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ALAN McMICHAEL A NEW ACTUALIST MODAL SEMANTICS 1. A PROBLEM FOR ACTUALISM ABOUT POSSIBLE WORLDS Suppose that we interpret necessity as truth in all possible worlds, possi- bility as truth in some possible world, and truth simpliciter, naturally, as truth in the actual world. Suppose that we take this interpretation seriously, not merely accepting it as a useful instrument, but believing that it reflects the very meanings of “necessary”, “possible”, and “true”. Then we are committed, in some sense, to there being many possible worlds, and at the same time, to the idea that just one world, our world, is actual. And this is an apparent contradiction, for what is being if not actuality? Of the many solutions to this puzzle, I shall be concerned with the one offered by actualists. Actualism is the view that there are in general no nonactual objects, and in particular no nonactual possible objects. Those actualists who believe in possible worlds, which certainly seem to be non- actual possibles, take them instead to be existing abstract entities. These abstract entities are chosen in such a way that just one of them corresponds, in an appropriate sense, to reality. For example, Alvin Plantinga says that worlds are maximal possible states-of-affairs, only one of which, the actual world, obtains. ’ In this way, actualists can maintain both that all possible worlds have being, in the sense that they exist, but that one is distinguished as actual, in the sense that it alone corresponds to reality. Thus our initial puzzle is solved. I shall argue, however, that at a deeper level, actualism is not compatible with standard possible worlds semantics. According to the standard seman- tics, a statement ‘It is possible that A’ is true if and only if there is a possible world which includes the state-of-affairs expressed by A. The statement “It is possible that Socrates have chosen exile” is true, for instance, if and only if there is a possible world which includes the state-of-affairs of Socrates’ choosing exile, a truth-condition that is not at all unreasonable. Difficulty is encountered, however, in the interpretation of statements containing iteruted modal operators, such as “It is possible that there have been a man Journal of Philosophical Logic 12 (1983) 13-99. 0022-3611/83/0121-0073$02.70. Copyright 0 1983 by D. Reidel Publishing Company, Dordrecht, Holland, and Boston, USA

A new actualist modal semantics

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Page 1: A new actualist modal semantics

ALAN McMICHAEL

A NEW ACTUALIST MODAL SEMANTICS

1. A PROBLEM FOR ACTUALISM ABOUT POSSIBLE WORLDS

Suppose that we interpret necessity as truth in all possible worlds, possi- bility as truth in some possible world, and truth simpliciter, naturally, as truth in the actual world. Suppose that we take this interpretation seriously, not merely accepting it as a useful instrument, but believing that it reflects the very meanings of “necessary”, “possible”, and “true”. Then we are committed, in some sense, to there being many possible worlds, and at the same time, to the idea that just one world, our world, is actual. And this is an apparent contradiction, for what is being if not actuality?

Of the many solutions to this puzzle, I shall be concerned with the one offered by actualists. Actualism is the view that there are in general no nonactual objects, and in particular no nonactual possible objects. Those actualists who believe in possible worlds, which certainly seem to be non- actual possibles, take them instead to be existing abstract entities. These abstract entities are chosen in such a way that just one of them corresponds, in an appropriate sense, to reality. For example, Alvin Plantinga says that worlds are maximal possible states-of-affairs, only one of which, the actual world, obtains. ’ In this way, actualists can maintain both that all possible worlds have being, in the sense that they exist, but that one is distinguished as actual, in the sense that it alone corresponds to reality. Thus our initial puzzle is solved.

I shall argue, however, that at a deeper level, actualism is not compatible with standard possible worlds semantics. According to the standard seman- tics, a statement ‘It is possible that A’ is true if and only if there is a possible world which includes the state-of-affairs expressed by A. The statement “It is possible that Socrates have chosen exile” is true, for instance, if and only if there is a possible world which includes the state-of-affairs of Socrates’ choosing exile, a truth-condition that is not at all unreasonable. Difficulty is encountered, however, in the interpretation of statements containing iteruted modal operators, such as “It is possible that there have been a man

Journal of Philosophical Logic 12 (1983) 13-99. 0022-3611/83/0121-0073$02.70. Copyright 0 1983 by D. Reidel Publishing Company, Dordrecht, Holland, and Boston, USA

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who does not exist in the actual world and who chooses death, although it is possible that he have chosen exile.” Two worlds other than the actual world are involved in the truth-condition for this statement, one which includes the existentially quantified state-of-affairs of there being a man who does not exist in the actual world but who chooses death, and a third world which includes his existing and choosing exile. Since the original statement is true, such worlds must exist, but this means that there is a world, the third world, which includes a state-of-affairs X exists, where X is some nonactual possible individual. This is a result that actualists, who don’t believe in nonactual possibles, are apparently unable to accept.?

An actualist might respond that in the second world where this nonactua. individual X exists, there is indeed a state-of-affairs X exists, and there are maximal possible states-of-affairs, worlds, which contain it. That is, the worlds which exist from the point of view of this second world are distinct from those which exist from the point of view of the actual world. But under this conception, the modal operators cannot be interpreted by quanti- fiers ranging over some one universal set of possible worlds, and to concede this is to abandon the original goal of possible worlds semantics, the goal of interpreting modal statements within an extensional world theory.3

Another response for the actualist, one championed by Plantinga, is to say that a state-of-affairs of the form X exists does not really have the individual X (actual or nonactual) as a constituent, rather it is a compound of the property of existence and an indiuidual essence. Thus, it is perfectly possible for there to be such a state-of-affairs even though there is no actual individual X, for what enters into the state-of-affairs is not the individual X, but instead an unexemplified essence.

I find this appeal to individual essences rather unsatisfactory. Although I must forego any detailed criticism here ,4 let me make two brief points: (1) It seems impossible to reduce individual essences to existing individuals, qualitative properties, and qualitative relations. The main argument against such a reduction is that it is possible for a complete world to be entirely symmetrical, so that an object on one side of the axis of symmetry has exactly the same qualitative intrinsic properties and qualitative relational properties as its correspondent on the other side of the axis. Thus, if indi- vidual essences were merely combinations of such properties, then an object and its correspondent would have the same individual essence. This is im- possible due to the individuality of the supposed essences. And it does not

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help to add actual individuals to the combinations, since in the symmetrical world, an object and its correspondent may bear exactly the same qualita- tive relations to whatever actual individuals exist there. (2) On the other hand, to introduce essences as primitive properties, some of which are specific to nonactual individuals, seems tantamount to abandonment of Actualism, since they are introduced solely in order to fill out a structure isomorphic to that described by the possibilists, and not because we have any independent reason to believe in their existence.

