Identity and possible worlds

Is the relation of numerical identity contingent or necessary? It seems that in many cases identity statements are contingent. Gottlob Frege explained how it is possible that true identity statements can be informative. We may understand the meanings of names “a” and “b” without knowing that they refer to one and the same object, hence the statement “a = b” tells us more than the trivial truth that an object is identical with itself. It is tempting to interpret Frege’s result as implying that the identity “a = b” could be false, i.e. that there is a possible world in which “a” and “b” refer to different objects while retaining their original meanings. Saul Kripke famously questioned that suggestion. Kripke claims that all identity statements are in fact necessarily true, and he presents a formal argument in support of his claim. The argument is based on two premises. (1) Every object is necessarily identical with itself (For all x, it is necessary that x = x). (2) If x has property P and y is identical with y, then y has P. Premise (2) is a variant of Leibniz’s law. We can now use the formula “It is necessary that x = x” and, given that x = y, we can substitute y for x, obtaining “It is necessary that x = y”. In conclusion, (1) and (2) lead to the statement: (3) For all x, y, if x = y, then it is necessary that x = y. From (3) it follows that if we take any proper names “a” and “b”, then if only it is true that a = b, it is necessarily so.

But how can we reconcile this formal result with the intuition expressed at the beginning of the previous paragraph? In the actual world the names “Hesperus” and “Phosphorus” refer to the same object: the planet Venus. But couldn’t it be the case that in some other possible world Hesperus and Phosphorus were different objects? Kripke explains away this intuition by pointing out that the possible situation in which we would be tempted to say that Hesperus is not identical with Phosphorus can be reinterpreted in such a way that the identity will be preserved. According to Kripke, terms “Hesperus” and “Phosphorus” are so-called rigid designators, i.e. terms that refer to the same object in all possible worlds in which they refer to anything at all. “Hesperus” is not synonymous with the description “The brightest star on the morning sky”, nor is “Phosphorus” synonymous with “The brightest star on the evening sky”. These descriptions are used contingently in the actual world to fix the reference of both names. In another world the descriptions may not pick the object which is the referent of both terms (i.e. the planet Venus), but if Venus exists in this world, both terms “Hesperus” and “Phosphorus” will continue referring to it.

It may be pointed out that when we restrict the thesis of the necessity of identity statements to rigid designators, its truth becomes quite trivial. However, Kripke claims that his thesis has non-trivial consequences regarding for instance the mind-body controversy. Without going into too much detail, let us consider the Identity Theory, according to which mental events are numerically identical with some physical events. The identity theorists maintain that their claim is true but only contingently, i.e. in some possible worlds there are beings that possess particular neurological states but lack mental states (so-called zombies), and in other possible worlds there may be disembodied minds. But according to Kripke’s analysis, if the statement “This pain is identical with this stimulation of the nervous system” is true, it is necessary, and hence neither zombies nor disembodied minds are possible. But couldn’t we explain away these possible scenarios in a similar way we have redescribed the Hesperus-Phosphorus case? Namely, couldn’t we just say that the rigid designators “this pain” and “this stimulation of the nervous system” are contingently associated with some descriptions, which fail to pick the same object in alternative possible worlds? Unfortunately, as Kripke points out, the terms referring to mental states have no associated descriptions, because their reference is fixed directly by the person that is in a given mental state. So the case of the mind-body identification is different from the case of the Hesperus-Phosphorus identity.

What is the ontological status of possible worlds? One radical answer to this question is known under the name of modal realism (possibilism), which has been proposed by David Lewis. Modal realism consists of several claims. One of them is that possible worlds are made up of concrete, spatiotemporal objects. Possible worlds are not fictions or abstract constructions, but real places with flesh-and-blood inhabitants. Things contained in other possible worlds exist in the same fundamental sense as things in the actual world. Thus it is legitimate to assume that the variables of the existential quantifier range over all possible worlds. For the sake of convenience we may want to relativise the notion of existence to a particular world (speaking about “existing-in-a-world”), but this relativisation does not imply any significant ontological difference. The second element of the doctrine of modal realism is that there is nothing fundamental and absolute about the notion of actual world. The term “actual” is indexical (analogously to terms such as “here”, “now”, “I”, whose meaning depends on the context of utterance), which means that each possible world is actual from the perspective of its inhabitants. Finally, modal realism assumes that possible worlds are spatiotemporally and causally separated from each other. One important consequence of these assumptions is that one object cannot exist in more than one world. Transworld identity is an empty notion in modal realism. But how can we interpret the modal statement “This two-metre high tree could be five metres high” without assuming that this tree can exist in other possible worlds? Lewis solves this problem by introducing the notion of a counterpart. This tree has many counterparts in other possible worlds – trees that are sufficiently similar to it, but not numerically identical. An object could have a given property, if one of its counterparts possesses this property in some possible world.

The main motivation for modal realism comes from its radically reductive character. Lewis subscribes to nominalism and he uses the concept of concrete possible worlds to give a reductive analysis of various abstract entities, such as properties, propositions or meanings. For instance a proposition is simply defined as a class of possible worlds (the proposition “Snow is white” is the class of possible worlds in which snow is white). According to this definition, proposition p is true in a world w, if w is an element of the class of worlds p. Properties, in turn, are defined as functions which assign a set of objects to each possible world. These sets are intuitively understood as consisting of objects that possess a given property in a particular world. This definition avoids the well-known difficulty resulting from the fact that two numerically distinct properties can nevertheless have the same extension in the actual world. The property of being an elephant and the property of being the largest land animal living on Earth now may have the same extensions in the actual world, but there are possible worlds in which elephants are not the largest living land animals. However, reductions offered by modal realists are often criticised as not entirely adequate. For instance, it is pointed out that all necessary true propositions become identical. But we believe that there is a difference between the statements “2+2 = 4” and “It is snowing or it is not snowing”. Similarly, it can be maintained that the property of being triangular and being trilateral are different, and yet in all possible worlds their extensions are identical.

Modal realism is often rejected on grounds of its extravagant ontology. An alternative position is offered in the form of actualism (moderate realism). Alvin Plantinga suggests that possible worlds are mere theoretical constructions which enable us to formulate non-reductive explanations of modal notions. The only genuine world is the actual world, and the quantification in our language should be restricted to objects in the actual world. Possible worlds different from the actual world can be defined as complete and consistent sets of propositions, or better as complete and consistent states of affairs (situations). Each proposition corresponds to a given state of affairs. All states of affairs are abstract objects which exist in the actual world, but only some of them obtain (those that correspond to true propositions). False propositions describe existing states of affairs which nevertheless don’t obtain. According to actualism, the expression “actual” has an absolute meaning: it refers to one and only world that truly exists.

One important difference between actualism and possibilism regards the notion of transworld identity. Actualism admits that one object can exist in many possible worlds. But how are we to understand this statement, if possible worlds do not exist literally, but are mere constructions out of abstract objects? Plantinga proposes the following solution. That an object a exists in a possible world w means that if w were actual, a would exist in it. Note that the explicans is a counterfactual conditional, but it cannot be interpreted in terms of possible worlds, since this would require an introduction of second-order possible worlds (a possible world in which another possible world would be actual). This fact shows that Plantinga’s conception does not offer a fully reductive analysis of modal notions, and that some modalities have to be taken as primitive.

Readings:

M.J. Loux, "The necesary and the possible", pp. 153-186, Metaphysics: A Contemporary Introduction.

E.J. Lowe, "Necessity and identity", pp. 84-95, "Possible worlds", pp. 120-133, A Survey of Metaphysics.