Modalities: Philosophical Essays

by Ruth Barcan Marcus

Based on her earlier ground-breaking axiomatization of quantified modal logic, the papers collected here by the distinguished philosopher Ruth Barcan Marcus cover much ground in the development of her thought, spanning from 1961 to 1990. The first essay here introduces themes initially viewed
as iconoclastic, such as the necessity of identity, the directly referential role of proper names as "tags", the Barcan Formula about the interplay of possibility and existence, and alternative interpretations of quantification. Marcus also addresses the putative puzzles about substitutivity and
about essentialism. The collection also includes influential essays on moral conflict, on belief and rationality, and on some historical figures. Many of her views have been incorporated into current theories, while others remain part of a continuing debate.

Genres: Philosophy, Logic

288 pages, Hardcover

First published January 1, 1993

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[Draft: editing in progress]
A most vexing problem in modal logic, regarding the notion of possible worlds, has been that of trans world identity. We normally take "=" as a relation between names which is true just in case those names refer to the same individuals. We also use this as the justification for being able to substitute one name for the other in any sentence without also changing that sentence's proposition. But it is questionable whether identities hold necessarily. So it is also questionable whether names can be substituted in modal contexts. This situation is sometimes called "referential opacity" because the referents of names fail to play the same role in modal context that they otherwise fulfill for other sentences. Should we take this view then we are accepting that identity is merely nominal; that identity holds only between names for individuals rather than individuals themselves.

Let's motivate our discussion of this problem with a simple example:

(1) Lois Lane believes Clarke Kent is weaker than Superman.

In this example "believes" is our modal operator and "Clarke Kent is weaker than Superman" is in a modal context. The names "Clarke Kent" and "Superman" refer to the same person, as most people know about this story already. Lois Lane does not know this though, and far from believing it, she does not even suspect it. At least this is her characterization in the story. But clearly no person can be weaker than their self and any statement affirming that would be a contradiction. The question posed here is: Does Lois Lane believe a contradiction?

The crux of the question is that if identity is merely nominal--holding between names rather than individuals--then we require the same names to refer to the same individuals in all accessible worlds in order for substitution to be valid (i.e. to maintain the proposition through substitution). And we require a little more too. We also require the same names to stand in the same identity relation. This is precisely what fails in the case of Lois' belief. Her belief includes a possible world in which the names do not refer to the same individual. Hence her believing it is not a case of her believing a contradiction. It is not a contradiction for her because for her it is a possibility that those names refer to different individuals.

To understand this a little more formally, let's consider the following distinction between between intentional and extensional sets. This discussion will appear off topic at first but it will come back to our main point in a round-about way. Below we have two ways of expressing the same set:

(2) S' = { 0, 2, 4, 8 }
(3) S = { the first four even naturals }

The first is extensional and the second, intentional. The intentional expressions are well known to admit paradoxes of the form

(4) The set of all sets that do not contain themselves.

The existence of these paradoxes has cautioned students of logic and math against the naive use of intentions in specifying sets. On the other hand, for an extension set to express a paradox it would have to be the case that something both is and is not a member of it. Some people might reasonably claim that paradoxical extension sets do not exist. Let us agree to call these "impossible" sets but leave aside the question of their existence for now.

What we want to ask is whether identity is an intentional or extensional set. Consider the binary relation "I" given as

(5) I = { (x, y) : x and y are names }

such that

(6) aIa
(7) if aIb then bIa
(8) if aIb and bIc then aIc

for any names a, b, c.


Now, to be able to substitute a for b using I under a modal operator, all individuals of the actual world must also exist in all accessible worlds, all names must denote the same individuals in them, and moreover, I must also exist and hold for all the same names. Our conditions on I do not guarantee this on their own. In fact, our conditions alone do not even guarantee that I denotes the same relation, since I is a universal relation, not an individual, and only individuals are said to exist between worlds. So should we also assume identity of relations between worlds? Such assumptions are hasty as we have not yet given identity to individuals. If we express it terms of a universal relation which also requires trans world identity then we have not gained anything on the concept of trans world identity. That is the characteristic of a mistake. So even I as a relation is opaque, not only the individuals.