Lewis and Accidental Intrinsics Sara Smollett
April 12, 2002

The Problem of Accidental Intrinsics

In this paper I will discuss the so-called problem of accidental intrinsics. I intend to show that there is no problem with accidental intrinsics. I will begin by briefly introducing modal logic and possible world semantics. I will then state the problem of accidental intrinsics as described by David Lewis. Finally, I will consider several reasons why the problem of accidental intrinsics is not, in fact, a problem.

Part 1

Possible worlds are useful for illustrating possibilities, and hence for illustrating modal propositions - propositions concerning possibility and necessity. A possible world is not a physical, material world, but a set of propositions about objects in the domain of the world and properties that the objects exemplify. There is no question, then, about the existence of possible worlds. Nor should there be any question about the existence of objects in the domain of a world. An object is in the domain of a world just in case that object could possibly exist.

We are now ready to begin translating modal propositions into the semantics of possible worlds. The proposition ``It is possible that it will rain tomorrow'' means ``there exists a possibility of rain'' or ``there exists a possible world, call it w1 which contains the proposition `it will rain tomorrow'''. This translation reduces a modal proposition to a non-modal proposition. Similarly, the claim ``Hubert Humphrey could have won the presidential election'' is true if and only if there is a world which contains the proposition ``Humphrey wins the election''. The clause ``there is a world which contains the proposition F'' is often abbreviated as ``there is a world according to which (or at or in which) F obtains truth (or is true). Thus the claim ``Humphrey could have won the election'' can be expressed by ``there is a world in which Humphrey won the election'' (Loux, 183).

Now that we have a way to represent the modal operator for possibility, it is easy to represent necessity. To say that a proposition is possibly true is equivalent to saying that that it is not necessarily false. It follows that a proposition about an object is necessary if it is true in all possible worlds containing that object. The proposition ``Necessarily 2+2 = 4'' is true if and only if the proposition is a member of, or is true in, all possible worlds.

In addition to necessity and possibility, it is useful to represent actuality and contingency. Of all the possible worlds, just one represents the actual state of affairs relative to us. This world is known as the actual world. A proposition is actually true, or simply true, if and only if it is true in the actual world. If a proposition is true in all possible worlds, it is necessarily true. If a proposition is actually true, but could have been false, it is said to be contingently true. For a proposition to be contingently true, it must be true in the actual world but false in at least one possible world (Loux, 185).

Part 2

Propositions can contain several different kinds of properties. One common way to sort properties is to divide them into essential and accidental properties. Essential properties are those properties which an object has necessarily. In contrast, those properties which an object has contingently are called accidental properties (Companion, 137). Properties are also frequently divided into extrinsic and intrinsic properties. Roughly, extrinsic properties of an object are properties that are expressed in terms of both the object and something external to the object. Extrinsic properties express relations between the object and its surroundings. Intrinsic properties, on the other hand, are those which are exemplified by the object alone (Companion, 162).

It is the latter kinds of both categories of properties, that is, accidental intrinsic properties, on which this paper will focus. According to Lewis, shape, size, and composition are examples of intrinsic properties. They are also, he says, accidental properties. They are properties that an object has in and of itself that can be expressed without use of external relations. They are also non-essential properties or properties that could have been otherwise (Lewis, 158).

Having now defined and given some examples of accidental intrinsics, we are ready to consider the so-called problem of accidental intrinsics described by Lewis. Lewis begins with the following scenario: Hubert Humphrey has (in the actual world) five fingers on his left hand. But it is not a necessary fact that Hubert Humphrey has five fingers on his left hand. That is, Humphrey could have had six fingers on his left hand. The property ``having exactly n fingers on one's left hand'' is an accidental intrinsic property. In particular, the property ``having five fingers on one's left hand'' (which I will later refer to as the property F) and the property ``having six fingers on one's left hand'' are both accidental intrinsic properties (Lewis, 158). It follows from the straightforward application of possible world semantics that Humphrey, H, has five fingers on his left hand (exemplifies the property F) in this world. It also follows that Humphrey has six fingers on his left hand (H fails to exemplify F in another possible world).

But this conjunction seems to be contradictory. How can it be that Humphrey has both five and six fingers on his left hand? The claim that Humphrey has both five and six fingers seems absurd. It seems just as absurd to suppose that the Humphrey in this world (or rather the object in this world picked out by the name ``Humphrey''), who is identical with the Humphrey in another world (again, the object named ``Humphrey''), has a different number of fingers than the other-worldly ``Humphrey''. The reason this seems absurd is because it contradicts the principle of indiscernibility of identicals, which states: Necessarily, for any objects, a and b, if a is identical with b, then for any property, f, a exemplifies f if and only if b exemplifies f. That is, identicals are indiscernible; alternatively, discernible objects cannot be identical (Loux, 194).

