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Metaphysical necessity: understanding, truth, and epistemology.

This paper presents an account of the understanding of statements involving metaphysical modality, together with dovetailing theories of their truth conditions and epistemology. The account makes modal truth an objective matter, whilst avoiding both Lewisian modal realism and mind-dependent or expressivist treatments of the truth conditions of modal sentences. The theory proceeds by formulating constraints a world-description must meet if it is to represent a genuine possibility. Modal truth is fixed by die totality of the constraints. To understand modal discourse is to have tacit knowledge of the body of information stated in these constraints. Modal knowledge is attained by evaluating modal statements in accordance with the constraints. The question of the general relations between modal truth and knowability is also addressed. The paper includes a discussion of which modal logic is supported by the presented theory of truth conditions for modal statements.

1. Problems and goals

The philosophical problem of necessity seemingly shares with the philosophical problem of consciousness and some problems in the philosophy of mathematics this distinction: that there is practically no philosophical view of the matter so extraordinary that it has not been endorsed by someone or other. In recent years, all of the following views have been taken of such a statement as "There could be a 500-floor skyscrapee': that its truth involves the existence of a genuine material 500-floor skyscraper which is spatially unrelated to any actual objects; that its utterance is the expression of a certain kind of imaginability; and that its correctness is dependent upon what conventions are in force. I will attempt to steer a middle course, developing a treatment which does not regard statements of necessity or contingency as made true by the mental states they express, or by any form of convention. But the account I propose does not buy the objectivity it claims at the price of postulating an inaccessible reality which determines the truth values of modal statements. The account aims to integrate the metaphysics and the epistemology of modality.

More specifically, I have two goals. The first is to give an account of what is involved in understanding discourse involving a necessity operator, where the necessity in question is what is usually called "metaphysical necessity". My second goal is to give an account of how propositions about what is necessary can be known. The two accounts must of course dovetail, and, on a truth conditional theory of meaning and understanding, they must do so very directly. Ways of coming to know that something is necessary must be appropriate to the kind of truth condition for statements of necessity which is underwritten by the account of understanding. Equally, the truth condition for statements of necessity must be one whose fulfillment is established by what we take to be ways of coming to know that something is necessary. When we have before us interlocking accounts of understanding and knowledge of necessities, we will be in a position to contrast them with the views of the modal realist and the theorist who holds that modality is mind-dependent.

In the past twenty years, significant advances have been made, particularly in the writings of Robert Adams and Robert Stalnaker, in elaborating a moderate view of modality which does not involve David Lewis's modal realism (Adams 1974; Stalnaker 1976, 1988). The views of Adams and Stalnaker differ from one another, but they agree in treating discourse about possible worlds as legitimate, and in explaining such discourse as talk (immediate or derived) about ways the world might have been. The possible worlds of their moderate view are, in David Lewis's terms, ersatz possible worlds. Their nonactual possible worlds are not things of the same kind as the actual spatio-temporal universe around you.

Anyone who wants to follow the middle course I described will find such a moderate view attractive. Does it give us an answer to the question of what it is to understand discourse involving necessity? The moderate view in itself -- as opposed to additional doctrines with which it might be combine -- simply takes for granted the notion of something's being a consistent (metaphysically consistent) way the world might have been. Indeed, Stalnaker himself goes further, and says "Lewis is right, I think, that if we reject modal realism, then we must give up on the project of providing a reductive analysis of modality" (1988, p. 123). Even if that is a little strong, it is certainly fair to say that nothing in the usual expositions of the moderate view gives any clue about what a reductive analysis of modality would be like. Now in other cases, the fact that an expression has to be treated as primitive is no bar to our stating what is involved in understanding it. The problem is rather that it is not at all apparent how any of the various extant accounts of understanding other primitive expressions provides a model which we could follow, if we aim to provide an accurate and credible treatment of the concept of necessity.

Some primitive concepts require for their possession that certain of the judgments involving them are, in favourable circumstances, appropriately caused by the holding of the content judged. Observational concepts, and possibly some others, fall in this class. It seems to be impossible to assimilate the concept of metaphysical necessity to this case. Judgements, like any other actual events or states, can be explained only by what is actually the case. The fact that something is necessarily the case is never part of the causal explanation for some temporal state of affairs. That is what is right in a quasi-Humean principle that there is no impression from which the idea of metaphysical necessity could be derived. When the word "of" bears a partially causal sense, an impression -- something impressed by the way the world is-cannot be of a metaphysical necessity. Any concept individuated in part by the fact that certain judgements involving it are responses to its instantiation would, it seems, fall short of being the concept of metaphysical necessity. The distinctively modal character of the concept could not have its source in that sort of causal role.

A second kind of model for understanding a primitive expression is provided by the logical constants. The condition for understanding a logical constant is plausibly given by alluding to some kind of grasp of certain introduction and/or elimination rules involving it. If, though, we take the inferential rules for necessity in some specific modal system, such as K or T, they are far from uniquely satisfied by the pre-theoretic concept of metaphysical necessity. They will for instance equally hold for certain notions of provability.1 Now it is true that various notions of provability will have their own distinctive axioms which distinguish them from metaphysical necessity. So could we perhaps develop an approach under which metaphysical necessity is characterized as the weakest operator meeting certain conditions? To this, though, there are three objections.

First, if we are really to capture the concept of metaphysical necessity, we will have to include such celebrated Kripkean principles as those of the necessity of origin and of constitution. The problem is not only that these do not appear to be broadly logical in character, but also that it is implausible that our understanding of metaphysical modality is simply list-like, given by a list of principles correct for metaphysical necessity. We seem to have some understanding of metaphysical necessity upon which we can draw to work out that the Kripkean principles are correct. The class of true general principles of metaphysical necessity is potentially open-ended.

Second, the point just made about the Kripkean principles actually applies to the more "logical" ones like "From the premisses necessarily p and necessarily if p then q, it follows that necessarily q". It seems that we have some understanding of metaphysical necessity from which it can be worked out that this is a valid principle of inference. If that is so, the principle should not just be stipulated as a primitive rule, part of an implicit definition. Rather, it should be derived from whatever is involved in our more fundamental understanding of metaphysical necessity, from whatever it is which allows us to work out the correctness of the principle.

Third, even if (perhaps per impossible) we had a notion of provability which precisely coincided with that of metaphysical necessity, we would still not have fully answered the questions about understanding metaphysical necessity. It is one thing to say that a proposition is provable; it is another to say that it is necessary. Saying that die proposition is provable is not yet to say that it holds in other possible worlds. It is only to say that its actual truth is establishable in a special way. There must always be a further task of showing that establishability in the special way ensures necessity. In carrying out that further task, we must be drawing on some feature of our understanding of necessity which in the nature of die case goes beyond anything in our understanding of provability.

Given that necessity operators behave somewhat like universal quantifiers, it may also be tempting to elucidate the understanding of metaphysical necessity in a way which follows the general pattern for universal quantifiers. The pattern for universal quantifiers I myself favour is one in which the distinctive feature of understanding a universal quantification is the range of commitments a thinker incurs in accepting it. So one might say that metaphysical necessity is that operator O understanding which involves the thinker's meeting this condition: that in judging Op he incurs the commitment that in any possible state of affairs, p. These possible states of affairs can be ersatz possible worlds. It seems, though, that a thinker can satisfy this condition for understanding only if he already has the concept of possibility. Only the failure of p to hold in a genuinely possible state of affairs should make a thinker give up his acceptance of "Necessarily p". The thinker must have some grasp of which specifications of states of affairs are relevant counterexamples to a claim of necessity, and which are irrelevant, if he is to possess the concept of necessity. But if we are taking grasp of possibility for granted, how is that to be explained without adverting back to something which involves the thinker's grasp of necessity? If we are allowed to take the thinker's grasp of possibility for granted, we might just as well have defined necessity in terms of possibility. To pass the burden of explanation back and forth between the concepts of necessity and possibility is not to answer the question of the nature of understanding in any explanatory fashion.

At this point, a theorist may object that what I am seeking is something it is not necessary to find. If we make optimal total sense of someone by interpreting him as meaning necessity by a given operator, then he has the concept of necessity, and we should not look for anything explanatorily more fundamental. Similarly, if another person's beliefs about necessities are reached by methods we recognise as ones which, when applied by us, yield us knowledge of necessities, then the other person knows the necessities in question too. Why should we ask for more?

I make two points in response. First, the following question seems to be in order: are our methods of reaching modal beliefs enough to justify belief in their contents, and if so, why? The question is neglected by the objector when he says, in giving his own position, that we should not ask for more. It does not require any heavy-duty general theory, or loaded assumptions about content, to accept the genuineness of the question of whether modal realism, or some form of non-cognitivism, or something else, is correct. The same applies to the corresponding questions of whether our methods of fixing modal beliefs are appropriate to the species of content identified as correct by a philosophical account. In asking these questions, we are asking questions of a kind which we can legitimately raise and address for any kind of content.

Second, I should emphasise that the tasks of explaining the understanding and epistemology of necessity do not presuppose some reductive account of meaning and content. It is true that for someone who believes that a concept is individuated by its possession condition, the issues could be formulated as ones concerning the nature of the possession condition for necessity and its bearing upon the epistemology of modality. But such a position on concept-individuation is not compulsory for the present enterprise. Even if the account of the role of a concept in thought can never exhaust what is involved in its identity, one can still raise these questions: "What would be the nature of a non-reductive account of what is involved in understanding necessity-discourse? What account of truth for modal statements would mesh with this account? And how would modal knowledge be possible on the correct account?" Here I am indeed presupposing that there is a connection between what are, a priori, good reasons for judging a content, and what is involved in understanding a sentence expressing that content. But this presupposition does not involve even the weakest reductionism.

I will address the issues in the following order. In the next two sections, I build up a positive account of the truth conditions of modal sentences and Thoughts, an account which involves neither modal realism nor thinker-dependence ([Subsection] 2, 3). I go on to discuss the relations between this account and modalism ([Sections] 4). [Sections] 5 outlines an epistemology for modal contents which aims to dovetail with the positive account of their truth conditions. The more technical questions of which modal logic is supported by the truth conditions, and of the effects of relaxing some of my simplifying assumptions, are addressed in Appendices A and B.

2. Admissibility, the principles of possibility, and the Modal Extension Principle

Let us return to the commitment model of understanding necessity. The problem we noted was that it presupposed the thinker's grasp of a distinction between those states of affairs which are possible, and those which are not. Suppose we could give a substantive account of what is involved in a state of affairs being possible. We might then argue that understanding a necessity operator involves some grasp of this substantive account of possibility. The problem we noted would then be overcome. That will be my strategy. I will try to state conditions which must be met if a specification of a state of affairs is to be a specification of a genuine possibility. I call these conditions "principles of possibility". The approach aims to identify a set of substantive principles of which it is true that a specification represents a genuine possibility only if it respects these principles.

These specifications are similar to the "ways for the world to be" discussed by moderate modalists (Salmon 1989, esp. pp. 5-17; Stalnaker 1976). There are ways the world might be, there are ways the world is, and there are ways the world could not be. My claim is that someone who understands modal discourse has a form of implicit knowledge of the principles of possibility. It is this implicit knowledge that allows him to discriminate between those ways which are ways the world might be, and those ways or specifications which are not. More generally, someone who understands modal discourse applies this knowledge of the principles of possibility in evaluating modal claims.

I begin by explaining in more detail two auxiliary notions I shall be using, that of an assignment, and that of an admissible assignment. The point of introducing the notion of admissibility is that it is a step towards. the elucidation of genuine possibility. Genuine possibility and admissibility are, I propose, related thus:

A specification is a genuine possibility if there is some

admissible assignment which counts all its members as true.

Assignments here are not assignments to uninterpreted schematic letters. Since specifications are given by sentences built from a set of meaningful expressions, or by Thoughts, our concern cannot be just with the uninterpreted. We have a choice as to whether to proceed with assignments to meaningful linguistic expressions, or to concepts which are built up into Thoughts. I will consider assignments to concepts. The changes needed to treat expressions are straightforward. An assignment s, then, does the following:

(i) s assigns to each atomic concept -- whether singular, predicative,

an operator on thoughts, or a quantifier -- semantic value of the

appropriate category. We will take the semantic values to be the

kinds of entity Frege assigned as the level of reference. So s

assigns an object to a singular concept; it assigns to a monadic

predicative concept a function from objects to truth values; and so

forth. Subject only to clause (iii) below, there are no restrictions

on what the assignments may be, provided that the category is

correct. We can call the assignment which s makes to an atomic

concept C "the semantic value of C according to s", and write it

"val(C, s)".

(ii) An assignment has an associated domain of objects, which is the

range of the quantifiers under that assignment.

For many purposes, these first two clauses are all we need in the notion of an assignment. They allow us, derivatively, to use the notion of the semantic value, according to assignment s, of a complete Thought. A complete Thought is built up from atomic concepts. The semantic value, according to assignment s, of a complete Thought is determined in the standard way from the semantic values of that Thought's constituent atomic concepts. Since we have set things up in a Fregean mode, the semantic evaluation of a complex Thought or concept from the semantic values of its constituents is just a matter of the application of a function to an argument (or n-tuple of arguments). If the semantic value, according to s, of the concept C is the function f, and the semantic value, according to s, of the singular concept is the object x, then the semantic value of the Thought Cm according to s is just the value of the function f for the argument x. If we were in a mood for formal and explicit characterizations, we could give a general inductive definition of the semantic value, according to a given assignment, of a complete Thought.