Given the difficulties with the appeal to individual essences, our original problem remains. Actualism is in conflict with standard possible worlds semantics. In particular, there is difficulty interpreting statements con- cerning the different properties which merely possible individuals might have had. Because of the presence of two modal operators in such state- ments, we might call this the problem of iterated modalities. A natural way to resolve the problem is to present a nonstandard modal semantics. This is what I shall attempt to do.5

2.GROUNDWORK

Under the standard semantics, we determine what an individual might have done by seeing what he does in various possible worlds. That is, we look at various possible worlds which include him as an existing constituent. The problem of iterated modalities arises because actualism prevents the exten- sion of this criterion to nonactual possibles. We can’t look for the nonactual possible himself in various worlds, since according to actualism, nonactual possibles are not constituents of anything.

One way to avoid the problem is to revise the criterion for deciding what an individual might have done. Instead of saying that what an individual might have done is what he does in some possible world, let us say that what an individual might have done is what any such individual does in some possible world, so that we emphasize not the individuals themselves, but rather the roles they play. To determine what Socrates might have done, we don’t look for worlds in which he appears, but instead we look for roles, in possible worlds, which are accessible to Socrates’ actual role. In one of these roles includes a certain property, then that property is one Socrates could have had; otherwise, it is not.

Several notions used in this account stand in need of explanation. On this

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view, it is natural to construe possible worlds as purely qualitative entities, entities which do not have individuals as constituents. That is, possible worlds are not maximal possible states-of-affairs, but maximal possible qualitative states-of-affairs. Plantinga calls these entities Ramsey worlds,6 explaining that for any maximal possible state-of-affairs M, there is a corre- sponding Ramsey world: Consider the set of all propositions which are true inM (that is, the set of all propositions which would be true ifM were to obtain). Conjoin all those propositions. For each individual which is a con- stituent of that conjunction, apply the rule of Existential Generalization. Call the resulting existentially generalized proposition, p, a Ramsey proposi- tion. The state-of-affairs of p’s being true is the desired Ramsey world.

A role is a maximal possible qualitative property. The role of an actual individual is the way that individual is - qualitatively. In addition to quali- tative intrinsic properties, it includes all the individual’s qualitative relational properties, such as being a descendent of some famous national leader, but no relational properties with individual constituents, such as being a descendent of Abe Lincoln. We can regard roles as being constituted from Ramsey propositions: A role is the result of replacing one of the wide-scope existential quantifiers of a Ramsey proposition p with the corresponding property abstraction operator. Whereas an expression for the proposition p would have the form ‘there is an X such that . . .‘, the expression for the corresponding role would begin, ‘the property of being an X such that . . .‘. Notice that a role “encodes” an entire Ramsey world, in the sense that if something were to have that role, then a certain Ramsey world would obtain. Let us say that the role includes the Ramsey world.

The notions of a role and of inclusion can be generalized to two or more places. A binary role is a maximal possible qualitative binary relation, and for any number i, an i-ary role is a maximal possible qualitative i-ary relation. An i-ary role R includes a j-ary role S (j < i) just in case it is necessarily true that if any individuals xi, . . . , xi exemplify role R, then the first j individ- ualsxr, . . . , Xj exemplify S.

I shall demonstrate the role semantics on some interpreted sentences of a modal first-order language:

OFs Fx : x is foolish s: Socrates.

This sentence is true just in case there is a unary role R’ which is accessible

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to the actual role of Socrates and which includes the property of being foolish.

0(3x) (TX & OSX) TX : x is a 300 lb canary 5’~: x is a 600 lb canary.

This sentence is true just in case there is a Ramsey world W such that for some unary role R’ which includes W, R’ includes being a 300 lb canary, and there is a unary role S’ accessible to R’ such that S’ includes being a 600 lb canary. Notice that in giving the truth-conditions for this iterated modal statement, there is no reference to individuals at all, actual or non- actual.

(w3YF-xY &O&Y) Pxy : x is a parent of y.

This sentence is true just in case there are individuals x and y such that the actual binary role of x and y does not include the parenthood relation, but for some binary role R2 which is accessible to the actual binary role of x and y, R2 includes the parenthood relation. This truth-condition may not be reducible to one involving only unary roles. That is, x and y might be such that the role (hx)(3y)R2xy of the parent is accessible to the actual role of x, and the role (hy)(3x)R2xy of the child is accessible to the actual role of y, yet the binary role R2 is not accessible to the actual role of x and y. This might be so simply because it is logically impossible for an individual to have parents other than those he or she actually has.

Statements asserting possible nonexistence give rise to special problems of interpretation. Consider:

O-ES Ex : x exists s: Socrates

Shall we say that this sentence is true just in case some unary role accessible to Socrates’ actual role includes nonexistence? There are strong reasons for avoiding such a truth-condition. In the first place, it clashes with actualist intuitions. A role is a way a thing could be. But for actualists, being is simply existence. Thus nothing could have a role without being, and so existing. This means that every role includes the property of existence. Since every role is possible, it follows that no role includes nonexistence, and so no role accessible to Socrates’ actual role includes nonexistence, Yet surely Socrates might not have existed.

Leaving intuitions aside, there remain theoretical difficulties with the

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postulation of roles that include nonexistence. Such postulation can take one of two basic forms. On the one hand, one could suppose that for each Ramsey world IV, there is an associated null role, and that any individual that fails to exist in such a world fills that role, The null role would be the conjunction of IV with the property of nonexistence (that is, the property not existing but being such that W obtains). On the other hand, one could suppose that different nonexistents can fill different roles in the same Ramsey world.

The null role view is more congenial to actualism, since it postulates no interesting differences between nonexistents. But because it implies that all nonexistents, in a given world, have the same role, it has the con- sequence - assuming the clauses of a role semantics - that whatever is is possible for one nonexistent is possible for all the others, a consequence that forces one either to adopt a strange sort of metaphysics or to conclude that nothing whatever is possible for nonexistents, a conclusion I come to without null roles (see Section 7).

The alternative view, that different nonexistents can have different roles, is harder to square with actualism. If different nonexistents have different roles, then there must be properties that are exemplified by some non- existents but not by others. But what could these properties be? What properties, for example, could distinguish a nonexistent Socrates from a nonexistent cockroach? The property being possibly human? It seems to me that Socrates is possibly human because of properties he actually has, and not because he possesses, even in worlds where he does not exist, some irreducible modal Froperty.

For these reaso,rs, 1 propose a more economical truth-condition for state. ments of possible nonexistence, one that does not involve roles that include nonexistence: O-Es is true just in case some role of rank zero (Ramsey world) is accessible to the actual unary role of Socrates. This seems to me the only natural actualist truthcondition. Socrates’ nonexistence is possible not because there is a role accessible to his actual role that includes, in Meinongian fashion, nonexistence, but because there is a role accessible to his actual role that is not really a role for him at all. It is not a role for him, because there is no sense in which Socrates could fill a role of zero argu- ments.