It follows, then, that if the object referred to by ``Humphrey'' in this world and the object referred to by ``Humphrey'' in that world are identical, they cannot differ in any properties. If one Humphrey exemplifies the property F of having five fingers while the other Humphrey does not, there is no single individual Humphrey, but rather two non-identical objects called ``Humphrey''. (Note: I will drop the quotation marks, as I believe the distinction between use and mention can be inferred from context).

The problem of accidental intrinsics, or rather the problem raised by accidental intrinsics, is the claim that the existence of accidental intrinsics and the principle of indiscernibility of identicals are incompatible (Loux, 195). Because the principle of indiscernibility is true, some philosophers argue, we must abandon the idea that accidental intrinsics exist. However, I am not convinced that there is an irreconcilable conflict between accidental intrinsics and the principle of indiscernibility, nor am I convinced that we should be worried if there is.

Does Lewis' example illustrate a conflict between accidental intrinsics and the principle of indiscernibility? If so, can this conflict be resolved? What is the danger of committing ourselves to an ontology devoid of accidental intrinsics? Or, what do we lose by conceding the principle of indiscernibility of identicals? Do we need individuals across possible worlds? These questions raise a number of issues which I will address in the remainder of this paper.

Part 3

I will now consider whether there is a conflict between accidental intrinsics and the principle of indiscernibility. If the existence of accidental intrinsics does not preclude the principle of indiscernibility of identity, then there is no problem of accidental intrinsics. (Or if there is a problem, it is a different problem from the one being considered here.)

For now I will assume that there is one object referred to by ``Humphrey'' in multiple worlds. Surely the claim ``Humphrey has five fingers in this world and six fingers in another world'' is not to be understood literally as Humphrey having the intrinsic properties of having five fingers and six fingers. Instead, the claim being made is really that Humphrey, Humphrey himself or Humphrey in every world, exemplifies the property ``having five fingers in this world'' and that he exemplifies the property ``having six fingers in another world''.

I will now represent this re-interpretation of ``Humphrey has five fingers but could have had six'' in symbolic notation. Let w0 and w1 be the actual world and another world, respectively. Let Fw0 be the property ``having five fingers in w0'' and Fw1 the property ``having five fingers in another world, w1''. Define V(p), where p is a proposition, to be 1 if p is true and 0 if p is false. We can now write: V(Fw0(H)) = 1 \land V(Fw1(H)) = 0 to mean that Humphrey has the property Fw0(H) but not the property Fw1(H). The conflict in question seems to disappear. There is no longer a claim that F(H) \land ~ F(H). There is no tension between these two new predicates being true of the same object because they are not contradictory; no appeal to the principle of Indiscernibility of Identicals will yield undesired results. The problem that Lewis' example seemed to illustrate has been dissolved.

Notice what has been done here, though. The proposition stating that Humphrey has five fingers or six fingers has been rephrased. The new proposition is not a proposition about the property F of having five fingers. Instead it is a proposition about two world-relative properties, Fw0 and Fw1. Or, to put it another way, it is a proposition about two relations, the relation Humphrey bears to this world and the relation Humphrey bears to another world. Recall that properties expressing relations between an object and something external are extrinsic, not intrinsic, properties (Lewis, 159).

It seems reasonable to postulate that all ``intrinsic'' properties in modal contexts are world-relative; they are really relations or extrinsic properties in disguise. Our attempt to preserve the principle of indiscernibility of identicals, at least in this example, has been at the cost of certain intrinsics. Are there any intrinsics? The answer, I believe, is that if there are intrinsics, there are few intrinsics. If an object exemplifies essential properties, then it can be said to exemplify essential intrinsic properties.

If, for example, Mozart has five fingers necessarily, then Mozart has the extrinsic property of having five fingers in this world, the extrinsic property of having five fingers in another world, the extrinsic property of having five fingers in still another world.... That is, "nV(Fwn(H)) = 1. It follows that Mozart has the property of bearing the five-finger relation to every world. Since the proposition ``Mozart has five fingers'' is true in every world, we can drop the relation to worlds without losing and meaning and say simply that Mozart has the intrinsic property of having five fingers. No obvious difficulty with the principle of indiscernibility arises from claims of essential intrinsic properties (Lewis, 161).