For any given assignment, there is its corresponding specification: the set of Thoughts it counts as true (maps to the truth value True). The notion of the specification corresponding to an assignment will play an important part in what follows.

Sometimes it is important to consider properties and relations of a sort which cannot be identified with anything at the level of Fregean semantic values. There are several good reasons for acknowledging properties and relations, understood as distinct both from concepts and from Fregean semantic values (Putnam 1970). For example, the most convincing way to understand the role of an expression occupying the position of F in a true statement of the form "The fact that x is F explains such-and-such" is as involving reference to a property. So we add:

(iii) An assignment may assign such properties and relations to atomic

predicative and relational concepts respectively.

When there are such assignments, assignments must also specify the extensions of these properties and relations. So we need several additional pieces of notation. When we are considering assignments of properties and relations, let us continue to use val(c, s)" for the semantic value at the level of Fregean reference. This must be distinguished from the property-value of C according to s, which we write "propval(C, s)". The property-value of a monadic concept according to an assignment must be a monadic property; for a binary concept it must be a binary property; and so forth. Finally we must also specify, for a given assignment s the extension of property P according to s. For a monadic property, this will be a function from objects to truth values; for a binary property, it will be a function from ordered pairs of objects to truth values; and so forth. We also stipulate that assignments must be such that these three notions, of semantic value, property-value and extension under an assignment be related in the intuitive way, viz. that the semantic value of a concept under a given assignment be the same at the extension of its property-value under that assignment. That is, for any C, s and f, we have that

val(C, s) = f iff ext(propval(C, s), s) = f.

(I use a notation that would have made Frege blanch.)

When assignments deal with properties and relations, we can correspondingly explain a notion of the truth value of a singular proposition, built up from a property and an object, according to a given assignment. The truth value of the singular proposition Px according to the assignment s is True iff the extension of P according to s maps x to the True. Similarly we can explain the notion of the truth value, according to an assignment, of a singular proposition consisting of an n-place relation and an n-tuple of objects. We can then extend the notion of the specification corresponding to an assignment. We can allow this specification to include not only the Thoughts, but also the singular propositions, which the given assignment counts as true.

I am initially going to make two simplifying assumptions about assignments, solely for the purpose of allowing us to walk before we run. The first assumption is that the assignments are total, for some presumed background range of concepts, properties and objects. Any given assignment, under this supposition, assigns a semantic value to every atomic concept in the background range. The reasons for wanting eventually to relax this assumption need not stem only from a questioning of bivalence. Even one not disposed to question it can still recognize that our ordinary talk of possibilities does not involve treating them as total at all -- otherwise we would hardly talk, as we actually do, of the possibility that it will rain tomorrow.

The second initial simplifying assumption is that assignments are restricted to Fregean semantic values. We will, to start with, not consider property-values. The issues involved in relaxing these two simplifying assumptions are considered in Appendix B.

In the standard Kripke-style model-theoretic semantics for modal logic, as we noted, assignments are made to uninterpreted expressions. In making assignments to uninterpreted expressions, rather than exclusively to meaningful expressions or interpreted sentences, the standard semantics does not deal, even indirectly, with the question of which Thoughts are true at a given world. The standard model-theoretic semantics does not need to. There are good reasons for saying that it is unnecessary to use the notion of which Thoughts or interpreted sentences are true at a world when one's fundamental concern is with the modal validity of a schematic formula. Since the atomic predicate letters and individual constants of the schemata do not have a sense, there is no question of respecting any constraints flowing from the concepts they express. But when our concern is with modal truth, rather than modal validity, matters must stand very differently. We may be interested in the question of whether the sentence or Thought that all material objects have a certain property, or that all conscious beings are thus-and-so, or that all fair arrangements are such-and-such, are each of them necessary or not. When we are asking such questions, we cannot evade the issue of whether it is genuinely possible for the concepts in these thoughts to have certain extensions in given circumstances.

In the case of those expressions of a modal logic which are not schematic, that is the logical constants, it is particularly clear that the standard modal semantics takes for granted points which a philosophical treatment should explain. For instance, the standard semantics takes for granted that if either A is true with respect to a possible world w, or B is true with respect to w, then the sentence "A or B" is true with respect to w. There is no gainsaying the correctness of the principle. But its correctness is in no way explained by the model-theoretic semantics, which simply writes that very same principle into the rules for evaluating complex formulae with respect to a world. If, however, someone asked why that rule of evaluation is correct, for any arbitrary possible world, he would not find any answer within the modal semantics. A philosophical account must attempt to answer this questioner.

The question is not one which arises only for the logical constants. It is a special case of the general question: what determines the restrictions on semantic values for given concepts and meaningful expressions at genuinely possible worlds? A correct answer to this general question should have as a special case an answer to what determines the conditions to be satisfied, at genuinely possible worlds, by the semantic values assigned to sentences and Thoughts containing the logical constants.

We will seek in vain for an answer to these questions -- even for the restricted case of the logical constants -- in the standard semantical theories of absolute truth, that is truth not relativized to a model. If the theory or definition is concerned with absolute truth, rather than truth relativized to a model, then it is not in itself, without supplementation, speaking to the question of which models correspond to genuine possibilities. Some theories of absolute truth do, certainly, use modal notions, and have as axioms such principles as "Necessarily, any sentence of the form "A or B" is true iff either A is true or B is true". Our question, however, is why this principle is correct -- we are looking for a rationale for the principle, not just a use of it.

On the other hand, if we turn to the notion of a model which has been developed for the elaboration and investigation of logical consequence, we also draw a blank in attempting to answer our imagined questioner. It is indeed true that intuitive expositions of the notion of logical consequence employ modal notions. This is true of Tarski's own intuitive expositions, which use the word "must". For instance, Tarski says of a sentence A which "follows in the usual sense" from a certain set of other sentences that "Provided all these sentences are true, the sentence A must be true" (1983, p. 411).(2) However, the models used in discussing logical consequence are confined to those in which varying assignments are made only to the non-logical vocabulary. It is indeed the case that A v B is true in any model iff either A is true in it or B is true in it; but this is simply a consequence of not varying the assignments to the logical vocabulary (or replacing them with variables, in the fashion of Tarski). The schema "If A, then A v B" would not be true in all models if we varied assignments to "v", as we do with the non-logical vocabulary. Equally, if we varied even less, a wider class of schemata would be true in all models. If we had some additional reason for saying that the usual kind of models employed in the investigation of logical consequence correspond to precisely the genuine possibilities, we might then have a rationale for the usual principles for evaluating formulae at non-actual worlds. But that additional reason would then be doing all the work in explaining the bounds of genuine possibility.

When, following the work of Kripke and Kaplan, we sharply distinguish the a priori from the necessary, it also becomes in any case much less plausible to try to explain aspects of metaphysical necessity by relating them to notions designed to elucidate validity. We expect validity to be a relatively a priori matter. An inference such as "From p, it follows that Actually p" is a priori. It is not necessarily truth-preserving. The demands of a theory of validity, then, are going to be rather different from those of a theory of metaphysical necessity. This makes model theory conceived as in the service of a theory of validity a less promising resource for the philosophical elucidation of necessity.

The positive strategy I said I would follow is that of trying to characterize the admissibility of an assignment in such a way that, for each genuinely possible specification, there is some admissible assignment which counts as true all the Thoughts (and propositions) in that specification. The goal is, in characterizing admissibility, to give some explanation of why an assignment which, for instance, assigns the classical truth function for conjunction to the concept of alternation is not an admissible assignment. If we can do this properly, we will have answered our imagined questioner.

The first constraint on admissibility -- my first Principle of Possibility -- is one which is basic to my whole approach. It is a two-part principle which I call the Modal Extension Principle. The name is doubly appropriate. First, the Modal Extension Principle constrains the extension a concept may receive from an assignment, if that assignment is to be admissible -- it gives a necessary condition for admissibility. Second, the Modal Extension Principle also extends to genuinely possible specifications the way the extension is fixed in the actual world. As always, the Principle can be stated either for concepts or for expressions; I will state it for concepts, in step with our practice so far. The Principle is best introduced by giving examples of assignments which would be in violation of the Principle.

Consider an assignment s which treats the concept of logical conjunction, &. Suppose s assigns to & a function which does not, when applied to the truth values True and False (say), taken in that order, yield the truth value False. Then s would not be assigning to & a semantic value which is the result of applying the same rule for determining the semantic value, according to s, of Thoughts containing & as is applied in determining the semantic value of Thoughts containing & in the actual world. For this reason, s would be counted as inadmissible by the Modal Extension Principle.

For a second example, consider the hoary case of the concept bachelor. Let us take it that the way the semantic value of this concept is fixed in the actual world is by taking the intersection of the concepts man and unmarried. Now consider an assignment s which is such that val(bachelor, s) is not the same as the intersection of val(man, s) with val(unmarried, s). This assignment would not be applying the same rule for determining the semantic value of bachelor as is applied in determining its semantic value in the actual world. Again, this is a violation of the Modal Extension Principle.

As a third example, consider the case of a binary universal quantifier ??x(Fx, Gx), meaning that all Fs are G. Suppose we have an assignment s with a domain which includes the three objects x, y, z. s gives the concept planet a semantic value which maps just these three objects to the True. It also assigns to spherical a semantic value which maps these three objects to the True. Now consider the Thought All planets are spherical. Under these suppositions, s is admissible only if the function s assigns to this universal quantifier ?? maps the pair of functions assigned to planet and to spherical respectively to the truth value True. If the assignment s were not to do so, it would not be applying the same rule in determining the semantic values, according to s, of Thoughts containing ?? which is applied in determining the actual truth value of Thoughts containing ??. That rule is just that the extension of F be included in that of G. So if s is as described, we would again have a violation of the Modal Extension Principle.

As a fourth illustration of the Main Part of the Modal Extension Principle, consider an assignment s which assigns to the concept F a function which maps only the particular object x, and nothing else, to the truth value True. Now consider the function which the same assignment s assigns to the definite description operator [Iota]. Suppose this function does not map the function s assigns to F to the unique object x. Then the assignment s would not, according to the Main Part of the Modal Extension Principle, be admissible. For the way in which the semantic value of a definite description [Iota]x(Fx) is fixed in the actual world is by applying the rule that it refers to an object iff that object is the unique thing which is F.

We are now in a position to state the Main Part of the Modal Extension Principle which is violated in these examples. This Main Part is concerned with concepts which are not, to extend the terminology of Kripke (1980) and McGinn (1982), de jure rigid. In cases of de jure rigidity, as Kripke explains the distinction, "the reference of a designator is stipulated to be a single object, whether we are speaking of the actual world or of a counterfactual situation". This contrasts with "mere `de facto' rigidity, where a description `the x such that Fx' happens to use a predicate `F' that in each possible world is true of one and the same unique object (e.g. `the smallest prime' rigidly designates the number two)" (1980, p. 21, fn. 21). Kaplan's operator "dthat" (1978) is another example of a de jure rigid expression, and so too would be the concept it expresses. The distinction between what is de jure rigid and what is not begins to bite only for expressions or concepts for which variation of reference between possible specifications is in question.

With all these preliminaries out of the way, we are at last in a position to formulate the Modal Extension Principle.

Modal Extension Principle, Main Part:

An assignment s is admissible only if: for any concept C which is

not de jure rigid, the semantic value of C with respect to s is the

result of applying the same rule as is applied in the determination

of the actual semantic value of C.

What, in the general case, is the rule which, when applied to the way things actually are, yields the semantic value of a concept? On the theory I once favoured, the rule is given by what I called the determination theory for the concept in question (Peacocke 1992b, ch. 1). On that approach, the determination theory takes the material in the concept's possession condition, and says how something in the world has to stand to it if it is to be the concept's semantic value. In the case of an empirical concept, it will be an empirical condition something has to satisfy to be its semantic value. On a different theory, the semantic value of a concept is determined by an implicit conception governing the concept, and the content of that implicit conception is what an assignment has to respect if it is to be admissible. But the details and general presuppositions of my own particular approach to concepts are not required by the present treatment of possibility. Whatever may be your favoured theory of how the actual semantic value of a concept is fixed can be used, in combination with the Main Part of the Modal Extension Principle, to formulate a constraint on the admissibility of an assignment. Provided that we can make some sense of the notion of the way the semantic value of a particular concept is fixed in the actual world, the Modal Extension Principle can get off the ground. (I will return at the end of [sections] 4 to consider whether this notion of a way in which the semantic value is fixed is a notion which is sufficiently independent of the thinker's understanding of modality to be used in its explanation.)

For someone who thinks that we can in fact make no sense of the idea that some particular rule contributes to the determination of the semantic value of some concept in the actual world, the Modal Extension Principle does not formulate any substantial constraint on admissibility, nor, therefore, on possibility either. The apparatus and theses I am developing are entirely dependent upon the applicability of such a notion of a rule. It is no accident that those who have been sceptical about the intelligibility of any such notion of a rule have also tended to be sceptics of one stripe or another about the notion of necessity. Rules and necessity sink or swim together.

The second part of the Modal Extension Principle deals with concepts which are marked as rigid. The second part does not amount to much more than an application of the definition of de jure rigidity.

Modal Extension Principle, Second Part:

For any concept C which is de jure rigid, and whose semantic

value is in fact A, then for any admissible assignment s, the semantic

value of C according to s is A.