The possibility of nonexistence must often be taken into account even when existence is not explicitly mentioned:

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o-Fs Fx: x is foolish s: Socrates

This sentence is true if either (1) some role of rank zero is accessible to the actual unary role of Socrates (that is, if Socrates might not have existed), or (2) some unary role accessible to the actual role of Socrates does not include foolishness (that is, if Socrates might have existed and been non- foolish).

3.AN OBJECTION

One might object that the role semantics is no advance over Plantinga’s system of essences. Uninstantiated individual essences have simply been replaced by other complex uninstantiated properties. What do we gain thereby?

First, it is clear that roles are not the same as individual essences. An individual’s role includes many properties that the individual has only contingently, whereas an individual’s essence includes only the individual’s necessary properties. In a symmetrical world, two individuals may fill the same role. No individual essence, on the other hand, can be possessed by more than one individual. Finally, the role of a contingently existing indi- vidual is merely qualitative and could have been filled by some other individual, but no individual’s essence could have been possessed by some other individual.

Secondly, roles are less suspicious entities than individual essences. An individual’s essence includes only properties that the individual has necessarily, yet those properties must necessarily characterize just that individual. In the case of “nonactual possible individuals”, it is unclear that we can find complexes of properties satisfying these conditions, so unclear that there are any uninstantiated individual essences. Indeed, the argument from symmetrical worlds shows that in certain cases, such complexes cannot be found. An individual’s role, on the other hand, is a complex of properties that the individual merely has. No condition is placed on the properties that would bring into question the existence of such complexes.

The role semantics is tailored to a certain view of properties:

QA Every property is analyzable in terms of actual individuals and qualitative properties and relations.

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Let us call this the doctrine of qualitative analysis. We should accept this doctrine because we have no reason for believing in the existence of any- thing else that could enter into the analyses of properties. Once we accept it, however, we are compelled to reject Plantinga’s system. Plantinga’s un- instantiated essences are properties that violate QA, as can be seen from the argument involving symmetrical worlds: In a symmetrical world, an object and its correspondent on the other side of the axis of symmetry share the same qualitative intrinsic and relational properties. So assuming that they bear exactly the same qualitative relations to whatever actual individuals exist in the world, these objects are indistinguishable in terms of actual individuals and qualitative properties and relations. If therefore there is an “essence” that one has and the other lacks, that essence is a property that violates QA.

4. ROLE SEMANTICS AND LEWIS’ POSSIBILISM

The semantics I am proposing is, as I shall soon make clear, structurally similar to the possibilist semantics David Lewis has developed.’ This simi- larity might lead some to believe that the two theories, despite the appar- ently grave metaphysical differences, are in substance the same, so that there can be no basis for preferring one to the other. I shall argue that this is not the case, that, on the contrary, there is good reason for preferring the theory I propose.

Lewis is a possibilist, that is, he believes there really are nonactual possible objects. Furthermore, he believes that among these nonactual possibles are some which are complete worlds. According to Lewis, these worlds are on an ontological par with our own world, so that its actuality is a merely relative property. This world is indeed the only actual world, but that is merely to say that this world is the only world which is actual with respect to this world. Inhabitants of other worlds are equally correct in asserting the actuality of their worlds. Finally, Lewis also holds that no individual is present in more than one world, a doctrine that is a natural accompaniment of the view on actuality, for how could we speak unam- biguously of “our world” if we also exist in some others?

It is this last doctrine, that individuals are worldbound, which gives rise to Lewis Counterpart Theory. Lewis cannot say, for example, that it is true that Socrates could have chosen exile just in case he is present in some

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world where he does choose exile, since Socrates, and every other individual, exists in just one world. Instead, Lewis says that it is true just in case there is some world in which some counterpart of Socrates (an individual suffic- iently like Socrates, and as much like Socrates as anything else in his world) chooses exile. The structural similarity between the role semantics and this counterpart semantics should now be clear: Lewis concrete possible worlds correspond to the abstract qualitative Ramsey worlds, his possible individ- uals correspond to unary roles, and the counterpart relation among possible individuals corresponds to the accessibility relation among roles.

There are a couple of minor structural dissimilarities: (1) The access- ibility relation among roles is not confined to unary roles. The counterpart relation, on the other hand, holds only between single possible individuals. There is no notion, at least in Lewis’ original theory, of the joint counter- parts of a pair of individuals. (2) The accessibility relation, as I shall charac- terize it, is both symmetric (over roles with a given number of arguments) and transitive. Symmetry and transitivity are each incompatible with Lewis’ description of the counterpart relation, so could not be added to his theory.

Many objections to Lewis’ Counterpart Theory are based on the features mentioned in (1) and (2).* However, by changing those features, it might be possible to meet the objections. In his recent work on de re belief, Lewis does introduce a joint counterparts relation, and it could be used to counter the objections based on (1). Perhaps also one could come up with a reason- able counterpart relation that satisfies the symmetry and transitivity requirements and so remove the objections based on (2). Since these modi- fications would increase the structural similarity between Lewis’ possibilist theory and the role semantics, the question naturally arises whether there would remain any good ground for deciding between them.

My main objection to Lewis’ view concerns not his use of counterparts, but rather his possibilist ontology. On grounds of ontological economy, I do not believe that we should settle for a possibilist conception of possible worlds unless there is no workable actualist alternative, and it seems to me that there is a workable actualist conception, the one presented in this paper. Against this, one might argue that I am not entitled to invoke con- siderations of ontological economy, since I myself have ignored them too often. I have appealed to a horde of abstract objects: states-of-affairs, properties, relations, and infinite combinations of these things. In reply, I would argue that we need properties, relations, and infinite combinations

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to account for the structure of this world. We are simply stuck with nbstructa. But given the existence of abstracta, I am arguing, there is no reason to believe in the existence of unreduced possibilia.

My project can be viewed, roughly, as an attempt to replace talk of nonactual possible objects with talk of complex qualitative properties, roles, It seems to me that only one circumstance could stand in the way of this replacement. It is compatible with possibilism that there be distinct possible worlds which differ only in the respect that different nonactual possibles have different roles in them. These worlds are qualitatively identi- cal, and actual individuals which have a certain role in one have exactly the same role in the other. Since actualism is incompatible with the existence of such “actualistically indiscernible” possible worlds, it would be in trouble if one could show that some acknowledged truth of the modal object lan- guage comes out true only under interpretations containing them (or their equivalent). I seriously doubt that any statement of this sort can be found, and unless such a statement is found, we have no reason to believe in the existence of such worlds. But their existence seems to me the one circum- stance that could prevent the replacement of nonactual possibles. So assum- ing that there are no actualistically indiscernible worlds, the replacement must go through.’

5.ATHEORYOFRELATIONS

The role semantics is based on a certain theory of states-of-affairs (including Ramsey worlds), properties (including unary roles), and relations (including binary, etc., roles). Or, construing states-of-affairs as relations of zero argu- ments and properties as relations of one argument, we may simply say that it is based on a theory of relations. The relations of the theory are relations of lowest logical type, relations of individuals. My purpose in this section is to give an axiomatic description of that theory. I shall begin with axioms that hold of all relations whatsoever, and go on to discuss qualitative rela- tions and roles.