We turn back to accidental intrinsics. The above reasoning cannot be applied to them. Because accidental intrinsic properties are not true at every world, their non-modal parts must be understood as relative to particular worlds. It seems that one object cannot be both intrinsically five-fingered and intrinsically six-fingered, or intrinsically green and not green, or intrinsically triangular and non-triangular. That is, it appears that all intrinsic properties are essential properties; there are no accidental intrinsics. But so what? Why should we want accidental intrinsics?

One explanation is as follows: Suppose there are no accidental intrinsics, that all intrinsics are essential properties. Then every property which an object has itself, intrinsically, must hold in every possible world. But if properties are fixed across possible worlds, the argument goes, the worlds do not represent possibility; instead, they represent inevitability (Loux, 196).

I do not see this worry about inevitability as a genuine problem. If some properties are necessary, then of course they will seem unavoidable (inevitable) from an abstract, omniscient, metaphysical perspective. But for the rest of us, failing to be omniscient, what is necessary can only be decided in retrospect. The past is determined, unchangeable. There is no immediate danger that our lives will seem limited in any way.

Putting arguments about inevitability and determinism aside, we have already seen that it is only when considering intrinsic properties, not extrinsic properties, that this problem is encountered. But why should we care about intrinsic properties at all? Or rather, what accidental intrinsics are there to care about? The ``intrinsic'' property that motivated this discussion, ``having five fingers'', turned out to be an extrinsic property. Perhaps all such accidental ``intrinsic'' properties can be similarly rephrased as extrinsic properties. If this is the case, as it seems likely to be, why does it matter that there are no accidental intrinsic properties? We are not removing accidental intrinsic properties from our ontology; it is simply that there are no accidental intrinsic properties to remove. No properties have been lost, only the label ``accidental intrinsics''. Surely, it seems, we can easily live without this name.

Part 4

But many philosophers are unwilling to give up accidental intrinsics in this way. Must accidental intrinsics be problematic then? Their existence, it seems, is inconsistent with the principle of indiscernibility of identicals as understood thus far. It seems that those who wish to accept the existence of accidental intrinsics cannot hold the principle of indiscernibility.

To this as well I say perhaps this is no great loss. What is the cost associated with dropping the principle of indiscernibility? If we really want to say that the Humphrey-named object with five fingers in one world is identical with the Humphrey-named object with six fingers in another world, then the principle of indiscernibility of identicals does not accurately represent what we mean by identity.

If we are to have choices in our everyday life, as I believe we do, we stand at every point in time at a metaphorical branching of paths. One path, the chosen path, represents the actual world. The many other paths not taken represent unactualized possibilities. Consider the following scenario: I have ten minutes of unexpected time before a class. I can read a book or strike up a conversation with someone sitting next to me (or do any of a number of other things). That is, there is one possible world in which I read and another possible world in which I converse. The ``reading me'' in one world is discernible from the ``conversing me'' in the other world. But I want to say that both are ``me'' and that whichever I decide to do, I am still me. If, however, reading the book completely changes my life or it is unfathomably contrary to my nature that I would choose to converse, it seems that the ``reading me'' and the ``conversing me'' are not just discernible, but fundamentally or essentially different.

Reasoning thus, it seems to follow that individuals are not necessarily self-identical. Or rather that there is no continuity of self over time. But it is not meaningless to say that I am the same person now as I was ten seconds ago. My thoughts have changed, my chemical makeup and structure has changed, but I do not believe there are not two physically different individuals: the me of ten seconds ago and the present me. It seems that what is sometimes meant by the word ``identity'' in common usage is not strict identity.

Perhaps, then, the principle of indiscernibility of identicals is stronger than it should be for ordinary purposes. I propose the following principle, which may be called the principle of near-indiscernibility of identicals or the indiscernibility of identical essences: For the object named by a to be identical with the object named by b, the two objects, a and b, must exemplify all of the same essential properties. If this new principle is satisfactory in place of the original principle of indiscernibility of identicals, then we can give up the original principle.

Part 5

However, I think few philosophers are prepared to give up or even modify the principle of indiscernibility of identicals. For those who want to hang onto both this principle and the existence of accidental intrinsics, I believe there is still a way to side-step the problem. If someone named Humphrey has an intrinsic property of having five fingers and if someone named Humphrey has an intrinsic property of having six fingers, since Humphrey is discernible from Humphrey, it follows by the principle of indiscernibility that the two beings are not the same Humphrey. There is no conflict at all between accidental intrinsics and the principle of indiscernibility of identicals here. Instead, there is a conflict between identity and transworld individuals - the idea that a single self-same individual exists in multiple worlds.