As before, both parts of the Principle should be understood as generalized to arbitrary categories of concept and their appropriate kinds of semantic value.

Alarm bells may have been ringing in the reader's mind for several paragraphs. If our aim is to give an account of the nature of genuine possibility, how can we help ourselves to the notion of rigidity? Must not the notion of rigidity directly or indirectly involve the notion of genuine possibility, either by way of the notion of a possible world, or via that of a counterfactual situation, or even via some notion of the truth conditions for sentences or Thoughts containing an operator for metaphysical necessity? I reply that I have alluded to the notion of de jure rigidity as a heuristic device for drawing attention to the class of concepts to which the Second Part of the Modal Extension Principle applies. My claim is that there is a class of concepts (and expressions) grasp (or understanding) of which involves some appreciation that, in their case, an assignment is admissible only if it assigns to each one of them its actual semantic value. That a particular concept or expression is in this class is something which has to be learned either one-by-one for the concept or expression, as with Kaplan's "dthat", or has to be learned by the concept's membership of a general class for which this constraint on admissibility holds, such as the classes of demonstratives, indexicals and proper names.

Of course when we are considering the Modal Extension Principle as formulated for expressions, and a particular expression is such that, in the language, only a semantic value and not a concept is associated with it, only the Second Part of the Modal Extension Principle can apply. It is no accident that proper names are de jure rigid.

The conception I am in the course of outlining sits well with the view that the distinction between de jure rigid expressions and the rest can, in the general case at least, be elucidated only in combination with some elaboration of modal notions. If we look at the role of expressions only in non-modal contexts, it seems that in some cases at least we could not distinguish in respect of sense between rigidified and non-rigidified versions of the same expression or concept (cf. Evans 1979, p. 202). The difference between "the capital of the United States" and the Kaplanian "dthat (the capital of the United States)" would be one such pair. The same goes for "the F" and "the actual F", for any predicate F. Expressions in each of these pairs do not differ in their contributions to cognitive significance of non-modal sentences in which they occur. If there are such pairs of cases, then it is not possible to regard the boundary between the de jure rigid and the rest, which is employed by the Modal Extension Principle, as something whose extension is already implicit in grasp of the concepts and expressions when they are employed in non-modal contexts. As their respective semantical clauses would lead one to expect, an account of what is involved in understanding "actual" and Kaplan's "dthat" can be given only simultaneously with, and not as prior to, the thinker's grasp of modal notions.

With this apparatus in place, we can return to doing some more substantial philosophy, and consider some very simple philosophical applications of the Modal Extension Principle.

(a) We can establish, without begging the question, the metaphysical necessity of (for instance) truths of propositional logic. Consider the propositional logical constants. Suppose it granted that these constants have truth functions as their semantic values. Suppose also that you hold one of two theories. One theory is that the truth function for any given constant is fixed as the one which makes certain inferential principles always truth-preserving. These are the principles a theorist might say are mentioned in its understanding condition. Here "always truth-preserving" does not mean something modal. It means "truth-preserving under all assignments of truth values to schematic letters", in the way in which, for example, the classical truth function for conjunction is the only truth function which makes truth-preserving the classical introduction and elimination rules for conjunction. Or suppose, alternatively, you hold that understanding a logical constant involves having an implicit conception stating the truth conditions of Thoughts or sentences containing it in terms of the truth conditions of its constituents.

On each of these theories, the semantic value of a logical constant with respect to an admissible assignment will always be the same as its actual semantic value. Under the first of the two theories I mentioned, this point holds because a logical constant's semantic value under any admissible assignment must be the one which makes certain principles of inference truth-preserving. The same semantic value will make them truth-preserving whatever the nature of the assignment. So the principles mentioned, according to this first kind of theory, in the understanding-condition for the logical constant will be truth-preserving in any admissible assignment. It follows that the inferential principles which individuate the meaning of a logical constant, such as the introduction and elimination rules for conjunction, are metaphysically necessary. The same result follows even more straightforwardly under the theory that the rule by which the semantic value of a logical constant is determined is given by the content of some implicit conception such as "A or B is true iff either A is true or B is true". To assign a semantic value -- a truth function -- to or other than the one fixed by the content of this rule would thereby be to depart from the way the truth value of alternations is actually determined.

For the propositional logical constants, this is the answer I offer to our imagined questioner, who asked how we could justify the usual principles for evaluating, at an arbitrary possible world, formulae containing propositional logical constants. The metaphysical necessity of such principles is a consequence of what is involved in a specification's being possible, when possibility is explained in terms of admissibility, and when the explication of possibility involves the Modal Extension Principle.(3)

It is crucial to note that this rationale neither explicitly, nor tacitly, relies on the premiss that the relevant inferential principles, on the first theory, or the content of the implicit conceptions, on the second theory, are metaphysically necessary. That those principles are necessary is the conclusion of the argument, not its premiss. In the rationale, reliance is rather being placed on the point that part of what makes an assignment admissible is its conformity to the Modal Extension Principle. Hence a respect for certain inferential principles, or alternatively the content of implicit conceptions, is guaranteed by the Modal Extension Principle. Conformity to the Modal Extension Principle helps, via the connection proposed between admissibility and possibility, to fix which specifications are genuinely possible. The Principle, and other principles of possibility are, on the conception I am offering, antecedent in the order of philosophical explanation to the determination of what is genuinely possible.

If we suppose that we have some conception of the genuinely possible specifications elucidated without reliance on the Modal Extension Principle, I think it will (rightly) seem to be an impossible task to explain why certain primitive logical principles are metaphysically necessary without some kind of question-begging. The impossibility of that task, according to the present approach, results from a wrong way of looking at the problem. We can explain why conformity to the primitive logical principles helps to fix which specifications are genuinely possible, rather than having some independently understood notion of possibility for which we then have to explain why genuine possibilities respect those logical principles.

Logical concepts contrast sharply, in respect of this first point, with many empirical concepts. in the case of the empirical concept expensive, say, nothing in the conditions on admissibility formulated so far, nor any to follow, excludes an admissible assignment s in which the concept expensive has an extension different from its actual extension. Such a difference in extension would be entirely in accordance with the Modal Extension Principle if s also assigns semantic values to other concepts -- in particular those on which truths about relative prices depend.

(b) One of our introductory illustrations of violations of the Modal Extension Principle supported the claim that no admissible assignment will ever count anything of the form a is a bachelor and a is married as true. Hence, no specification corresponding to an admissible assignment will count such Thoughts (or their corresponding sentences) true. This places me in disagreement, by implication, with one of David Lewis's claims. He says that it is a serious objection to what he calls ersatz possible world -- possible worlds which are not things of the same kind as the actual world -- that an approach involving them must take modality as primitive (1986, p. 150ff). In particular, he claims that a theorist who endorses ersatz possible worlds and tries to explain them in terms of logically consistent sets of propositions "would falsify the facts of modality by yielding allegedly consistent ersatz worlds according to which there are unmarried bachelors, numbers with more than one successor, and suchlike impossibilities" (p. 153). Using sets of propositions which are consistent in a strict logical sense is, though, not the only way of developing a treatment of modality with ersatz possible worlds. The worlds I have been speaking about are certainly ersatz worlds in Lewis's classification -- they are nothing more than sets of Thoughts and/or propositions; and I noted earlier that the present approach does not need to be reductive of modality. But I have also just argued that the Main Part of the Modal Extension Principle does rule out the existence of possible worlds at which there are unmarried bachelors. Only by not applying the same rule which determines the actual extension of bachelor can an assignment count something of the form a is a bachelor and a is married as true. So we can meet part of Lewis's objection to ersatz possible worlds if we accept the Modal Extension Principle. A theory employing ersatz worlds can and must go beyond logical necessity in any strict or narrow sense having to do only with the logical vocabulary. More generally, I think we should also draw the conclusion that an account of what other worlds are possible must draw on information about how concepts come to have their semantic values in the actual world.

The Main Part of the Modal Extension Principle can be seen as an implementation and generalization of the widely held intuitive point that we should not, when concerned with fundamental philosophical explanation, regard the one-place concept x is happy as just a special case of the two-place relativized concept x is happy in world w, viz., the special case in which w is assigned the actual world as its value. The two-place relativized predicate is not explanatorily prior. Rather, world-relativized concepts have a general relation to their unrelativized versions, and there is a general rule stating how the extension of the two-place x is happy in w is fixed from the rule determining the actual extension of the predicate x is happy. The Main Part of the Modal Extension Principle is just such a general rule.

Whether the meaning of the unrelativized predicates is absolutely prior must depend in part on the resolution of a metaphysical issue about the actual world -- the issue of whether it is to be conceived in a fully nonmodal fashion, or in a partially modal fashion. (There will be more on this in [sections] 6 below.) The Main Part of the Modal Extension Principle does, however, exclude the hypothesis that understanding world-relativized concepts is absolutely prior to understanding unrelativized versions. It is, then, in apparent contrast with the view of Hintikka (1969), who holds that for first-order languages "'meanings' are bound to be completely idle", and that "in order to spell out the idea that the meaning of a term is the way in which its reference is determined we have to consider how the reference varies in different possible worlds, and therefore go beyond first-order languages" (p. 93). If the Modal Extension Principle is correct, the meanings for the expressions in the first-order language, far from being idle, actually serve to determine the world-relative extensions.

This brings us to the more general issue of the correct way to conceive of the relation between meaning and de dicto necessities. In some earlier work on the a priori, like many other writers I rejected the applicability of the notion of truth purely in virtue of meaning. But I did defend the idea that meaning can explain the special status of a priori truths (1993). I argued that a priori truths are ones whose truth in the actual world can be derived from the understanding-conditions (and associated determination theories) for their constituent expressions. An analogous link between meaning and the a priori arguably still holds if we replace possession conditions with implicit conceptions. So the question arises: is there any analogous link between meaning and necessity?

The idea that meaning can explain the special status of necessary truths has historically certainly been found attractive. Not surprisingly, the temptation to succumb to that idea has been strong in times in which the notions of necessity, the a priori and analyticity were not sharply distinguished. Carnap, for instance, in Meaning and Necessity (1956), offered "L-truth" as an explication of necessity. He wrote that "A sentence [Sigma] is L-true in a semantical system S if and only if [Sigma] is true in S in such a way that its truth can be established on the basis of the semantical rules of the system S alone, without any reference to (extra-linguistic) facts" (1956, p. 10, second edition, notation altered). So can we give a qualified endorsement of the view that meaning plays a special role in the explanation of certain de dicto necessities, analogous to the way in which the treatment of the a priori gave some qualified endorsement of the view that meaning plays a special role in the explanation of the a priori?

The present approach suggests a positive answer. I already noted that the necessity of certain de dicto sentences is derivable from the Modal Extension Principle for expressions, together with whatever feature of the meaning of those expressions determines their extensions. Just as the treatment I earlier endorsed of the a priori the present does not legitimate any application of the notion of truth-purely-in-virtue-of-meaning, so the present explanation of de dicto necessities does not underwrite the applicability of the notion of necessity-purely-in-virtue-of-meaning. It would hardly be consistent to do so. Since necessity entails truth, any endorsement of necessity-purely-in-virtue-of-meaning would require endorsement of at least some cases of truth-purely-in-virtue-of-meaning. There is no endorsement in the present framework of the application of necessity-purely-in-virtue-of-meaning, because logic, the Modal Extension Principle and other principles are used in the derivation of the necessity of a de dicto truth. Everything I have said is consistent with the principle that what makes a statement of the form Necessarily A true is just its truth condition. The substance of what I have said lies in giving some necessary and sufficient conditions for that disquoted truth condition to hold.

(c) The Modal Extension Principle does not count all a priori principles as metaphysically necessary. I mention two cases familiar from discussions in modal semantics.

Suppose that the whole set of principles of possibility ensures that there is a possible world -- genuinely possible specification -- at which p is not true. Suppose also that "actually", and equally the concept it expresses, is marked as a rigidifier. "If (actually p) then p" will be a priori. By the Second Part of the Modal Extension Principle, however, it is not necessary.

Another example, made familiar by David Kaplan's writings, is given by "I am here now". The thought that I am here now is a priori but is not necessary. The present apparatus secures this result, given that "here" and "I" are de jure rigid, and given the Second Part of the Modal Extension Principle. Provided the whole set of principles of possibility ensures the existence of a world at which you are somewhere else now, your present thought "I am now here" will be false with respect to some genuinely possible specification, but has an a priori status (at least if preceded with the antecedent "If this place here exists").

3. Other principles of possibility and the truth conditions of

modal statements

I turn now to a set of principles of possibility which can be called constitutive principles. I will deal with them briefly, because they are more familiar than the Modal Extension Principle. These constitutive principles concern not concepts and the level of sense, as the Modal Extension Principle does. They are rather about objects, properties and relations. Intuitively each such principle states that an assignment is admissible only if it respects what is constitutive of the objects, properties and relations it mentions. Given the connection we have proposed between admissibility and possibility, this implies that a specification is possible only if it respects what is constitutive of the objects, properties and relations it mentions.