There is an ordinal number associated with each relation, called the rank of that relation, which is the number of its argument sequence. Binary relations have rank two; states-of-affairs, rank zero. Some relations are of infinite rank, The rank of a relation will be indicated by a superscript: A’ is of rank i.

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Defined over all relations of individuals, there is a binary relation of relations called inclusion. Informally, a relation A’ includes a relation B’ just in case:

Cl Necessarily, for any individuals x0, . . . , Xi-r, if x0, . . . , Xi-1 bear relation A’, then x0, . . . , xi-1 bear relation B’.

For example, the binary relation is cz mother of includes is u parent of These relations are of equal rank, but a relation can inclu.le a relation of lower rank. For example, the binary relation is a mother (!fincludes the property has a child.

Can a relation A’ include a relation B’ of higher rank? Let us apply condition Cl to such A’ and B’. Call the resulting instance Cl*. Notice that Cl* must contain free variables xi, . . . , Xi-1 . As I understand such a condition, Cl* implies that if any individuals x0, . . . ,+r were to bear relation A’, then arbitrary individualsxi, . . ,xjsl would exist. But no relation, except an impossible one, can include the existence of arbitrary individuals. Thus A’ is either impossible, in which case no individuals x0,. . . ,Xi-l could bear it, or its does not include B’.

This result is expressed in the Rank Axiom:

Al Rank - If A’ includes B’ and i < j, then A’ includes every relation.

Saying that a relation includes every relation is just another way of saying that it is impossible. Notice that I arrived at the Rank Axiom via a particu- lar understanding of Cl. Someone with a different understanding might have arrived at a different result. Thus the Rank Axiom may appear to be only an arbitrary stipulation. However, it is actually a vital part of the semantics I shall develop.

Condition Cl can also be used to justify the less controversial axioms of Reflexivity and Transitivity:

A2

A3

Reflexivity - A’ includes A’.

Transitivity - If A’ includes B’ and B’ includes Ck, then A’ includes Ck

It is natural to postulate also that relations of the same rank which include one another are identical:

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A4 Antisymmetry - If A’ includes B’ and B’ includes A’, then A’=@.

I say that it is natural, but some philosophers think it is wrong. The Anti- symmetry Axiom says, in essence, that necessarily equivalent relations of the same rank are identical. These philosophers deny it on the basis of the argument:

(1) The meaning of an n-adic predicate is the n-ary relation it expresses. (2) N-adic predicates which have the same meaning can be substituted,

with preservation of truth-value, in all nonquotational contexts. (3) Some n-adic predicates which express necessarily equivalent relations

cannot always be substituted, with preservation of truth-value, in contexts of belief.

(4) Belief contexts are nonquotational. (5) Therefore, some n-adic predicates express necessarily equivalent

relations which are not identical. I reject premise (1). I believe that the meanings of predicates are more finely individuated than the relations predicates express. It may be that predicates have the same meaning only if they are intensionally isomorphic in the sense defined in Rudolf Carnap’s Meaning and Necessity. However, to elaborate Carnap’s theory, or any alternative, is beyond the scope of this paper.

From the Antisymmetry Axiom, it follows that each rank contains at most one impossible relation, one relation which includes every relation. And it is reasonable to suppose that each rank contains at least one impos- sible relation, since for any i, it is easy to construct a contradictory i-adic predicate of individuals. The i-th impossible relation is denoted by “O’“, and it includes any relation:

A5 Impossible Relations - 0’ includes A’.

Since it is also easy to construct tautological predicates, we may postu- late the existence of a necessary relation at each rank. The necessary rela- tion of the i-th rank in denoted by “1 i’r, and it is included by every relation of equal or higher rank. Necessary relations include nothing except other necessary relations:

A6 Necessary Relations -

(A) If i > j, then A’ includes lj.

(B) If I’ includes A’, then A’ = 1’.

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The necessary relation of rank one, 1’ , can be regarded as the property of existence, since according to actualism, it is necessarily true that every individual exemplifies it.

Any two relations have a conjunction. For example, the conjunction of the properties of being red and being round is the property of being red and round. This is an example of a conjunction of relations of the same rank, but there are also conjunctions of relations of different ranks. For example, the conjunction of being a z such that z is happy and the relation of being an x and y such that x is a parent of y is the relation of being an x andy such that x is happy and a parent ofy. Notice that argument places are associated starting from the least (so we don’t get the relation of being an x and y such that y is happy and x is a parent of y). A conjunction is the weakest relation which includes each of its conjuncts:

Al Conjunction - For any A’ and Bi, there exists a conjunction A’ *B’ such that

(1) the rank of A’ *B’ is the maximum of i and j, (2) A’ *B’ includes A’ and it includes B’, and (3) for any relation Ck such that k = max (i,j), if Ck in-

cludes A’ and it includes B’, then Ck includes A’ *B’.

For any i-ary relation A’, there is a relation B’-’ which results from A’ by applying existential quantification at its i-th argument place. For example, the property bus a child is the result of applying existential quanti- fication at the fmal argument place of the relation is a parent of- Existential quantification is governed by the axiom:

A8 Quantification - For any A’, there exists an existential quantification 3 A’ such that

(1) rank@A’)=i-1, (2) A’ includes 3 A’, and (3) ifk<i,then3(Ai*Bk)=3Ai*Bk.

Notice that an existential quantification of A’ is the strongest i-1-ary relation which is included by A’. (Proof: By A8(2), A’ includes !I A’. Let B’-’ be any other i-l-ary relation which is included by A’. By A8(3), B (A’ * B’-‘) = 3 A’ * B’-‘. But since A’ includes B’-’ , A’ * II’-’ = A’, and we have: 3 A’ = 3 A’ * Bib’. But this is equivalent to the proposition that 3 A’ includes Bi-l.j10

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In order to obtain the effect of applying existential quantification to initial argument places, and to express inclusions between relations and their converses, we need to postulate operations which shuffle argument places. Each such operation corresponds to a function p which permutes the ordinals less than some ordinal i:

A9 Shuffling - For any permutation p of the numbers of an ordinal i (that is, of the ordinals less than i), there is a shuffling operation p which maps i-ary relations to i-ary relations. Such oper%ions obey:

(1) If p is an identity permutation (that is, p(n) = II for all n <i), then p is an identity operation (that is, p ‘&‘LQ). -

(2) p q A’ = p o q A’, where p and 4 are permutations of iaid p o q is the composition of p and q, that is, the function such that p o q(n) = p(q(n)).

(3) If A i includes B’ and i < i, then for any permutation p of i such that p(n) < j iff n < j,

p A’ includes p I j B’, - -

where p 1 i is the restriction of the function p to j.

