vagueness is a bit shocking at first. Why should someone so deeply involved in metaphysics, especially someone whose project is specifically a refutation of a larger semantic theory, seek the solution to the problem of vagueness in a semantic theory? If nothing else, this is strikingly counterintuitive.
Much like others who found vagueness to be a problem of language and not of the world, Lewis seeks to eliminate vagueness by precisifying problematic language:
"If a sentence is true over an entire range, true no matter how we draw the line, surely we are entitled to treat it as simply true. But also we treat a sentence more or less as simply true, if it is true over a large enough part of the range of delineations of its vagueness. (For short: if it is true enough)" (Lewis 1983, 244)
Here Lewisʼ position on how to deal with problems of vagueness is clear. Because vagueness is “semantic indecision,” the proper manner by which it should be dealt with is to look at the way sentences admit of vagueness and re-evaluate how they are interpreted. Vague sentences are “true enough” to be considered true when they are true over some sufficient range (the “large enough part of the range”) of precisifications. Lewis avoids dealing with the question of what counts as “true enough” by calling this itself a vague matter. This is for Lewis, however, not an important enough issue to pursue, as is evidenced in the next paragraph of the text:
When is a sentence “true enough”? Which are the large parts of the delineations of its vagueness? This itself is a vague matter. More important for our purposes, it is something that depends on context. What is true enough on one occasion is not true enough on another. The standards of precisions in force are different from one conversation to another (Lewis 1983, 244-245).
One should assume, at risk of otherwise creating a straw man argument, that Lewis expects his own semantic prescription for handling vagueness should be sufficient to explain away the vagueness he himself admits is inherent in how he tells us to handle such problems. However, in at least one case, Lewisʼ account is not a sufficient method of explaining away vagueness as semantic indecision.
Consider the following scenario: there is a series of possible worlds in which at one end there is one and only one possible counterpart of Lewis, and he is a spitting image of Lewis in every single possible relevant manner. At the other end of the series, there is a world populated entirely by just one single rooster. This set-up is consistent with how Lewis assumes possible worlds work. Now, assign to this series, in the search for counterparts, the highly precisified set of criteria for counterparthood that is exactly and only the breadth of wingspan and the volume of caw. Lewis would acknowledge that his spitting image meets all the relevant criteria to be his counterpart. He would agree as well that he is not a counterpart of the rooster at the far end of this sequence, being that he meets no relevant criteria for being its counterpart and thus is not, in any way, a viable candidate of counterparthood.
If one observes the series of worlds that starts in the world of the lonely rooster and ends in the world in which there is an exact duplicate of Lewis illustrates a possible worlds sequence of a Lewis/rooster chimera that runs in reverse. Actual-world Lewis would admit that there is one world in this series in which the counterpart relation ceases to be sufficient enough under the relevant criteria for the thing in that world to count as a counterpart of the rooster. By extension, in virtue of the nature of the relevant criteria of this sequence, it is logically true (assuming that counterparthood is cardinal) that there is also some point in this sequence at which the Lewis/rooster chimera ceases to meet any relevant criteria for Lewis-counterparthood.
The problem that Lewis is forced to acknowledge, by his own justification of the counterpart relation, is that there remain penumbral cases of counterparthood in this series. Specifically, there are at least some worlds in which it is indeterminate whether or not the most Lewis-like thing in that world can be rightly called a counterpart of Lewis qua the relevant criteria, but are still, definitely, the counterparts of other non-Lewis counterparts in the series qua those same criteria. Every case in this series in which the chimera is more like a non- counterpart of Lewis, but is still a Lewis/rooster chimera, yields an indeterminate counterpart relation for Lewis. Each further attempt to create more precise semantic boundaries for vague predicates such as “is a counterpart of Lewis” serves only to shift the problem up by one degree of order; there is no way in Lewisʼ treatment of vagueness for there to be enough precisification to eliminate vagueness as a problem of semantics alone.
III. Saving Modal Realism
The basic premises Lewis works from are as follows*:
- The world is made of material things,
- Modal realism is true,
- All possible worlds are spatiotemporally and causally isolated,
- Spatiotemporal location is a necessary quality of identity,
- Counterpart theory is the means of measuring counterfactuals across spatiotemporally isolated worlds,
- Vagueness is semantic in nature.
My thought-experiment shows that modal realism is philosophically unsound if Lewis accepts a theory that uses counterpart theory and eschews metaphysical vagueness, because his prescribed semantic treatment of vagueness results in an infinite regression. However, the counterpart relation is essential to Lewisʼ modal realism, and thus cannot be given up without losing the larger theory. It is the only way he can account for being able to assess counterfactuals. In the above example, I have shown that even under Lewisʼ strict semantic treatment of vagueness, there no precise cut off point at which a thing in the example series is still be enough like Lewis that, were there no other Lewis-like things in its world, it could be determined whether or not that thing would still be Lewisʼ counterpart.
Meeting all of Lewisʼ criteria for hyper-precisification of the relevant criteria fails to eliminate vagueness of the counterpart relation in the above example. I submit that this is the case because the notion of counterparthood and the counterpart relation are based in comparative overall similarity. The problem with such evaluations is that the terms of these evaluations are themselves vague predicates. Thus, we enter an infinitely reiterating argument that fails to address the issue at hand (that being the location of vagueness as being in language or the world). It is a necessary consequence of precisely this is that Lewisʼ system does not function to evaluate modal properties with any hope of being other than accidentally correct. As such, Lewisʼ theory of modal realism with vagueness as a semantic problem is less than a sufficient account of de re modality.
However, I propose that if Lewis were to add to Premise 6 that vagueness can be semantic and metaphysical, he would no longer face an issue of infinite regression. Lewis argues against metaphysical vagueness because he conceives of identity as a most basic property, which he effectively argues to be unique self-sameness, and his argument against vague objects is based on the fact that he assumes there is a finite answer to the question “how many objects are in the world?” Lewis is correct to state that if vagueness is metaphysical, then it would be a fact of the world that there is no finite number of objects in the world. However, why should it be assumed that there is a finite number of objects in the world? I can think of no compelling reason to assume this to be the case, and Lewis certainly fails to give a sufficient account supporting his personal predilection for there being a finite number of things in the world. At every opportunity, he fails to address the issue as he turns questions of metaphysical vagueness into questions of semantic vagueness.
For Lewis to accept vague objects, he is not required to give up any other part of modal realism. Revising Premise 6 as I have prescribed has no impact on the previous five premises. However, the real issue at hand here is that extending Premise 6 to acknowledge the existence of vague objects might be the only way for Lewisʼ modal realism to survive. Rather than supporting modal realism, his position on vagueness as a problem only of semantics actually undermines his project of modal realism. There is no sufficient argument provided as of yet that can semantically explain away problems of vagueness. The problem of infinite regression that Lewisʼ solution entails makes it unjustified and therefore philosophically unsatisfactory. The existence of vague objects is not only simply the more logical solution, but may possibly be the only way he can salvage modal realism from its problem of infinite regression.
*These, I mean to say, are implied by Lewisʼ arguments for and concerning modal realism and the counterpart relation. Nowhere, to my knowledge, does Lewis state these explicitly.
1. Kripke, Saul. Naming and Necessity. Cambridge: Harvard University Press. 1972, ! 1980
2. Lewis, David. On The Plurality of Worlds. Malden MA: Blackwell Publishing. 1986.
3. Lewis, David. “Counterparts of Persons and Their Bodies.” The Journal of Philosophy ! Vol. 68 (1971). 203-221.