One plausible constitutive principle concerns the fundamental kind of an object, where for instance your fundamental kind is the kind human being and the fundamental kind of New York is the kind city. This notion of an object's fundamental kind is the one David Wiggins aims to elucidate in saying that it is the highest sortal under which the object falls (Wiggins, 1980). The plausible constitutive principle is this:

If P is a property which is an object x's fundamental kind, then an

assignment is inadmissible if it counts the proposition x is P as


This principle is to be understood as universally quantified, as propounded for all properties and objects. I have formulated the principle in such a way that, even if the property P is x's fundamental kind, the principle allows that an admissible assignment may make no pronouncement on the truth value of the proposition that x is P. This is in accordance with our rubric that assignments need not be total. What the principle does require is that if an assignment pronounces on the truth value of the proposition x is P at all, it must count it as true.

Properties are essential to formulating this principle correctly. It is not a mode of presentation which is the fundamental kind of an object. The fundamental kind must be something at the level of reference, not of thoughts. Nor can the fundamental kind be merely a Fregean semantic value, a function from objects to truth values. The fundamental reason for this is the list-like character of such a function. The kind-essence of an object is supposed to contribute to a metaphysical (not an epistemological) account of what individuates the object. If the kind-essence is to succeed in doing that, it cannot be something whose individuation involves the object itself, as the Fregean function does. The same point applies to sets and to Fregean courses-of-values.

Similarly, if there are individual essences, admissible assignments must respect them. Suppose, as is also plausible, that it is constitutive of person a that she originated in the particular sperm b and egg cell c from which she actually developed. Then this should be another of our constitutive principles:

An assignment is inadmissible if it both counts the proposition a

exists as true and counts the proposition a develops from b and c

as false.

More generally, in any case in which it is constitutive of the object x that it bear R to the object y, we should endorse the principle that

An assignment is inadmissible if it both counts as true the

proposition x exists and counts the proposition x bears R to y as false.

There will also be constitutive principles for properties and relations, stating what is constitutive of them.

There is obviously a huge amount to be said about which constitutive principles are true, and why. My concern here is not with which particular constitutive principles are true. It is rather to note the apparent existence of this class of principles of possibility, and their role in our understanding of possibility. We certainly cannot, though, totally bypass the issue of how the constitutive is related to the modal.

Some constitutive properties of objects have a life -- that is, a significance -- outside the sphere of the modal. The kind-essence of an object illustrates the case. The kind-essence of an object helps to determine the conditions under which a particular continuant object persists. If F is an object's fundamental kind, the object x persists from one time to another only if there is at the later time something which is the same F as x. The notion of the persistence condition of an object should be elucidated by saying that it is the condition which specifies what it is for the object to persist. If we omit the constitutive phrase "what it is", we will not have excluded overlapping persistence conditions which happen, given the way the world is, to give the right answer, but which do not answer the constitutive condition. Since, though, it is not at all obvious that the constitutive has to be explained in terms of the modal, this may still be an application of the notion of kind-essence which is not fundamentally modal -- though it will of course have modal consequences.(4)

It is, however, much harder to see essence of origin as having applications outside the modal. On the contrary, the fundamental role of truths about the individual essence of an object seems to be that of constraining which genuinely possible states of affairs are ones in which the object exists. I am not sure whether truths about individual essence have any role beyond this. An objector may now say that this point threatens the whole project of elucidating modality in terms of a set of principles of possibility with a fatal circularity. This objector may say: "It is one thing to try to elucidate our understanding of modality by stating the conditions required for a specification to be possible. But if those conditions themselves employ the notion of possibility, explicitly or implicitly, your whole approach is doomed to circularity."

This may sound plausible, but it is a mistake to suppose that these points about essence of origin mean that the principles of possibility have themselves to use the notion of genuine possibility. They do not, and there is no circularity of that sort. The constitutive principle implied by a true statement of the individual essence of a particular object amounts simply to a further axiom placing a condition on what has to be the case for any given assignment to be admissible. We do not need to use the notion of possibility to state what that condition is: it is just that the assignment must not deny the object in question what are in fact its actual origins. Indeed, we could even formulate the constraint in accordance with the austere demands of the A(C) form of A Study of Concepts (1992): possibility is that concept C, predicated of specifications, to possess which a thinker must have implicit knowledge of certain principles [P.sub.1](C) ... [P.sub.n](C), where one of these principles states that any specification falls under C only if any continuant object which exists, according to that assignment, has its actual origins. An account of possession of the concept of possibility along these lines does not take for granted that the thinker already possesses the concept.

Still, even if that is granted, it may be objected that we have a circularity if we try to specify the general class of constraints to be included in the restrictions on admissible assignments, if individual essence has to be explained in terms of its connections with the modal. Whether this objection stands depends on whether we can make sense of what is constitutive of a particular object independently of the modal. There is some plausibility in the idea that constitutive matters are explanatorily prior in fixing which object is in question. They then contribute, via principles of possibility, to fixing what are genuine possibilities. Some of Fine's work tends in this direction (Fine 1994). In the worst case, if it turns out that a notion of the constitutive explanatorily prior to the modal cannot be sustained, there may be still be a fall-back position in characterizing the class of constitutive principles. Forbes, for instance, has argued that there is an objectionable kind of bare identity which would have to be swallowed if necessity of origin were rejected (Forbes, 1985, ch. 6, [subsections] 3-4). So another option which might be explored as a fall-back position in characterizing the class of constitutive principles is that they are principles which can be justified in a certain fashion. But I will leave this fascinating diversion for now. All that matters at present is that constitutive principles must be included in the constraints upon the admissibility of an assignment.

So far, we have two kinds of principles of possibility: the Modal Extension Principle and the constitutive principles. Though these have very different subject-matters, they are not just arbitrarily slapped together. Both the Modal Extension Principle and the constitutive principles require that what holds according to a genuine possibility must respect what makes something what it is -- whether it be a concept, an object, a property or a relation. On any plausible theory of concepts, the identity of a concept depends on the rules which determine its semantic value in the actual world. The Modal Extension Principle then requires that what is involved in this identity be preserved in a certain way across genuinely possible worlds. So the Modal Extension Principle draws out the consequences of what is individuative of a concept for what is genuinely possible, while the constitutive principles for objects, properties and relations draw out the consequences of what is individuative of objects, properties and relations.

A third kind of principle of possibility is a principle of plenitude, and it has, within the framework I am proposing, the effects of Lewis's principle of recombination (1986, pp. 87-92). This principle of plenitude is a second-order principle. Unlike the principles of possibility given so far, it makes reference to the other principles of possibility. This principle also, unlike all the other conditions, which to date have been only necessary conditions, formulates a sufficient condition on admissibility. The principle states simply that

An assignment is admissible if it respects the set of conditions on

admissibility given hitherto.

We can call this "the Principle of Constrained Recombination". All sorts of recombination of properties and relations amongst individuals are allowed as possibilities, as long as they respect the full set of principles of possibility. Since we have said all along that a specification is genuinely possible iff there is some admissible assignment which counts all its members as true, this principle of plenitude implies that

A specification is possible iff it corresponds to some assignment

which satisfies the set of constraints on admissibility formulated

in the earlier paragraphs.

If "respecting", as it occurs in the Principle of Constrained Recombination, meant the same as "metaphysically consistent with", we would have a serious circularity problem. But it does not. The Modal Extension Principle and the constitutive principles each state necessary conditions for a specification to be possible. The recombination principle states that conformity to these principles -- if indeed they do exhaust the principles of possibility -- are sufficient for a specification to be possible. So all that a specification's "respecting" the principles of possibility involves is its conforming to the conditions they lay down. In explaining what is involved in that, all that is involved is the notion of joint satisfaction of the conditions, period.

It would indeed be open to a theorist to follow a different approach. He could introduce some notion of a canonical description of a specification. One could then raise the question of whether, under some consequence relation, it is a consequence of the principles of possibility that a specification so described is not possible. Finally one could propose that a specification is possible if its impossibility is not such a consequence. Such an approach in effect takes the propositions and thoughts which are true according to a given specification, and then considers the consequence relations in which those propositions and thoughts (or their canonical specifications) stand. It seems unnecessarily roundabout as compared with the approach which considers simply whether a specification meets the principles of possibility. It is also not clear that our intuitive notion of possibility fixes on one particular relation out of the many consequence relations there are, and is to be individuated in terms of that one particular relation.

With the possibility of a specification something now explained in terms of the principles of possibility, as formulated in terms of admissible assignments, we are in a position to state the contribution made to truth conditions by the standard modal operators:

A Thought or proposition is possible iff it is true according to

some admissible assignment.

A Thought or proposition is necessary iff it is true according to

all admissible assignments.

We call these two displayed biconditionals "The Characterization of Possibility" and "The Characterization of Necessity" respectively. When we evaluate modal claims in our ordinary modal thought and reasoning, we draw on our implicit grasp of the body of information stated in the principles of possibility, understood as including these Characterizations. (I give this formulation to respect the point that one and the same body of information may be captured in many different axiomatizations.) Hence forth, I will call the conception I have been outlining the principle-based conception of metaphysical modality.

The principle-based conception is an actualist conception. Given our approach, it has to be. An unreduced notion of a merely possible object is not explicable in terms of a predicate of possibility which applies to our specifications, which are sets of Thoughts and propositions built up from actual objects properties and relations. It would take us too far off course to tackle here the issues raised by actualism. I just note, as a hostage to fortune, that the obligations and challenges faced by any actualist theory are equally faced by the principle-based conception.

4. Modalism, understanding, reduction

What is the relation between the principle-based conception and what is sometimes called "modalism"?

Modalism has received various nonequivalent characterizations, but one idea central to it is that we have some grasp of the notions of possibility and necessity which is explanatorily prior to any understanding of possible worlds. At first blush, the principle-based conception, with its use of the Characterizations of Necessity and Possibility, may seem incompatible with modalism. For implicit knowledge of these characterizations is said to be partially explanatory of understanding, and the characterizations explain modality by mentioning specifications which are possible. And are not these genuinely possible specifications, to all intents and purposes, possible worlds? They are not the Lewisian modal realist's possible worlds, to be sure, but they are fine "ersatz" worlds.

Such a claim of incompatibility would, however, be superficial. A central motivation for modalism was the idea that there are constraints involving possibility and necessity that determine which worlds are possible: the possibility of a world is dependent upon its satisfaction of these constraints. The principles of possibility I have been outlining are precisely such constraints. They are just what is needed to make specific one central idea of modalism.

We should, then, distinguish between what we can call constraint modalism and ontological modalism. Constraint Modalism is the doctrine that there are constraints involving the notion of possibility which are explanatorily prior to whether a world is possible or not. There is no prima facie contradiction in a constraint modalist quantifying over possible worlds of an ersatz kind, and using such quantification in the explanation of modal discourse, so long as the possible worlds are conceived as derivative from the satisfaction of various constraints involving the notion of possibility. Ontological modalism, by contrast, denies that possible worlds have any part to play in the elucidation and understanding of modal discourse. The ontological modalist will insist that "necessarily" is never in any way to be understood as a quantifier. The letter of the principle-based conception I have presented is indeed in conflict with ontological modalism, for I have quantified over worlds, of a sort, in talking of the genuinely possible specifications. But the letter of the formulations could be altered. If we become convinced, for one reason or another, of the truth of ontological modalism, we could adapt the main ideas I have put forward to square with it. For instance, in assessing whether two propositions are compossible, we would have to ask whether they could both be true only in circumstances in which the semantic value of some concept is not fixed in the way it actually is; and so forth. In short, we could maintain the core ideas of the principle-based conception without quantifying over worlds at all.(5) Ontological modalism is really orthogonal to the main claims of this paper. I do myself, though, think that once we have accounts of the understanding of possibility, it would be straining at a gnat to object to possible worlds set-theoretically constructed using the principle-based conception of possibility.

It cannot be said that all of the literature of the seventies and early eighties was completely clear on these matters (I certainly suspect that I was confused myself). Disquotational truth theories for necessity were developed in that period, by Davies (1978), Gupta (1978), and myself (1978). Some of the literature of that period reads as if these disquotational truth theories are all that is required by way of a theory of meaning and understanding for modality. I would now say that a disquotational truth theory can never by itself give us a full account of what is involved in understanding an expression, even though it is a necessary part of a full account, and constrains the additional parts of the required full account. The disquotational theory cannot by itself answer (for instance) the questions of $1 above, the questions raised by the inapplicability to modality of models of understanding available and appropriate for other primitive expressions. The question then arises: from the perspective of the principle-based conception, how should we understand the relation between a truth theory for necessity and the substantive account of understanding involving implicit knowledge of the principles of possibility?

Any substantive account of understanding a particular expression should aim to give a dovetailing account of its contribution to truth conditions. Failing this, we will not have explained how the expression, when understood in the way specified in the account of understanding, is also capable of occurring in truth-evaluable sentences. So it would be absurd to regard a substantive theory of understanding a particular expression as somehow in competition with a truth conditional theory for it. The theory of understanding requires the existence of a dovetailing truth conditional theory. What is involved in the relation of dovetailing is one of the central issues in the theory of intentional content. It must involve at least the kind of integration of epistemology with the truth conditions discussed as a desideratum back in $1. For the principles based account of understanding of necessity, the dovetailing takes a particular form. The conditions which are implicitly known in understanding necessity are just those which do determine the truth conditions of modal statements.

For any substantive account of understanding, issues of soundness and completeness also arise (see Peacocke, forthcoming). For our present case of modality, soundness involves the following: that the intuitive property of genuine possibility, which is true or false of specifications, respects the principles of possibility I have formulated. I hope this is made plausible by inspection of the various principles.