The solution to this problem is simple: give up identity of transworld individuals as commonly understood (Lewis, 167). We turn to the Humphrey example again. When we say that Humphrey has six fingers in a possible world, w1, we do not mean that the Humphrey in this world has six fingers. What we means is that there is a possible Humphrey who has six fingers. Nor do we mean that there is one object Humphrey (H) in the domains of two worlds and one property, F, such that the proposition F(H) is a member of the set w1 while F(H) w2. Rather than indexing properties so they are relative to worlds, we can relativize objects to worlds. We can now express the statement ``Humphrey has five fingers in this world and Humphrey has six fingers in a possible world'' as V(F(Hw0)) = 1 \land V(F(Hw1)) = 0. Clearly there is no conflict with the indiscernibility of identicals, because there is no claim of strict identity; Hw0 Hw1.

I believe it has been the idea of transworld individuals that has been puzzling all along. How can the same object exist in two different worlds? How can Humphrey, for example, lead multiple lives? That is, how can there be objects named Humphrey in different worlds that do different things and have different properties, but are identical with each other?

So we abandon the idea of double lives. Fortunately, there are other ways to represent concepts of modality and transworld individuals. One such way is counterpart theory, as outlined by Lewis. According to counterpart theory, one object is not contained in the domains of multiple possible worlds. Instead, the object is represented as having possibilities by having counterparts in the other worlds. Just as a painting or statue is a replica of an original, counterparts of Humphrey are Humphrey-like objects that play the role of, or represent, Humphrey. Humphrey's counterparts are similar, even almost identical, but they are not strictly identical. To say that Humphrey might have won the presidential election is to say, in counterpart theory, that there exists a possible world, and a counterpart of Humphrey in that possible world who won the presidential election (Lewis, 166).

One primary objection against counterpart theory, or against any theory which is thought to reject transworld identity, is that it cannot adequately capture the concept of having possibilities. That some other Humphrey has six fingers, the argument goes, does not tell us anything about this Humphrey. The intuition is that the actual Humphrey wouldn't care at all if some other individual, even another individual very much like him, had six fingers or won a presidential election. He wouldn't care because such a statement would be irrelevant to him (Lewis, 156). To have possibilities, we need to be able to say things such as ``I could have done otherwise'', not ``Someone like me, but not me, could have done otherwise''.

I believe this argument, applied to counterpart theory, misunderstands the work being done by the counterpart relation. Counterparts, though not identical to each other, are not independent individuals. Counterparts represent classes or families. Each member of the set of counterparts of Humphrey is similar or congruent to every other member of the set. While there is no strict identity relation, there is an identity-like relation that must not be overlooked. A statement about a counterpart of Humphrey does have some bearing on Humphrey. The relation between actual and possible Humphreys is built into the counterpart relation.

Counterpart theory can, I believe, capture the important features of modal theory. It is not necessary for Humphrey to be in several possible worlds to express possibilities. If Cw1(H) is the counterpart of Humphrey in w1, previously referred to as Hw1, then the proposition V(F(Cw1(H))) = 0, that the counterpart of Humphrey in w1 does not have five fingers, is a representation of the possibility that the actual Humphrey could have had some other number of fingers.

Counterpart theory rules out transworld identity, but only as it is commonly understood. A single individual is not in the domain of multiple worlds; by appeal to counterparts, however, individuals do exist according to multiple worlds (Lewis, 167). I believe this is a feature, not a failing, of counterpart theory. It does not make sense to think of transworld individuals as individuals in the same ways as intraworld individuals. Counterpart theory does not really rule out transworld identity; in fact, it provides a framework and gives us a way to speak of transworld individuals as sets or sums of counterparts. Counterpart theory, then, clarifies the important distinction between intraworld and transworld identity and allows us to identify the source of the conflict known as the problem of accidental intrinsics.

In this paper I have addressed several ways in which the problem of accidental intrinsics can be resolved. It seems clear that at least one of accidental intrinsics and transworld individuals must be conceded. I have found no convincing reason to consider giving up either of these to be a costly concession. I have shown that several common objections and problematic consequences can be avoided. If there is a genuine problem with accidental intrinsics, I have yet to find it.


Kim, Jaegwon and Ernest Sosa, Ed. A Companion to Metaphysics. Blackwell Publishers, 1999.

Lewis, David. ``Counterparts or Double Lives?'' Metaphysics: An Anthology. Ed. Jaegwon Kim and Ernest Sosa. Blackwell Publishers, 2000. 154-171.

Loux, Michael J. Metaphysics: A Contemporary Introduction. Routlege, 2002. 176-214.

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