Of special interest are those shuffling operations which simply exchange a single pair of argument places:

Dl Given an ordinal i, let (i, k) be that permutation of the ordi. nals less than i which maps! to k and k to j and leaves all other numbers unchanged. Then (i,k) denotes that shuffling operation on i-ary relations which switches argument places numberedi and k and leaves all other places unchanged.

For any binary relation R2, there is a property which a thing has just in case it bears the relation R2 to itself. This property can be denoted using existential quantification and the relation of identity: (Ax)(Zty)(y = x & R2xy). In the symbolism of the relation theory, this becomes: 3 (I2 * R’), where ‘I2 ’ is the symbol for the identity relation. The special properties of the identity relation are expressed in an axiom:

A10 Identity - (A) 3 I2 = 1’.

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(B) I2 * A’= (o,I) (f2 * A’), where (0,is assumed to operate on relations of the appropriate rank, namely max(2,i).

Relations divide into those which are qualitative and those which are not. A relation A’ is qualitative if and only if either (1) A’ is the i-th impossible relation, O’, or (2) for any contingent individual x (individual whose exist- ence is not included in the necessary state-of-affairs lo), Ai does not include the state-of-affairs of x’s existing. On this definition, the property of being red, for example, is qualitative, since something’s having that property does not include the existence of any particular contingent individual. The property of being identical to Socrates, on the other hand is not qualitative, since nothing can have that property unless Socrates exists.

In order actually to formalize the definition of a qualitative relation, we need to be able to talk about relations which do have individual constitu- ents. This can be accomplished by associating with each individual x an operation x+ that maps i-ary relations, for all i > 0, to appropriate i-1-ary relations. Axioms and definitions involving the plus-operations (Al 1, D2, and A17) are not crucial to the role semantics, but, for the sake of logical completeness, they are included in the notes.”

A natural closure condition on qualitative relations is that the conjunc- tion of any qualitative relations is qualitative:

Al2 Closure - If F’ and G’ are qualitative, so is F’ * G’.

Ail other closure conditions follow from the definition: 1’ is qualitative, and if A’ is qualitative, so are 3 A’ and p A’, for any permutation p of i. Another important fact about qualitative relations is that each has a Boolean complement :

Al3 Complements - For every qualitative relation F’, there exists a complement -F’ such that

(1) -F’ is a qualitative relation of rank i, (2) F’ * -F’ = of, (3) ifF’*G’ = O’, then G’ includes -F’, (4) _ - F’ = F’,

Clauses (2) and (3) tell us that - Fi is the weakest relation which contra- dicts F’. Clause (4) tells us something about the symmetry of the system of

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complements. Taken together with the other axioms, Al3 implies that the qualitative relations of any given rank i form a Boolean algebra.

Under the conception of relations outlined here, it is not true that every relation has a Boolean complement. For example, the state-of-affairs of Socrates’ existing (a relation of rank zero) is nonqualitative, but it is con- tradicted by the qualitative state-of-affairs of there being no people. Thus if there were a complement of Socrates existing, it would, by A13(3), be included by the qualitative state-of-affairs of there being no people. But anything which is included by a possible qualitative state-of-affairs is itself qualitative, hence the complement of Socrates existing would be qualitative This is impossible, since its complement by A13(4), namely Socrates exist- ing, is not qualitative. (By A13(1), a complement of a qualitative relation must itself be qualitative.)

We may now define the key notion of a role, a maximal possible qualita- tive relation:

D3 F’ is an i-ary role iff F’ is a qualitative relation, F’ # O’, and for any qualitative G’, either F’ includes Gi or F’ includes -G’.

A consequence of the axioms is that if i > 0 and R’ is a role, then its exis- tential generalization, 3 R’, is a role, in fact it is the unique i-I-ary role included by R’.

Crucial to the modal semantics is the postulate of Atomism:

A14 Atomism - For any qualitative F’, if F’ # O’, then there is a role R’ such that R’ includes F’.

If an i-ary qualitative relation is possible, then it is included in some maxi- mal possible i-ary qualitative relation. Al4 implies that the Boolean algebra of i-ary qualitative relations, for any i, is an atomic Boolean algebra.

Each n-tuple of individuals fills a unique actual role:

A15 Actual Roles - (A) For any individuals x0, . . . , Xi-1 ,

@(x0,..., Xi-l) = the actual role of x0, . . . , Xi-r .

(B) For any permutation p of i,

p%xo,. . . , Xi-*) = @(Xp-‘(O), . . . 3 Xp-l(i-*)),

where p-‘(j) = the number mapped to j by p.

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(C) If j < i,

@(x0,. . . , xi-i) includes @(x0, . . . 3 Xj-1).

In the case i = 0, @ is the actual Ramsey world, the way the world is - qualitatively. A15(B) tells us that a permutation of the argument places of an actual role is identical to the actual role of the same individuals taken in the permuted order. A15(C) says that the actual role of a sequence of individuals includes the actual role of any initial subsequence of them.

Finally, we need to introduce an accessibility relation among roles. I suggest the following basic principles:

Al6 Accessibility - (A) Every role is accessible to itself. (B) No role is accessible to a role of lower rank. (C) For any roles R’ and S’ such that j < i, and any permu-

tation p of i such that p(n) < j iff n < j, p 1 j S’ is access- -. ible to p R’ if and only if S’ is accessible to R’.

(D) If a role,!? is accessible to a role R’, then if j < i, then Sj is accessible to 3 R’, and if j = i, then 3 S’ is access- ible to 3 R’.

A16(A) expresses the reflexivity of the accessibility relation, and so vali- dates the principle that whatever relations individuals actually bear are relations which it is possible for them to bear. A16(B) expresses the idea that i many individuals cannot fill the role of j many individuals if j > i. Al6 does not rule out the existence of roles R’ and S’ such that S’ is accessible to R’ and j < i. If such a relationship holds between roles R’ and S’, I take this to mean that any individuals x0, . . . , Xi-l which fill role R’ are such that it is possible for ~0, . . . , Xi-1 to bear S’ and Xi, . . . , xi-i not to exist. A16(C) says that mere permutations of argument places do not disturb accessibility relationships. Al 6(D) says how an accessibility relationship between roles is passed on to roles they include.

The most popular modal logic is undoubtably S5, and not without reason, since S5 seems to give the right answers to a wide variety of modal questions. I do not believe that it gives the right answers to all modal questions, so I shall be presenting a semantics which does not validate all of S5. Nevertheless, I agree that the fragment of purely general state- ments, at least, is governed by an S5 modal logic. This is so because the accessibility relation is symmetric and transitive over roles of rank zero.

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In fact, I believe that relation is symmetric and transitive in a still more general sense:

A18 Minimal S-5 - (A) If an i-ary role S’ is accessible to an i-ary role R’, then R’ is accessible to S’. (B) If a role S’ is accessible to a role R’ and a role Tk is accessible to S’, then Tk is accessible to R’.