We should also aim for completeness: that the truth of any proposition involving possibility is determined by the principles of possibility. For the moment, I simply note this as an important commitment of the approach I am advocating. It will be crucial in $5, when we come to discuss epistemological matters. The commitment is one which sharply differentiates the principle-based conception from Lewisian modal realism, under which it is not apparent why any such claim of determination should be true.

This general treatment in terms of the principles of possibility is in itself neutral on the issue of whether there exists a genuinely reductive analysis of necessity and possibility. It would plausibly contribute to a reductive account if two further conditions were met. First, we would have to have a set of principles of possibility which we could reasonably believe to be exhaustive. Second, we would have to be convinced that none of the auxiliary concepts used either in formulating or in deploying the principles of possibility rely on modal notions. If these two further conditions were met, then an enthusiast for reduction could define a possible specification as one which respects the allegedly complete set of principles. This is a big "if". The principles of possibility include constitutive principles, and it takes hard philosophical thought to discover what is constitutive of a given property or relation. Who is ever in a position to say that there are not more constitutive principles to be discovered? It does not seem at all necessary to believe that both those two conditions are met in order to regard the principle-based conception as promising. Even someone who is sceptical of the possibility of reduction will still be making progress if he identifies certain central principles of possibility and argues that implicit knowledge of those principles is involved in our understanding and knowledge of modal truths. He can then go on to say that our very general grasp of the notion of possibility involves some implicit grasp of the idea of a set of principles of possibility of which it may always be an open question whether it is complete.

A more specific doubt that the principle-based conception can be reductive, one that might even be elevated into an objection of circularity, is that the notion of identity of rule, or of identity of concept, has to be explicated modally. For example, it may be objected that the only reason that we say the same rule is applied in determining the extension of bachelor as is applied in determining the extension of unmarried man is that it is necessary that all and only bachelors are unmarried men.

In this formulation, the objection misunderstands the principle-based conception. The identity of rules and concepts is fixed at a level having to do with informativeness and epistemic possibility, the level of sense. The position of the principle-based conception is that we do not in general (de jure rigid cases aside) need to appeal to the truth or falsity of object-language modal sentences to explain what is involved in someone's employing one concept rather than another, nor to explain the identity of concepts and rules.

A more persistent objector may say that even in the principle-based conception as here explicated, modality is still in the wings. For in the classical neo-Fregean explication of identity of concepts C and D, do we not say that there is no possible circumstance in which a pair of thoughts A(C) and A(D) differ in cognitive significance? So the objection is that the very notion of concept we are employing, and the notion of a way of fixing a semantic value which goes with it, already involves the notion of metaphysical necessity. To this there are two responses. The first is that identity of concepts may be explained in constitutive terms, in terms of what it is to possess one concept rather than another. Modal facts about informativeness can then be regarded as consequential upon this constitutive level. That is the position to which I am most tempted.(1) 6 It has to be granted, though, that we need a better understanding of the constitutive and the nature of its links to the modal. Even if this first response were quite unavailable, however, there is a different response to the challenge. The use of modality in the identity condition for concepts is a use of modality by us as theorists. It is not a use by the thinker whose mastery of concepts is being characterized. The distinction is important. It means that the alleged circularity is no more objectionable than, for instance, our using logical vocabulary -- as we can hardly avoid doing -- in giving an account of a thinker's understanding of the logical constants. These uses of modality outside the scope of the thinker's own attitudes would not at all reduce the principle-based conception to vacuity. We have already seen how the principle-based conception can be used in the explanation of the truth of some necessities. The logical, epistemic and understanding-theoretic uses to which I am about to put the principle-based conception equally do not require a definitional reduction of modality. They would not be undermined by our using modal operators in the metalanguage in our account of understanding.

5. The epistemology of metaphysical necessity

We can divide questions about the relations between modality and knowledge into two broad kinds. There are questions about how certain particular methods can succeed in producing knowledge of modal truths. There are also questions about the correct way to conceive of the general relations between modal truth and knowability. Let us consider some issues about particular methods first.

The principles of possibility fix the concept of possibility. Suppose that understanding modal discourse does involves implicit knowledge of these principles of possibility. We can then see why some common methods of establishing or refuting particular modal claims are appropriate to the content of those claims. "Necessarily, p" is sometimes established by giving a proof of p. There may be an outright proof of proof p -- in which all premisses are discharged -- or p may be proved from other propositions whose necessity is already established. The following is a sufficient condition for proofs of each of these kinds to be sound methods of establishing a necessity: that each of the inferential principles relied upon in the proof, besides being valid, contains essentially only expressions whose semantic values are constant across admissible assignments. We argued that the propositional logical constants are just such expressions.(1) When the inferential principles in the proof contain essentially only such expressions, the validity of the inferential principles for the actual world will carry over to any other genuinely possible specification. So such proofs also justify belief in the necessity of their conclusions.

Such constancy across admissible assignments is, though, only a sufficient and not a necessary condition for the success of those methods of establishing a necessity. Another kind of case is that in which the expressions occurring essentially in a proof's principles of inference do not have constant semantic values across admissible assignments; and yet the proof can still establish a necessity. It is enough if the conditions fixing the semantic values of the expressions, together with the Modal Extension Principle, ensure that the principle of inference is truth-preserving in every admissible assignment, even when the semantic values vary across assignments. Such is the humble case of the inference from t is a bachelor to t is a man. This kind of case, together with the other two outlined in the preceding paragraph, between them cover a large part of the territory in which we establish a necessity by means of a demonstration.

When our concern is rather to refute a claim of necessity, we have to invoke more than the Modal Extension Principle. Suppose we are concerned with the a priori content "I am here", and that we want to show that it is not necessary. Given that "I" and "here" are de jure rigid, the Second Part of the Modal Extension Principle applies. But the Modal Extension Principle by itself does not imply that there is a genuinely possible specification according to which the actual utterer of "I" is located somewhere else. For that, we have to rely on the Principle of Constrained Recombination, that a specification is possible if it respects the full set of principles of possibility. This need to appeal to Constrained Recombination accords with our pretheoretical awareness that any claim of the existence of a genuinely possible specification according to which the actual utterer is somewhere else can always be defeated by showing that it violates some constraint on possibility.

If it is granted that implicit knowledge of the principles of possibility is appropriately employed in reaching a modal judgement, I would argue that we can also see how the judgement so reached can be knowledge. In other work, I argued for a certain relation between judging in accordance with a method mentioned in the conditions for possessing a concept, and the status as knowledge of a judgement so reached (Peacocke 1987, 1992b). Suppose a person makes a judgement, and does so for a certain reason. Suppose too that being willing to make such a judgement, for that reason, is part of the possession condition for one of the concepts in the judgement's content. Then the judgement is knowledge. For convenience, let us label this "the Epistemological Principle". So, for instance, this principle counts as knowledge a perceptual belief that a perceived object falls under an observational concept, when the belief is formed on the basis of an experience with a certain appropriate nonconceptual representational content (cf Peacocke 1992b, ch. 3). The appropriate general type of experiential content is that type mentioned in the possession condition for an observational concepts. Another example would be a belief containing a logical concept. The Epistemological Principle will count as knowledge a belief in a content which is properly inferred by a primitive introduction rule from something already known (cf. Peacocke 1987, pp. 176-8), provided the rule is one mentioned in the possession condition for the logical concept in question. I take it that both these examples have some intuitive plausibility.

The Epistemological Principle links the theory of possession conditions with the theory of knowledge. This link was given a rationale in some earlier work (see Peacocke 1992b, chs. 1 and 5). The determination theories for different concepts all have the following property. The semantic values they assign (given the way the world is) to the concepts to whose possession conditions they apply ensure that judging in accordance with the possession condition results in true beliefs. If this is correct, judgements made in accordance with some particular clause of a possession condition are guaranteed to be true, as an a priori matter of considerations in the theory of conceptual content.

It is, though, not at all necessary to believe that concepts are individuated by their possession conditions to find it plausible that there is a link between epistemology and the theory of mastery of a given concept. The connection is plausible on almost any theory of concepts. Consider judgements involving a given concept, in given circumstances, which are not merely discretionary, but are rationally required of anyone who possesses that concept. Rational judgements aim not just at truth, but at knowledge. If a judgement made in given circumstances were not knowledgeable, it could hardly be required by possession of the concepts it involves, for someone could reflect that it would not be knowledge, and so refuse to make the judgement. The general principle that judgements required by possession of a concept will also be knowledgeable is one with some force whatever particular theory of concepts one favours.

If we accept that point, together with the preceding account of the understanding of metaphysical necessity, we now have the resources to explain why certain modal judgements amount to knowledge. Consider, for instance, a judgement of necessity, such as "It is necessary that if A, then A or B", where this judgement is explained by the thinker's implicit knowledge of the Modal Extension Principle, and so is made in accordance with the Main Part of the Modal Extension Principle. A judgement so made will be true, and it will be required by the account of what is involved in understanding necessity. If we accept that judgements required by the account of understanding are also knowledgeable, we thereby explain why this thinker knows that modal truth. Indeed, in this special case, the judgement of the modal truth is explained by the thinker's implicit grasp of what makes the modal truth hold. So we have in a strong form the fulfillment of the condition that the thinker is not just accidentally right in his modal beliefs.

Under this treatment, knowledge of particular necessities and possibilities, in the principle-based conception, no more requires dubiously intelligible faculties connecting the thinker with some obscure modal realm than does, for instance, knowledge of logical principles, which can equally be certified as such by corresponding means. So it seems that we have objectivity of modal discourse and a means of knowing modal truths, without being modal realists in Lewis's sense. This is how the principle-based conception of modality integrates its epistemology with its metaphysics.

Implicit knowledge of the principles of possibility seems to be rather closer to the operations of reason at the personal level than is the classic Chomskian case of tacit knowledge of the principles of grammar of our natural language. There are certainly respects in which die two cases are on a par, in particular in respect of their role in content-involving psychological explanation. But there are different subspecies of implicit knowledge. Your judgement that a string of words is grammatical does not result from conscious inference. If asked why a sentence that seems to you grammatical is so, you may or may not be able to come up with reasons. But even if you can, the reasons you come up with will not have been operative in the production of your initial perceptual impression of the grammaticality of the sentence. By contrast, when an ordinary thinker makes a judgement of possibility or necessity for himself, not on the basis of testimony, he will have reasons. In the case of a judgement of possibility, there will be some conception of a possible state of affairs in which the possibility obtains; in the case of a reasonable judgement of necessity, there will be some informal demonstration or proof of the proposition, or some reasonable belief that one exists. The ordinary user of modal notions cannot state the principles of possibility, but nevertheless the principles of possibility bear a much closer relation to his personal-level modal thought than do the principles of grammar to his thought about his own language. It is a question worth further thought what that relation is.

In some very unproblematic cases, an attribution of implicit knowledge is justified simply because it systematizes, and does no more than make explicit, the general principles of which a thinker has explicitly used instances, even if he has not stated the generalizations himself. Such would presumably be the case when we justifiably attribute implicit knowledge of the Peano axioms to nineteenth-century arithmeticians prior to their explicit formulation. Attribution of implicit knowledge of the principles of possibility certainly goes beyond that very simple case, since the principles of possibility employ concepts which are not simply generalizations of instances the thinker is already using at the personal level. The talk, for instance, of the way in which semantic value is fixed, in the Modal Extension Principle, goes far beyond such generalization. On the other hand, it is arguable that some inchoate appreciation of the Modal Extension Principle is responsible for our judgements that certain specifications are not genuine possibilities. It is a significant point that, even for a thinker unschooled in modal semantics, there is such a rational activity as reflecting on the axioms of a proposed modal logic, and coming to appreciate that they are (or that they are not) correct. There seems to be a rational transition in thought from features of this thinker's understanding of necessity to his evaluation of the axioms. There is, in the ordinary case, no such rational transition in hearing a sentence as having a certain meaning and structure. In order not to divert our attention too far from the topic of this section, the epistemology of modality, I will not pursue these important issues about implicit knowledge here. I confine myself to noting that to elaborate the nature of the implicit knowledge involved in modal understanding, we will have to consider other areas where implicit knowledge interacts with reason-guided capacities.

I now turn to what seems to me to be the more difficult of the two epistemological questions distinguished at the start of this section, the question of the general relation between the metaphysical modalities and knowability. On this topic, I wish to propose three general theses. I will try to make the three theses plausible by consideration of examples. The theses are all principles whose truth would be explained if possibility is constrained by the principles of possibility I have formulated.

Thesis (I):

In every case in which a modal truth involving metaphysical

necessity or possibility is unknowable by us, its unknowability is

wholly explained either by the unknowability by us of some truth

not involving metaphysical modalities, or by the unknowability

by us of one of the principles of possibility.

Let me first illustrate Thesis (I) in various ways, before discussing its significance. Take Goldbach's Conjecture, that every even number is the sum of two primes. If this Conjecture is true but unprovable, it is nonetheless necessary. Its necessity is unknowable (by us) if its truth is. The unknowability of its necessity, if it is unknowable, is wholly explained by the unknowability of its truth. The general principle that any truth of arithmetic is necessary is knowable and known. What we do not know is the totality of thoughts to which this known general principle can be applied.

A wide range of examples with empirical elements also falls straight-forwardly within the ambit of Thesis (I). It may now be unknowable whether the colourless liquid in the bottle on the kitchen table two years ago was water or vinegar. Suppose it was water. So, concerning the liquid in that bottle then, it was necessarily [H.sub.2]O. "Concerning the liquid in that bottle then, it was necessarily [H.sub.2]O" is now unknowable, but its unknowability is wholly explained by the unknowability of whether it was water in the bottle.

It is worth noting at least three other subspecies of unknowability consistent with the present framework. First, there is no obvious guarantee that everything which is constitutively involved in possession of a particular concept, even one of our concepts, must be discoverable by us. Undiscovered features may, via the Modal Extension Principle, generate necessary truths.

Second, what has just been said of concepts applies to everything else. It is not obvious that we should be able to know the constitutive properties and relations of each item in our ontology. It takes hard philosophical investigation to discover them, and we may be intellectually limited in our ability to attain knowledge of them. Unknowable constitutive properties will equally generate unknowable necessities, via the constitutive principles of possibility.(8)

Third, it will be recalled that a specification of a genuine possibility was required to respect the principles of possibility. Whether a specification respects the full set of principles of possibility (even if they are all known to us) is not, outside specially restricted cases, a decidable relation. After all, the specification may mention propositions of mathematics, set theory, or of higher-order logic. So in some cases a genuinely possible specification cannot be known by us to be so. This too will generate modal truths which cannot be known by us. But the explanation of the unknowability involves a general logical phenomenon which applies to any subject-matter whatever. It has nothing to do with the existence of any inaccessible modal realm.(9)

If Thesis (I) is correct, it has a twofold importance. Its first important consequence is that the existence of unknowable modal truths cannot, without further argument, be used to support a position according to which modal truth goes beyond what is fixed by a set of principles of possibility of at least the general kind I have given. (I write "general kind", because of course it is entirely possible that I have omitted some principles which ought to be included with the principles of possibility.) In particular, it is fallacious to conclude, from the fact that the particular principles of possibility I have suggested are knowable, together with the existence of some unknowable modal truths, that the present approach must be incorrect. In fact, the onus is rather on one of the other parties to this discussion, the Lewisian modal realist. If Thesis (I), or anything very close to it, is correct, it presents a problem for the Lewisian modal realist. How a modal realist could account for Thesis (I), or anything close to it, I do not know. He cannot do so by regarding his possible worlds as merely reflections of what is permitted by a set of principles of possibility. That would be to give up the modal realism. But if he does not so regard the possible worlds, he leaves Thesis (I) unexplained.

The other significant feature of Thesis (I) is the strong contrast it points up between modality and tense, despite the many well-known and undisputed formal parallels between them. We explained the existence of, for instance, unknowable arithmetical necessities by remarking that we know a priori that, if p is a true arithmetical content, then necessarily p. If Thesis (I) is correct, there will be similar conditionals for other domains about whose members there are unknowable truths. Absolutely nothing analogous holds for the past. There is no nontrivial class of propositions p, not already about the past, for which we know a priori conditionals of the form: if p, then in the past p. What makes some thoughts about the past presently unknowable cannot be explained satisfactorily by citing the unknowability of some truths not about the past, which in turn are sufficient for truths about the past. Similarly, there is no set of a priori "principles of the past", analogous to the principles of possibility, which collectively distinguish a past-tense operator from other operators, and grasp of which is fundamental to understanding past-tense discourse. No set of purely a priori principles can, together with purely present-tense truths, settle the truth value of a statement like "Yesterday, there was a place in Greenland with a temperature of -- 100[degrees] C".

The present unknowability of some statements about the past has a source which is different in kind from the two potential sources of unknowable modal truths, viz. unknowable non-modal truths and, perhaps, some unknowable principles of possibility. We can conceive of the truth of some past-tense statement being unknowable because we can conceive, in principle, of some past state of affairs leaving no traces accessible to us now, or in the future. By contrast, any unknowable modal truths are not such that, if only they were embedded in the world in such a way that they left the right traces accessible to us now, we would be able to know them. On the contrary, even the known modal truths are not known because of their causal interactions with us. The defence of the intelligibility of unknowable past-tense truths would also, correspondingly, draw on resources not present in the modal case.

I also suggest a second thesis.

Thesis (II):

In every case in which a content containing a metaphysical

modality is known, any modal premisses in the ultimate

justification which underwrites the status of the belief as knowledge are a


Thesis (II) implies that any a posteriori premisses in the ultimate justification for a piece of modal knowledge will not themselves be modal. A paradigm, and maximally simple, example of Thesis (II) is knowledge of the necessity that Hesperus is Phosphorus. The justification for this belief consists of two parts. One part is the a posteriori, but non-modal, knowledge that Hesperus is Phosphorus. The other part is the modal, but a priori, general knowledge of the necessity of the identity relation, in cases in which it holds. There is an analogous division into two parts of our justification for believing in particular instances of Kripke's a posteriori necessities of constitution and of origin. Thesis (II) is, of course, wholly consistent with Kripke's famous discussion and treatment of his examples.

One of the substantive consequences of Thesis (II) is that principles of constitution and origin always have an a priori component in their ultimate justification. On the conception Thesis (II) is elaborating, though it is a posteriori that humans originate in sperm and egg cells, it can be known a priori that whatever the origins of any given individual human, they are essential to that individual. This implication of Thesis' (II) deserves independent consideration in its own right. One of the attractions of Thesis (II) is its potential for making the epistemology of constitutive claims less problematic. A full defence of this thesis would also need to draw some more distinctions involving the a priori. Some philosophers have held that horses necessarily descend from a certain stock of animals, and that animals with a physiology and genetics identical with that of ordinary horses, but descending from a different stock, would not be horses. Let us suppose that this is so. (If it is not so, consider a natural-kind concept for which the corresponding claim of necessity does hold.) It seems also that a particular thinker could have attitudes properly described as involving the concept horse without being in a position to know a priori that horses necessarily descend from a certain stock. This thinker may have seen many horses, and correctly apply the word to them. He may not know, though, that they are biological entities -- to adapt a remark of Putnam's, he may even believe that they are remotely controlled devices. So this thinker is certainly not in a position to know a priori that horses necessarily descend from the stock from which they actually descend -- he does not even know that they descend from some stock. How is the defender of Thesis (II) to respond to this challenge?

He might react by saying that we already knew that merely partial understanding suffices for attribution of attitudes whose content involves the concept horse. The defender of Thesis (II) might, then, aim to use a notion of the a priori upon which we can say that a Thought is a priori if someone -- not necessarily every possessor of the concepts it involves -- can know it a priori. Some merely partial understanders of the expression would not be in a position to reach such a priori knowledge, but full understanders would.

This reaction does not, though, seem to me entirely adequate to the nature and the extent of the phenomenon. Consider another Putnam -- inspired example. We may explore the remains of a newly discovered ancient civilization, and come across brightly decorated objects we take to be ancient ornamental items attached to clothing. We may introduce a word for these objects. It may turn out later that these objects are actually a species of brightly coloured hibernating beetles. If they are so, then they necessarily come from the actual stock of beetles from which they are descended (or, if "beetle" does not work like that, we can introduce some notion which does). It is quite implausible, though, to say just after the word was introduced and before the true nature of the objects was discovered that someone who fully understood the word would be in a position to know this modal truth. No one could have known a priori that these objects were descended from any stock at all, as opposed to having been designed by human beings. Learning that they are biological objects is simply an empirical discovery.

A better reaction is to note that even in this case, someone who holds that organisms of a given species necessarily come from the actual stock from which they are descended will also hold the following proposition to be a priori: that if some kind of organism is self-reproducing, members of that kind necessarily come from the stock they do. What is empirical, for these theorists, in this most recent example is that objects of the newly discovered sort are organisms of a kind which reproduce. The (claimed) modal and a priori proposition is just that organisms of kinds whose members reproduce necessarily come from the stock they do. Such, at any rate, would have to be the position of one who wants to defend those modal claims in the context of a principle-based treatment of modality.

Thesis (II) is a consequence of the treatment in terms of principles of possibility, if the principles of possibility are a priori. The particular principles of possibility I have suggested have an a priori status. It is true that I have drawn back from claiming that we know all the principles of possibility, so an argument from case-by-case enumeration of the principles of possibility actually cited will not establish that all are a priori. But I do suggest that we have no conception of how some being, intellectually more powerful than us and who knows principles of possibility which we could not, could know them in anything other than an a priori manner. If we agree that something's being necessarily the case cannot causally explain anything, we already have an explanation for the impossibility of such a posteriori knowledge. A posteriori methods can produce, and rationally produce, various beliefs in us, but they cannot by themselves produce knowledge that something is necessary. We cannot at this point apply the model of our knowledge of the a posteriori but necessary truth that Hesperus is Phosphorus. That model worked only because there was also a priori knowledge of the general principle of the necessity of identity. In the present case we are ex hypothesi concerned with fundamental principles of possibility, which are not inferred from anything else. If the fundamental principles were a posteriori, there would in the nature of the case be no further a priori principles to which we could back up in an account of justification, on pain of their not after all being fundamental.(10)

If this line of thought is sound, we have prima facie reason to make a claim not merely about actual cases of modal knowledge, as Thesis (II) does, but about all possible cases. We can propose

Thesis (III):

Thesis (II) is necessary, and holds for all possible knowers.(11) How would the present account respond to broadly neo-Humean or neo-Quinean problems about the concept of metaphysical necessity? Having said several times that necessity itself cannot be causally influential, I am not about to find after all a suitable impression of necessity from which the idea could be derived. Nor is there any need to do so, on the principle-based account. The principles of possibility must be implicitly known by someone who possesses the concept of metaphysical necessity, but it is not at all required that this knowledge result from some causal influence of modal facts. The principle-based account and its attendant theory of understanding are both consistent with the principle that there is no impression from which the idea of metaphysical necessity is derived.

A full answer to all the major sceptical doubts about the concept of metaphysical necessity would require defence of all the pieces of apparatus I have deployed. I will not take on that task here. I will, though, briefly consider what conception of the relations between the actual and the possible should go along with the principle-based account. There is, first, on the present account, a systematic and general connection between necessary truths and thought about the actual world. The principles of possibility fix the truth values of modal contents by relating them to the conditions which individuate particular concepts, and the principles of individuation for the objects and properties we think about. These are the concepts employed, and the things thought about, in thought and discourse about the actual world that does not explicitly involve metaphysical necessity and possibility. The present treatment ties an account of the modal to these aspects of such thought about the actual world. In particular, the operators of metaphysical necessity and possibility, because of their relations to the Modal Extension Principle, can then be described as operators which are dependent upon the individuation of concepts. An operator or relation is dependent upon the individuation of concepts when its conditions of application are fixed, in the most fundamental cases, in part by the individuation-conditions of the concepts which comprise the thought-contents to which it applies, or which form one term of the relation. Besides the metaphysical modalities as applied to thoughts, another example of a concept which I would argue to be dependent upon the individuation of concepts is the relation of a priori justification of one content by another, or by suitable perception or thought (Peacocke 1992a).

A question it is sometimes tempting to raise is this: "Why do we need the concept of necessity at all? Why should we ever need to talk about anything other than the actual world?" One answer is that it is certainly desirable to know of a class of principles which can be legitimately employed when reasoning within the scope of any counterfactual supposition whatever. These principles will be the necessary truths, and we must be able to identify some of them if reasoning within the scope of counterfactual suppositions is to proceed. Counterfactuals are also indispensable in practical reasoning. So we can expect some identification of necessary truths to be practically, as well as theoretically, indispensable.

That is a first-pass answer to the question. Whether we can then go on to say more depends in part on which of two metaphysical conceptions we adopt.

We can distinguish between a fully non-modal conception of the actual world and a partially modal conception thereof. The two conceptions are agreed on the central role of causal relations in the way the world actually is. Many properties are individuated by their causal powers -- some have even defined identity of properties in terms of identity of causal powers. It is also arguable that the very idea of a material object is a causal one, perhaps for more than one reason. To be composed of matter is (at least) to stand in certain causal-explanatory relations to the magnitude of force. Material objects also have the property that their later states are explicable in part by their earlier states. If material objects necessarily involve causation, so too will all events individuated in part by their relations to material objects. We could continue in this vein at some length -- we have not, for instance, even begun to talk about the role of causation in the mental realm. But these points should already be enough to make it overwhelmingly plausible that causal relations must be inextricably involved in any plausible account of the actual world. The fully non-modal conception and the partially modal conception diverge not on these agreed points, but rather on the nature of causation itself.

The divergence is over the issue of whether causation itself has to be explained in terms of counterfactuals, or not. This need not involve divergence over whether there is a link between causation and counterfactuals. Both sides may accept that there is a link; but they will regard it very differently. For the partially modal theorist, the link is written into the nature of causation, which cannot itself be properly elucidated without reference to counterfactuals. For the fully non-modal theorist, the truth conditions for counterfactuals must be elucidated in such a way that they involve conditions on causal relations, whose nature is not dependent upon counterfactuals. Now the partially modal theorist will insist that we need to employ counterfactuals even in the description of the actual world. To try to describe the actual world without using counterfactuals would, for him, be to try to describe it without making reference, direct or oblique, to causal relations; and that would be impossible.