Notice, however, that symmetry only holds for roles of the same rank. The structure which these axioms describe is a nine-tuple (Rel, rank,

includes , -, 1’) Ind,+, @, cIccess), where Rel is the set of “relations of individuals,” rank is a function which maps each member of Rel to an ordinal number, includes is a binary relation over Rel, _ is a mapping from permutations of ordinals to operations on members of Rel, I2 is the “iden- tity relation”, Znd is the set of “individuals “,+ is a function from members of Ind to operations on Rel, @ is a function from n-tuples in Znd to n-ary maximal possible qualitative members of Rel, and access is a relation defined over these “roles”. Let us call any such nine-tuple satisfying Al-Al 8 a role algebra. In the modal semantics, symbols will be given an interpretation by assigning them objects in some given role algebra. For purely formal purposes, this need not be the true algebra of all roles and relations.

6,THE ROLE SEMANTICS

In this section, I give a role semantics for a first-order language L containing an identity sign and a possibility operator. l2 I begin with the syntax of that language :

Syntax of L I. Basic Symbols

A. Constants: a, b,c,al, . . . B. Variables: x,y,z,xl, . . . C. Predicates: F, G, H, F1, . . . and E (existence)

F2, G2, H2, F:, . . . and = (identity) F3,G3,H3,F;, . . . and so on.

D. Logical Symbols: -. &. 3.0, (, ).

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II. Terms: A term is a constant or a variable. III. Wffs,

A. If P is an n-place predicate and ti, . . . , I, are terms, then ‘Pt,. . . tnl is a wff.

B. If tl and t2 are terms, then ‘tr = tz is a variant form of the wff r= t1t21.

C. If A and B are wffs and z, is a variable, then the following are wffs:

-A, (A &B), (%)A, OA.

Semantics of L An interpretation is a pair (A, V), where A is a role algebra (Rel, rank, includes, -, I2 ,Ind,+, @, access) and V is a valuation function. V assigns an i-ary qualitative member of Rel to each i-place predicate and a member of Znd to each constant. Also, V(=) = 1’) and V(E) = 1’ .

The definition of truth under an interpretation, which contains one notion yet to be defined, is:

T Let A be a formula whose constants are k,, . . . , ki and whose free variables are pl, . . . , Vj, then A is true under I iff for any members x1, . . . , Xj of Ind, @(V(k,), . . . , V(k,),

, Xj) satisfies A under I and ordering of terms Si,‘.‘.‘. 9 ki, Zll, . . . , Vj).

Now Z may define the notion of a role R’ satisfying a formula A under an interpretation I and an ordering of terms (u,, , . . , Ui-1). The definition is inductive, and it resembles the Tarski definition of satisfaction by a sequence:

Sl A = rPt,,...t,,-7

(A) Suppose one of the terms to, . . . , t,-, does not occur in the ordering (ue , . . . , u~-~). Then R’ does not satisfy A under I and (I(~, . . . Y ui-l>. (B) Otherwise, let se, . . . , s,-i be the terms of A in the order of their first occurrence. Let p be any permutation of i such that p(i) = k whenever Uj = sk . Then R' satisfies A under I and (~0, . . . , Ui-1) iff p R’ includes the relation denoted in abstraction notation by:

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@so). . .(Xs,-,)PhJ.. . t,-,.

(Any constants must be construed as variables in this expres- sion.)

Or, using the notation of the theory of relations, R’ satisfies A under I and (uo, . . . > Ui-l) iff p R’ includes the relation

3 3 . . . ldho<m (h, lst-occurdd>t =9, Iii, kl W> n-m . i (The expression “1[i, ,I” denotes an operation on relations that is defined

by

D4 I[i,k] A’ = (O,j)(l,k)(I’* l’)*A’

This operation conjoins a relation A with an identifying link between its argument places numbered by j and k. The expression

0 r[i,kl tj= tk

denotes the composition of all operations 1[i, k] such that tj and tk are occurrences of the same term and j # k (that is, the operation which results from successive application of all these I-operations). The expression

1 <ho<, (h, 1 st-occurs(sh))

denotes the composition of those shuffling operations which exchange argu. ment place h with the argument place numbered by the first occurrence of the term sh in the formula ‘P t,, . . . tnell. The order of composition is important here. The shuffling operations should be applied in numerical order, beginning with h = 1, whereas the composition of I-operations des- cribed above is order-indifferent.)r3

The ordering of terms (uo, . , . , ui-r) tells us which argument places in the role R’ correspond to which terms. Hence, in the base clause of the satisfaction definition, I use this ordering to determine the permutation p which matches up the argument places in R’ with the appropriate argument places in the relation expressed by A.

s2 A = r-+1

R’ satisfies A under I and (u. , . . , Ui-l) iff R’ does not satisfy B under I and (uo, . . . , ui-r).

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s3 A = ‘B&C1

R’ satisfies A under I and (u,, , . . . , Ui-r) iff R’ satisfies both B and C under I and (me , . . . , ui-r).

s4 A = ‘(3v)E1

R’ satisfies A under I and (ue , . . . , uiWI) iff for some variable V’ not in (uot.. . , Ui-1) and some role Si+l such that Si+’ includes Ri, Si+r satisfies B v’/v under I and (ue , . . . , +r, v’). (Occurrences of a variable which are not bound by the same quantifier should not be associated with a single argument of a role, so it may be necessary to add a new variable to the ordering and substitute that variable for v in B.)

A = ‘OB’

R’ satisfies A under I and (ue, . . . , Ui-r) iff either (A) for some S’ accessible to R’, S’ satisfies B under 1 and (~0, . . . , uj-,), or (B) for some permutation F of i, there is an S’, j < i, accessible to p R’ such that S’ satisfies B under I and (u~-I(~), . . . , Up-‘(i-1)).

Clause S5(B) is needed to deal with possible nonexistence. This is best seen by means of example. Consider the interpreted formula:

Xl 0(- Eb & Fa) Fx: x is foolish a: Socrates b: Plato.

This expresses the proposition that it is possible that both Plato not have existed and Socrates have been foolish. Let’s consider its evaluation accord- ing to the semantics, step-by-step:

(1) X1 is true iff the binary role @(b, a) satisfies 0(-Eb & Fa) under (‘b’, ‘a’). (I delete references to the interpretation, which is fixed.)

(2) X1 is true iff either (a) for some binary role R2 accessible to @(b, n), R2 satisfies - Eb & Fu under (‘b’, ‘a’), or (b) for some permutation p of 2, there is a role Sj, j < 2, accessible to p @(b, n) such that S’ satisfies -Eb & FQ under fand (Us-‘, . . .-, z+-~e-J, where (uo, ul) = (‘b’, ‘a’).