It would take us too far from our principal topic of metaphysical modality to try to decide between the fully non-modal and the partially modal conceptions (even if I were capable of doing so). One could expect there to be many more rounds to the argument. The partially modal theorist may complain, with some historical justice, that those who have tried to employ a notion of the actual world which is purged of counterfactual notions have had to use devices which seem clearly inadequate. Nelson Goodman once wrote that "(t)he fictive accident to a given train under the hypothetical circumstance that a given rail was missing can be taken care of, for example, by saying that the train at that time was "accidentable", or, more fully, `rail-missing-accidentable'" (1965, p. 54). Such hyphenated predicates are a paradigm case for applying Davidson's demand that it be explained how the meaning of the complex depends upon the meaning of its constituents (1965). The first-pass attempt to do so would reintroduce the counterfactual locution. That is a challenge to which the fully non-modal conception must respond if it is to be a viable option.

One challenge the fully non-modal theorist faces is that of giving an account of causal laws consistent with his general position. Important steps have been taken towards explaining counterfactuals in terms of causation, notably by Frank Jackson (1977). But the developed and relevant explanations of counterfactuals known to me all employ the notion of a causal law, and whether this is a modal notion remains moot. The fully non-modal theorist must supply some account which treats the notion of a law non-modally. He has to do this for probabilistic as well as for deterministic laws. One might, at this juncture, distinguish two varieties of non-modal theorist. One variety declines this challenge for laws, and so confines non-modalism to the particular. The bolder variety aims to defend non-modalism without restriction.

If the partially modal conception is correct, one important consequence is that we could not regard grasp of contents that do not explicitly contain the concepts of metaphysical possibility and necessity as not tacitly involving any grasp of the notion of possibility. We could not then regard a person's understanding of apparently "categorical" sentences as explanatorily wholly prior to any understanding of the concept of possibility. This point does not of course undermine the principle-based account of necessity, and its correlative theory of understanding. It means only that we have another example of a local holism. Possession of the concepts of necessity and possibility, and of a large family of "categorical" concepts, can be elucidated only simultaneously. Neither is prior to the other in the order of philosophical explanation, if the partially modal conception is correct.

If, on the other hand, the fully non-modal conception is defensible, then stronger claims are possible. We could then defend the view that the categorical is fully explanatorily prior to the counterfactual. Correlatively, we could sustain the claim that unrelativized predicates are wholly prior, in the order of explanation, to their world-relativized counterparts. This option remains attractive; but it clearly needs to be earned by further argument.

6. Conclusion and prospects

I have been arguing that we can retain the objectivity of statements of necessity without accepting that they concern an inaccessible modal reality; and that we can retain their knowability without regarding them as thinker-dependent. I am more wedded to the general approach than to the particular development of it I have attempted here, and other and better developments may be possible. If the present version, or some variant, can be successfully defended, there are two kinds of area in which further investigation may prove fruitful.

The first is that of modalist treatments of at least the more elementary parts of mathematics, and possibly other domains of abstract objects. Modalist treatments of those areas have often run up against a cluster of objections to the effect that the metaphysics and epistemology of necessity are totally obscure, and that we do not really possess a genuine notion of necessity distinct from such tamer notions as logical truth. If this paper is correct, those objections can be overcome.

The second area is defined by a very general question. Can the general features of the principle-based approach to modality be applied to other areas where we need to steer a middle way between a realism which involves inaccessibility, and a position which secures accessibility at too high a price? Moral discourse is perhaps the most salient area for which the approach merits further investigation.(12)

Magdalen College Oxford OX1 4AU England

Appendix A: Modal logic and the principle-based conception

This Appendix addresses the question: which modal logic does the principle-based conception underwrite as correct? Does it guarantee the absolutely minimal axioms and inference rules that should be delivered by anything which is recognizably an account of metaphysical necessity? And if so, how far beyond the minimal axioms does the principle-based conception go?

I will proceed by presenting a series of numbered observations which build up to establishing that the principle-based conception, together with some intuitively acceptable principles, implies the correctness of the propositional modal system T, as it is called in Hughes and Cresswell (1968). This is the system called "M" by Kripke (1971). We follow essentially Kripke's concise formulation of the axiom schemata and rules for the system in question:

A0 All truth functional tautologies.

A1 NA [implication] A

A2 N(A [implication] B) [implication] .NA [implication] NB

R1 From A, A [implication] B, we can infer B.

R2 If A is a theorem, so is NA.

I think it would be widely agreed that a philosophical account of modality which is unable to support the correctness of the principles of T would be unacceptable. (Constructivists will of course not accept A0, but when they replace it with something more restrictive, the remaining distinctively modal part of T should be uncontroversial for them.) It must then be a task for the defender of the principle-based account to show that on his conception, the axioms, inference rule and rule of proof in T are correct.

In what follows I assume that we have fixed on a given background set of objects, properties, relations and concepts, so that we have a determinate range of specifications involving Thoughts and propositions built up from these entities. I also write

[A.sub.1], ... [A.sub.n]/B

to indicate that B can be inferred from [A.sub.1], ...,[A.sub.n].

Observation 1: Under the principle-based conception, the inference

NA / A is truth-preserving for any Thought or proposition A. To the actual world corresponds a certain assignment and its corresponding specification. This is the assignment [s.sub.@] which assigns to each concept its actual extension. Corresponding to [s.sub.@] is the specification @ of Thoughts and propositions which are true according to [s.sub.@]. A Thought or proposition built up from elements in our background set is a member of @ just in case it is true (really true, not relative to an assignment or anything else). The reader can quickly check that [s.sub.@] satisfies all the conditions on admissibility we have enumerated. Hence @ is a specification which is possible, on the principle-based account of possibility. (It had better be.) If A is necessary, it holds in all possible specifications; so it holds in this distinguished specification @. By the choice of this specification A is true outright.

Observation 2: We have arbitrary finite necessitations of the Main Part of the Modal Extension Principle. Consider first the necessitation of the Main Part. It is hard to see any contingent feature of the actual world upon which the truth of the Main Part of the Modal Extension Principle depends. Take a possible world w at which (per impossibile) the semantic value of a nonrigid concept C with respect to some world [w.sup.*] accessible from it is not fixed in accordance with the Main Part of the Modal Extension Principle. That is, it is not fixed by applying to the specification [w.sup.*] the same rule which, when applied to the actual world, yields its actual semantic value. There could not then be any good answer to the charge that we have lost touch with what is involved in making something a possibility for that very concept C. If this reasoning is accepted, it establishes the necessitation of the Main Part of the Principle.

Similar reasoning seems to apply to any merely possibly possible world; and to any possibly possibly possible world; and so forth. For any such world, if the semantic value there of C were fixed by a different rule than is applied in the actual world, we would not really have the same concept any more. Again, if this reasoning is sound, we have arbitrary finite necessitations of the Main Part of the Modal Extension Principle.

Observation 3: Suppose that from the Main Part of the Modal Extension Principle together with a statement of the rules for fixing the semantic values for the expressions in A we can derive NA. Then, given the following three other acceptable principles (1) -- (3), NNA is also true.

Principle (1) is that the derivation relation is itself necessary. So, if [A.sub.n] is derivable from {[A.sub.1], ... , [A.sub.m]}, we have N([A.sub.1], & ... & [A.sub.m][implication] [A.sub.n]). By "derivability" all we need to mean for these purposes is derivability in first-order logic. The necessity of this relation of derivation is also ensured by the Modal Extension Principle.

Principle (2) is that the Main Part of the Modal Extension Principle is itself necessary, something for which we have just argued in Observation 2.

Principle (3) is something to which we have already tacitly appealed, viz. that the rule by which the semantic value of a concept is determined is something constitutive of it.

With these three principles in place, we can argue as follows. We are given that from the Main Part of the Modal Extension Principle and the relevant rules for fixing semantic values we can derive NA. The Main Part of the Modal Extension Principle is necessary, and so are the rules for fixing semantic values; and the derivation relation holds necessarily. Hence it is necessary that NA, that is we have NNA.

Observation 4: The Characterization of Necessity, that

A is necessary iff A is true according to all admissible assignments is plausibly itself necessary. The grounds for accepting it do not rest on any contingent particular features of the actual world. It is equally plausible that if things had been different in various possible ways, it would still be the case that A would be necessary iff A were then true according to all (then) admissible assignments.

More generally, it would seem to be impossible to justify the claim that it is necessity which is in question unless we have arbitrary necessitations of the Characterization of Necessity.

Observation 5: If the Characterization of Necessity -- we can abbreviate it to "Chzn" -- entails NA, then we also have NNA. What is entailed by a necessary truth is also necessary; and by Observation 4, the Characterization of Necessity is a necessary truth. Hence if the Characterization of Necessity entails NA, it will also be true that NNA.

We can also conclude that the necessitation of the Characterization entails NNA. The argument of the preceding paragraph establishes the conditional "If N(Chzn), then NNA". But we can necessitate this conditional, as we relied only on necessary premisses and transitions. So we have "N(If N(Chzn), then NNA)", i.e. N(Chzn) entails NNA.

Observation 6: In the preceding observations, we have the resources to show that, under the principle-based conception, the axioms, inference -- rule and rule of proof of the system T are all correct. The proof proceeds by showing that every formula provable in T has all of the following three properties:

(a) it is true under all interpretations of its schematic letters;

(b) it is necessary, under the principle-based conception of necessity;

(c) its necessity is entailed by one or both of the Characterization of Necessity and the Main Part of the Modal Extension Principle, or by some finite necessitation of one or both of these principles. (The finite necessitations of A are NA, NNA, NNNA, ... .) The proof is by induction on the length of the proof of the formula.

We take it for granted that the truth functional logical constants have the usual classical truth functions as their semantic values.(13)

Induction: Basis Step

We now argue, for the basis of the proof by induction, that the axiom schemata have properties (a)-(c).

Axiom A0. Property (a) is trivial. For (b) we note that the necessity of truth functional tautologies follows from the Main Part of the Modal Extension Principle, as we argued back in [Sections] 2. In any specification which is possible, the semantic value of a sentence with a propositional logical constant as its main connective will result from application of the same function of the semantic values of its constituents as is applied in fixing its semantic value in the actual world. This ensures that all truth functional tautologies are true according to any specification which is possible. This argument establishes properties (b) and (c) for axiom schema A0.

Axiom A1. The truth of (instances of) this follows from the argument of Observation 1. Since, as we argued, (Chzn) is necessary, (instances of) A1 are also necessary. So we have properties (b) and (c) for Axiom A1.

Axiom A2. By the Main Part of the Modal Extension Principle, A [implication] B will be true according to an admissible assignment s iff this condition is met: if A is true according to s, B is also true according to s. Hence if both A [implication] B and A are true according to all admissible assignments, so is B. Thus (instances of) A2 are true. Again, since this result is obtained by a derivation from the Modal Extension Principle, the conclusion is necessary; so we have properties (b) and (c) for this case.

Induction Step:

Our induction hypothesis is that any theorem with a proof n lines or fewer has properties (a)-(c). We are required to prove that any theorem with a proof of length n+1 also has properties (a)-(c). If line n+1 of a proof is obtained by rule R1 from earlier lines of the forms A and A [implication] B, then it will trivially have property (a), since RI is truth-preserving. Line n+1 will also be necessary, since by the induction hypothesis we have NA and N(A [implication] B), which entail NB. Line n + 1 will also have property (c), since whatever follows by modus ponens from premisses entailed by a certain set of principles is also entailed by that set of principles.

The other case within the induction step is that in which line n + 1 results from the application of the rule R2, the rule of necessitation. Since line n has property (c), by application of the principle that what is derivable from a set of necessities is also necessary, line n + 1 will also have property (c). This concludes the proof that the principle-based conception endorses the correctness of the system T.(14)

Observation 7: If any of the principles of possibility is not necessary, then the characteristic principle of S4, the schema NA [implication] NNA will not be correct; and otherwise, it will.

We can illustrate this by one proposed principle of possibility. This is the statement that if a particular table in fact originally came from a certain particular quantity of matter m, then according to any genuinely possible specification at which that table exists, it originally comes from a quantity of matter overlapping to some specified degree with that of m. Nathan Salmon observes that on this approach to the origins of artifacts, it can be very reasonable to claim that although a particular table could not actually have come from a certain quantity of matter [m.sup.*] which differs too much from its actual origins, the following may be true: had the table had somewhat different origins, it could then have come from [m.sup.*] (1981, [Sections] 28). Something is possibly possible for the table which is not actually possible for it. If we accept this, then the principle of possibility restricting the genuinely possible specifications to those in which the table in question comes from matter overlapping to a specified degree with m is, though true, not necessary. So S4 would not then be counted as correct by the approach in terms of the principles of possibility. Salmon conjectures that T, and not S4, "may well be the one and only (strongest) correct system of (first-order) propositional modal logic" (Salmon 1989, p. 4).

I do not mean to endorse Salmon's approach to the example, but only to point out the connection between the status of S4 and the modal status of the principles of possibility. S4 will be counted as correct as a matter of the meaning of metaphysical necessity if and only if each of the principles of possibility is itself necessary.

In special cases, such as those in which we are considering only a restricted vocabulary, like that of arithmetic, it is relatively uncontroversial that we have arbitrary necessitations of the principles of possibility. In those cases, we could give an argument for the principles of S4, as restricted to the bounds of that special case. More generally, on the basis of what we have already said, though, there is a major subset of sentences for which S4 is guaranteed, a subset identified in the following observation.

Observation 8: For any sentence A whose necessity follows from the Main Part of the Modal Extension Principle and/or the Characterizations of Necessity and Possibility, we also have NNA. That follows from preceding Observations.

Observation 9: It is worthwhile to take the natural reasoning one would offer for certain propositions involving iterated necessities within the framework of the principle-based conception, and then to compare that reasoning with the model-theoretic arguments one would offer for the same propositions within the standard Kripke-style semantics. Consider in particular a proposition NNA, where A is a tautology of the propositional calculus. Within the Kripkean semantics, we would argue that NA holds at any world (and a fortiori, the actual world), from the rules for evaluating propositional calculus formulae at arbitrary worlds. We would then argue that for any world w accessible from the actual world, NA holds there; and so NNA holds at the actual world. This argument relies on the standard evaluation rules holding not just at possible worlds, but at possibly possible worlds. As we noted earlier, there is no gainsaying the claim that they do (for "total" worlds, at least). It would be quite wrong to see the principle-based conception as in any way incompatible with the Kripke-style semantics. We can recall the question of why the standard evaluation rules for complex formulae hold for possibly possible worlds. A plausible answer to the question would say that the Modal Extension Principle constrains which specifications are possible; what makes a specification possibly possible is that, according to some possible specification, it is possible; and the constraints on something's being possibly possible involve the necessity of the Modal Extension Principle. The cor rectness of the standard evaluation rules for the possibly possible follows from the necessity of the Modal Extension Principle. If the Modal Extension Principle, and its necessitations, are fundamental for the elucidation and understanding of modal truths, then a derivation of NNA from the necessitation of the Modal Extension Principle is a derivation from what is explanatorily more fundamental.

Appendix B: Relaxing the assumptions

B.1 Completeness relaxed

The first assumption that we need to relax is that of the completeness of assignments and specifications. This was the assumption that, for each of the Thoughts and propositions of the range in question, each admissible assignment counts the Thought or proposition as true or counts it as not true. As Martin Davies noted, this is at variance with our normal conception of "ways the world might have been", if possible specifications are meant to be such ways. Davies's having straight hair is a way the world might have been, but this "way" leaves open virtually everything else about the world.(15) So we should admit partial, genuinely possible specifications, if they are to capture the genuinely possible ways the world might be. When assignments and specifications are partial, however, we cannot hold on to the letter of the Modal Extension Principle.

The Modal Extension Principle implies, for instance, that any admissible assignment which counts A [disjunction] B true must either count A as true or count B as true. The motivation for allowing partial specifications means that this consequence of the Modal Extension Principle cannot be required of partial assignments, for that would be to demand a form of determinacy after all. Let a refinement of s be any assignment t such that anything (Thought or proposition) counted true by s is also counted true by t. When we have partial assignments, all that we can require in respect of alternations is that if s is to be possible, and A [disjunction] B holds at s, then for any refinement t of s, there is a refinement at which A holds, or there is a refinement at which B holds. (We cannot require that any refinement of s distinct from s be one at which either A or B holds, since the refinement may consist in settling the values of thoughts or propositions other than A or B.)

What is needed is a qualification only of the letter of the Modal Extension Principle, and not of its spirit. The idea underlying the Modal Extension Principle can still be adapted to take account of the distinctive features of the framework of partial specifications. We can continue to illustrate with the example of alternation. Let us take it as granted that whatever our favoured theory of understanding alternation, it is constitutive of the concept that it fixes two conditions, each separately sufficient and jointly necessary, for the actual truth of A [disjunction] B (viz. the truth of A or of B). The appropriate clause in the partial framework then says that for A [disjunction] B to hold at a possible specification is for any refinement of this specification to have a refinement in which one or other of these two conditions holds. This clause is still anchored in the way the semantic value of an alternation is fixed in the actual world. It is just that it is modified in precisely the respect needed to accommodate the possibility of partial specifications.

In this treatment of partial specifications, I follow Humberstone (1981), who has shown that the natural semantical clauses for partial specifications, with the natural conditions on the refinement relation between specifications, will validate the modal system K, and the modal system T if accessibility is reflexive. The natural conditions include what Humberstone calls "Refinability": if a sentence is undefined at a specification s, there is a refinement of s according to which it is true, and another refinement of s according to which it is false.(16) The reader is urged to study Humberstone's paper for illuminating details.

B.2 Properties and tethering

The other assumption to be relaxed is that we have only concepts and Fregean semantic values in our semantical apparatus. The apparatus of this paper as we have developed it so far must represent an unstable middle position. For we have properties featuring in propositions, and we also refer to them in the constitutive principles. If there are such properties, why should there not be atomic expressions which refer to them? But once these expressions are recognized, the Modal Extension Principle again needs further examination.

A theorist who treats predicates, and perhaps higher-level operators, as referring to properties may well doubt whether there are any nonrigid (atomic) expressions. When operating earlier within the Fregean framework, I classified as nonrigid a wide range of atomic predicates. The theorist who employs properties in his semantic theory will simply say that these allegedly nonrigid expressions are simply ones which always pick out the same property, and what varies from world to world is merely which things have that property. The precise nature of this theorist's approach will depend on many decisions, including whether a sense/reference distinction is to be recognized and used; on the nature of the properties treated as semantic values; on the handling of compositionality; and much else. But however these matters are resolved, it may seem that as long as the theory says that all atomic expressions are rigid, the Main Part, at least, of the linguistic version of the Modal Extension Principle never applies.

We must, however, be cautious. When properties are taken as semantic values, we have to distinguish between semantic value and extension. The Modal Extension Principle was formulated above in the context of a Fregean theory for which semantic value and extension coincide. But for theories on which they come apart, what matters to the Modal Extension Principle is a claim (as one might expect) about extensions. Even for a theory which treats a property as the semantic value of a predicate, there is an important and wide class of expressions and concepts for which it remains true that their extension with respect to an arbitrary world is determined by applying to that world the same rule which, in the actual world, determines its actual extension. For the theorist of properties, that rule about the actual world has to be formulated in (at least two) steps. First, something determines which property is the semantic value; and then the actual extension is determined by which things actually have that property. The extension of such an expression or concept with respect to another world is fixed by applying just the second step to another world. The spirit of the Main Part of the Modal Extension Principle is then preserved for this class of expressions and concepts. The plausibility of the Main Part of the Modal Extension Principle for one class of cases is not proprietary to just one semantical framework.

There are, though, cases outside this class, and for which we need some analogue of the Second Part of the Modal Extension Principle. Let us first consider an example or two. Very roughly, the condition for something to have, in the actual world, the property picked out by the word "red" is for it either to produce a certain kind of experience in a certain sort of person in normal circumstances, or to have a certain kind of physical quality which is the ground of the disposition to produce certain experiences in those circumstances. (The roughness here will not matter for the present point.) However, the following counterfactual seems true: if humans were not to have colour vision and were to see only in shades of grey, there would still be red things in the world. This counterfactual is true, because the extension of "red" at other possible specifications is fixed by the following rule. Take the physical quality Q of surfaces and solids which is in the actual world the ground of objects' disposition to produce a certain kind of experience of them in normal circumstances. Then an assignment s is admissible only if, in the terminology of [sections] 2, an object is in the extension of the property-value s assigns to "red" iff it has the physical quality Q. If humans were to see only in shades of grey, there would still be objects in the world having that quality Q. We can say that the concept red, and the predicate "red", are marked as tethered. They are tethered to the quality Q, which is related in a certain way to what fixes the extension in the actual world. Other concepts are not marked as tethered. As is often noted, the concept poisonous is not: if our digestive systems were different, certain substances which are now poisonous would not then be poisonous. It seems clear that there can be two concepts or expressions which differ only in that one of them is marked as tethered, while the other is not. (There may of course in particular cases be very good reasons for operating with one of the concepts rather than the other.)

Does being marked as tethered coincide with being de jure rigid, at least within expressions which refer to properties? That depends in part upon our treatment of terms which express higher-order properties. Consider the predicate "poisonous" again. On one treatment, this refers to a higher-order property of objects, viz. the property of having some first-order physical property whose possession by an ingested substance kills humans. Using lambda-notation for properties, we can say that under this treatment the predicate refers to the property [lambda] x[exists]P(Px & H(P)]. Under that treatment, it could be held that "poisonous" is de jure rigid, referring rigidly to the higher-order property. Alternatively, "poisonous" may be taken to refer, at any given world, to the physical, first-order property which satisfies at that world the condition H( ) on properties. Then the point of the preceding paragraph implies that on that alternative treatment, "poisonous" is not seen as de jure rigid. Under the former treatment, being de jure rigid comes apart from being marked as tethered: for the former treatment takes "poisonous" as de jure rigid, though it is not tethered in the way "red" is. Under the latter treatment, "poisonous" is not rigid at all. The choice between these treatments is analogous to that which arises in the semantics of other terms in which higher-order conditions are involved, functional terms and (according to some) theoretical terms. There is a classical discussion by Lewis (1970).

When we are acknowledging properties and relations in our semantical theories, the Second Part of the Extension Principle should be revised to read thus, for the case of concepts:

Modal Extension Principle, Second Part:

For any concept C marked as tethered in a specified way to a property Q, any specification s is possible only if val(C,s) = Q. Once again, this can be seen as no more than a definition of what it is for a concept to be marked as tethered.


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(1) See Boolos (1979), and the references therein to Kripke's early thoughts on the matter.

(2) The modal "must" also features in his requirement (F) on the consequence relation (1983, p. 415).

(3) With an eye on later epistemological issues, we can also note at this point that if the Modal Extension Principle is a priori, then principles whose necessity is deducible from it in the way illustrated are also a priori. I will return to the relations between necessity and the a priori in

(4) For the view that a theory of individuation should quite generally be given in non-modal terms, with modal propositions as consequential, see Wiggins (1980, ch. 4, [sections] 1).

(5) For an exchange which is primarily concerned with ontological modalism, see Melia (1992) and Forbes (1992).

(6) My own previous writings have not consistently stuck to the position endorsed here. In some of them, talk of the modal would be better replaced with talk of the constitutive.

(7) For quantifiers, the formulation must be revised to take account of varying domains of existents for different specifications.

(8) The provision of a general theory of the constitutive, as opposed to the modal, seems to me to be an urgent task for philosophy. We certainly do not want all the initial puzzlement about modality simply to be transferred to the domain of the constitutive. Only a satisfactory general theory of the constitutive, and an attendant epistemology, can allay this concern.

(9) Suppose consistency is explained in model-theoretic terms, which involve set theory. If set theory were itself to be explained in modal terms, then the unknowability of some instances of the relation of consistency would trace back after all to the unknowability of certain modal truths -- thus contradicting Thesis (I). So: either set theory is not to be understood in modal terms (or at least not those of "metaphysical" modality); or the relation of consistency is not to be understood in model-theoretic terms. If both of these alternatives are untenable, but the approach to understanding modality in terms of a set of principles of possibility is correct, then Thesis (I) will be false.

(10) Thesis (II), like the principle-based conception itself, has a highly Leibnizian flavour. In the New Essays, Leibniz's protagonist, Theophilus, is asked how he would respond to the challenge to provide some examples of innateness. Theophilus answers: "I would name to him the propositions of arithmetic and geometry, which are all of that nature; and among necessary truths no other kind is to be found" (1981, p. 86). What really matters to Leibniz in what he calls innateness is justifiability, possibly involving definitions, in terms of logical truths which can themselves be seen to be true given an understanding of the terms they contain; and thus are true a priori. Leibniz had, of course, a pre-Fregean conception of logic, and in pre-Kripkean fashion, he also did not sharply distinguish the a priori and the necessary. When we factor out these historical differences, there remains between Leibniz and the present position a core of agreement on the explanation and the epistemology of necessary truths, a core which involves Thesis (II) and the a priori status of the principles of possibility.

(11) The conception articulated in these three Theses, and the underlying motivations for them, can be used in support of the view of the relations between the a priori and necessity developed in Forbes (1985).

(12) Versions of this material have been presented at New York University, the Universities of Oxford, Sheffield, St. Andrews, UCLA and (in a much earlier version) at the 1993 Fodor-LePore NEH Summer Institute on Meaning at Rutgers University. This material has been developed in regular lectures at Oxford since 1992. I have been greatly helped by valuable discussions on all these occasions, and particularly by the comments of David Bell, Paul Boghossian, Bill Brewer, Tyler Burge, John Campbell, Mark Crimmins, Michael Dummett, Hartry Field, Jerry Fodor, James Higginbotham, Stephen Schiffer, Jason Stanley, David Wiggins, Timothy Williamson, Crispin Wright and Takashi Yagisawa. I thank the referee for Mind and Mark Sainsbury for valuable editorial and expository advice, and for particularly probing comments. This paper is already long, and I will have to leave for some future occasion consideration of the bearing of the present approach on both fictionalist and the many varieties of thinker-dependent treatments of necessity. Finally, I acknowledge with gratitude the extraordinary support provided by a Leverhulme Trust Research Professorship, which was crucial in providing the time to write up this approach.

(13) This is for ease of exposition. The main points of the argument below could be adapted to a modal logic which had as its non-modal basis a set of nonclassical axioms.

(14) What is established here is the truth of all instances of the axioms of T, and the truth-preservingness of its inference rules. Validity is a further step. Strictly speaking, we must introduce a notion of accessibility between possible specifications, and a corresponding model-theoretic characterization of validity, to take that further step of showing that all principles of T are valid.

(15) The point was made in Davies (1975); quoted in Humberstone (1981, p. 314).

(16) There are also conditions linking the refinement relation and the accessibility relation: see Humberstone (1981, pp. 324-5).
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