(3) 2a is true iff some such R2 satisfies both - Eb and Fa under (‘b’, ‘a’). But in order for R2 to satisfy - Eb , it must not satisfy Eb . Sl(B)

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tells us that R2 satisfies Eb under (‘6’, ‘a’) iff R2 includes V(E) (all the terms are already in proper order). But V(E) = 1’ , so R2 must include it (every binary relation includes existence). Hence, R2 satis- fies Eb, does not satisfy -Eb, and so 2a cannot be true.

(4) Xl is true iff 2b is true. 2b is true iff for some S’ accessible to a per- mutation of @(b, a),i < 2, S’ satisfies both -Eb and FQ under the appropriate readjustment of the ordering (‘b’, ‘a’). By Sl(A), no role So can satisfy I;h (since the ordering of terms for a zero-ary role is empty). Hence the truth condition is fulfilled only if it is fulfilled in the case j = 1. Suppose now that p is the identity permutation. To fulfill the truth condition in this way, we must have an S’ which satisfies -Eb and Fa under the ordering (‘b’) (since p-‘(O) = 0 when p is the identity). By Sl(A), it is impossible for S’ to satisfy Fa under an ordering which does not contain ‘a’. So to fulfill the truth- condition, p must be the permutation (0, 1). That is: Xl is true iff for some S’ accessible to (0,1) @(b, a), S’ satisfies both - Eb and Fa under (‘a’).

(5) (O,l)@(b, a) = @_(a, b). So X1 is true iff for some role S’ accessible to @(a, b),S’ satisfies both -Eb and Fa under (‘a’). By Sl(A), no 5” satisfies Eb under (‘a’), so any S1 satisfies - Eb under (‘a’). Thus X1 is true iff for some S’ accessible to @(a, b),S’ satisfies Fa under (‘a’). By Sl(B), Xl is true iff for some S’ accessible to @(a, b), S’ includes V(F), the property of being foolish.

(6) Xl is true iff for some unary role S’ accessible to the binary role of Socrates and Plato, S’ includes the property of being foolish.

?'.SOMELOGICALCONSEQUENCESOFTHESEMANTICS

The Barcan schema

0(3x)A 3 (3x)0.4, (not valid)

where “3” is an abbreviation for the material conditional truth-function, is not valid in L, This follows from the fact that there are nontrivial inter- pretations in which the domain of actual individuals is empty. By nontrivial, I mean that the formula 0(3x)Ex, where E is the existence predicate, is true under that interpretation. But since the domain of actual individuals is empty, (3x)OEx is false.

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The converse Barcan schema

(3x)&t 3 O(3x)A (not valid)

is also not valid in L. This is clear from the fact that (3x)0-& is true under some interpretations, but 0(3x)-Ex is true under none (it is impossible for there to exist something which doesn’t exist). Nevertheless, the schema is valid when restricted to atomic A :

(3x)VP’x 3 V(3x)P’x (valid)

Every property includes existence, so if something exists which possibly has the property denoted by “Pl”, then it is possible that something exist which has that property. There is also another valid variant of the converse Barcan schema:

(3x)O(Ex &A) 3 0(3x)4 (valid)

The characteristic schema of S4

oA 3 OOA (valid)

is valid in L, due to the transitivity of accessibility among roles (Al8(B)). The Brouwerian schema

A 3 nOA, (not valid)

which yields S5 when added to S4, is not valid in L. This can be seen from the instance: Ea 3 q OEa, where “u” stands for Socrates. Eu is true, but 00Ea is not. Since there is a possible world where Socrates does not exist, and since there Socrates is not such that he could exist (there he is not such that anything), it is not necessarily true that Socrates could exist, that is, 00&z is false. This consequence may seem bizarre, but I don’t think an actualist should try to evade it. There are no nonexistent possible individ- uals, according to actualism. But shouldn’t the actualist take this to be a necessary truth, that there indeed could not have been any nonexistent possible individuals? If so, it follows that had Socrates not existed, he would not have been such that he could exist - else he would have been a nonexistent possible.

The failure of the Brouwerian schema has important implications for the theory of identity. Since the role semantics assigns to the symbol “=” the relation I* , and since I* includes the property E of existence, “=” is a

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symbol for what Kit Fine calls strung identity. Consequently, if 9~” denotes a contingent individual under some interpretation, Oa #a is true for that interpretation. It is true because I have stipulated that only existing individ- uals can enter into the identity relation, yet the individual denoted by “8 might not have existed. There is nothing absurd about this consequence, since it holds in all strong identity systems. Fine points out, however, that in systems that include S5, we can use strong identity to define weak identity: x = *v =df Ox =y. l4 In such systems, weak self-identities are valid; there is no interpretation under which Oa f *a is true. But because of the failure to the Brouwerian schema, this maneuver is not available in the role semantics. If “LZ” denotes Socrates, then not only is Oa # CI true, but also Ooa f a, which is equivalent to Oa # *12.

The Brouwerian schema does hold for purely general statements, state- ments without constants or free variables:

A 3oOA “A” purely general (valid)

The failure of the unrestricted Brouwerian schema is a result of (1) the treatment of possible nonexistence selected at the end of Section 3 and (2) axiom Al 6(B), “No role is accessible to a role of lower rank”, which rules out complete symmetry of the accessibility relation. Since it is hard to make sense of the idea of a role being accessible to one of lower rank, it is un- likely that A16(B) can be modified. It seems feasible, however, to validate the Brouwerian schema by changing the treatment of possible nonexistence. If in considering a world where an individual does not exist we retain the individual’s original role as an index, then we can sensibly go on to speak of what is possible for that individual by making use of the index. I have chosen not to introduce this complication into the role semantics because I am not convinced of the intuitive validity of the Brouwerian schema.

8. CONCLUSION

While the logical consequences of the role semantics are interesting, they are not the only standard by which it should be judged. I have not con- structed the semantics simply to segregate the valid and invalid formulas of a first-order modal language. I have tried in addition to give a semantics which, for any given first-order modal statement, reveals the form of its truth conditions. In this endeavor, actualism is directly relevant. It rules out,

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for example, the usual Kripke semantics. A Kripkean interpretation includes a function which maps each member of a set of possible worlds to the set of objects existing in that world, objects which need not exist in our world. Thus truth-conditions are given in terms of a totality of all possible objects, including nonactual possibles, and no actualist can seriously countenance truth-conditions of this form. Thus an actualist should ultimately reject the Kripke semantics, even though, as I am ready to admit, the notion of validity which it yields is very nearly that which the actualist wants.”

But, one might persist, why is it necessary to have a correct opinion about the form of the truth conditions? Why not employ a Kripke seman- tics and not take its apparent ontology seriously? Why not view the seman- tics as an artificial construct, and the first-order statements themselves as an ultimate mode of expression?

A good semantics gives a comprehensive view of the facts which the object language is intended to express; in this case, of modal facts. It does not seem right to say that the first-order modal object language is an ulti- mate mode of expression. Certainly the object language L is not. For example, we cannot express within it: “There could have been a star larger than the largest star actually is.” Or: “There is an infinity of logical possi- bilities.” To be sure, we could enrich the object language. But assuming the existence of an adequate modal semantics, this piecemeal approach would leave us with a disturbing puzzle: Why is it that we can enrich the object language in just those ways that are sanctioned by the semantics? The answer, it seems to me, is that the semantics is not merely an artificial construct, that there are real entities which exhibit the given semantic structure .16

Of course, many questions remain concerning the intelligibility of modal predicate formulas, as well as the associated statements of natural language, statements of de re possibility. The role semantics tells us, for example, that “It is possible that Socrates have been foolish” is true if and only if some role accessible to Socrates’ actual role includes the property of foolishness, but it does not tell us whether there is such a role. It does not tell us whether wisdom is essential to Socrates, or indeed whether any interesting property is essential to any individual. It does not even give us a clue how to answer such questions.

My purpose in presenting the role semantics is relatively modest. I have argued that certain statements of iterated modality present a problem for

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actualist versions of possible worlds semantics. The role semantics over- comes that difficulty. I cannot claim that in the light of the role semantics, modal predicate formulas become perfectly intelligible, but they are a little more intelligible.”

Virginia Polytechnic Institute and State University

NOTES

r Plantinga, The Nature of Necessity, Oxford University Press, 1974, IV.l. * This objection, and the considerations following it, are presented in a more elabor- ate form in my ‘A Problem for Actualism about Possible Worlds’, 77ze Philosophical Review 92 (January, 1983). As I point out in that paper, three worlds are needed in the objection. It is compatible with actualism that there be a maximal possible state-of- affairs W which includes the stateofaffairs of there being someone who does not exist in the actual world. It cannot contain any instance of the stateof-affairs, but since the actualist denies the existence of any such instance, this does not contradict the maxi- mality of W. J Kit Fine thinks we can do without an extensional world theory. See his Postscript to Worlds, Times and SeZves by Fine and A. N. Prior, University of Massachusetts Press, Amherst, Massachusetts, 1977. Actually, Fine says that one of his world theories is “extensional”, because its quantifiers range only over individuals and sets (pp. 134- 135). But it is not extensional in the fullest sense, since it contains primitive inten- sional operators.

Also interesting in Christopher Peacocke’s attempt to do without world theory altogether and construct a homophonic modal semantics (‘Necessity and Truth Theories’, Journal of Philosophical Logic 7 (1978), 473-500). 4 A more thorough criticism of the essences view is contained in Robert Adams’ paper ‘Primitive Thisness and Primitive Identity’, Z%e Journuf of Philosophy 76 (1979), 5-26. Also see his critical study of Plantinga’s IIThe Nature of Necessity in Nous 11 (1977), 175-191. For a view that evades these criticisms, see Graeme Forbes’ ‘On the Philosphical Basis of Essentialist Theories’, Journal of Philosophical Logic 10 (1981), 73-99. On Forbes’ view, mere possibles are individuated by means of actual parts or ancestors. This has the unusual consequence, however, that there can be no mere possibles whose parts or ancestors are absent from the actual world. s A less convincing approach would be to retain standard modal semantics but con- strue it nonrealistically. That is, one might adopt a semantics which contains “dummy” objects, such as numbers, in place of nonactual possibles. This would allow the stan- dard treatment of iterated modalities. Problems are: (1) There is difficulty determining the required number and relationships of the “dummy” possibles, and (2) even if a nonrealistic semantics is constructed, we will want to transform it into a realistic semantics by “factoring out” its artificial features. For these reasons, I have chosen a more direct approach. 6 Plantinga, op. cit., 91. ’ Lewis, ‘Counterpart Theory and Quantified Modal Logic’, Journal of Philosophy 65 (1968), 113-126.

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* See Allan Hazen’s ‘Counterpart-Theoretic Semantics for Modal Logic’, Jour?ral of Philosophy 76 (1979), 319-335. 9 By similar reasoning, if there are no distinct possible situations which differ only in the roles which actual individuals have in them, then actual individuals should be reducible to qualitative properties and relations. I accept this consequence. However, I am inclined to believe that there are distinct possibilities of the sort described. lo In formulating this axiom and the next one, I was aided by Paul Hahnos’ article, ‘Algebraic Logic (II)‘, Fundamenfa Mafhemaricae 43 (1956), 255-285. I1 Axioms and definitions involving the plus-operations: Al 1 Individuals - For any individual x, there is an operation X* which maps each i-ary relation A’ to an i-l-ary relation x+A*. Such operatjons pbey:

(1) If j < i, then xc (A’ *B’) = x+Ai * Bi. (2) If j Q i, then

x+Ai * X+Bi = x+(~i * (j - 1, i - 1) (lf * Bj)), (3) x+oi = o’-1 (4) x+ 1’ = I’-’ * x+ 1’ . (5) y+x+A’= x+y* (i - 1, i - 2)Ai, (6) 3x+Ai=x’ 3 (i-l,i--)A’.

D2 A’ is qualitative iff either (1) A’ = O’, or (2) for any individual x, if lo does not in&de x+1’ , then A’does not include x+1’ .

Al7 Correspondence - (A) For any individuals x, , . . . I Xi-1 , an i-ary role Ri is accessible to,@(x,, . , . , xi-l) iff xi. . xl-, R’ f 0”. (e) A jary role S’, j < i, is accessible to @(x,, . . . , xi-,) iff xi. . .~j’-~ S’ # 0’ and for no xk such that j =G k < i does x0’. , . xl-l Si include xi 1’.

‘l For the sake of simplicity, 1 have not included the “actually” operator, A, in the language L. It is not a trivial matter to give a semantics for A. For example, suppose that we wish to evaluate a formula 0(3x) -AFx. In evaluating the part (3x)-AFx, we must retain the actual Ramsey world as an index (the “previous world”). Then, in assessing this existential formula, we must take account of all the various possible roles that the object x might have filled in the “previous world”, as well as the possibility that x Wed no role in that world. Despite these complications, however, I am confi- dent that a semantics could be given. I3 Example: The relation denoted in abstraction notation by “(hy)(hx)(hz)A’ yyxyzzx”is denoted in the theory of relations by “3333(2,4)=1[0, I][[(), 31 1[2,6]1[4,5] A’.” I4 Fine, op. cit., 133-134. Is See Kripke, ‘Semantical Considerations on Modal Logic’, Acra Philosophica Fennicu 16 (1963), 83-84. I6 Obviously I cannot defend this answer here. The contrary view, namely, that the modal semantics is of purely instrumental value, has many adherents. ” I am gratefuf to Graeme Forbes and Eleonore Stump for their comments on earlier versions of this paper.