The Nature of Epistemic Space (2001 version)

David J. Chalmers

Department of Philosophy
University of Arizona
Tucson, AZ 85721.

chalmers@arizona.edu

[N.B. A very rough draft, with no notes or citations yet, and almost certainly lots of mistakes.

Note added 2013: This html version was put online in April 2001 and has remained unchanged since (apart from this note and the qualification in the title). In mid-2008, I uploaded a revised pdf version with added sections on scenarios vs worlds, subsentential intensions, transcenario identity, and infinitary scenarios, and an abbreviated section on applications. I later revised this further for publication in the 2011 OUP book Epistemic Modality (edited by Andy Egan and Brian Weatherson), adding a good deal of extra material on infinitary scenarios and Kaplan's paradox (pdf version). I have left the 2001 version online because it has attracted published responses and because it contains some material that is not in the 2011 version.]

1 Ways the world might be

There are many ways the world might be, for all I know. For all I know, it might be that there is life on Jupiter, and it might be that there is not. It might be that Australia will win the next Ashes series, and it might be that they will not. It might be that my great-grandfather was my great-grandmother's second cousin, and it might be that he was not. It might be that copper is a compound, and it might be that it is not.

There are even more ways the world might be, for all I know with certainty. It might be that there are three chairs in this room, and it might be that there are not. It might be that water is H2O, and it might be that it is not. It might be that my father was born in Egypt, and it might be that he was not. It might be that I have a body, and it might be that I do not.

We can say that it is epistemically possible for a subject that P, when it might be that P for all the subject knows. So it is epistemically possible for me that there is life on Jupiter, or that copper is a compound. One can define various different standards of epistemic possibility, corresponding to various different standards for knowledge. For example, one might say that it is epistemically possible in the Cartesian sense (for a subject) that P when it might be that P, for all a subject knows with certainty. So in the Cartesian sense, it is epistemically possible for me that water is not H2O, and it is epistemically possible for me that I do not have a body.

A natural way to think about epistemic possibility is as follows. When it is epistemically possible (for a subject) that P, there is an epistemically possible scenario (for that subject) in which P. We can think of a scenario as a maximally specific way things might be: a sort of epistemically possible world, in a loose and intuitive sense. On this picture, corresponding to the epistemic possibility that Australia will win the next Ashes series are various epistemically possible scenarios in which they win in all sorts of different ways. And corresponding to the Cartesian epistemic possibility that I have no body are various scenarios in which I am disembodied, each epistemically possible by the Cartesian standard: e.g. scenarios in which I am a brain in a vat, or in which I am a disembodied Cartesian mind.

To fill out this picture a bit further, we might imagine that there is an overarching space of scenarios. We can think of these scenarios as constituting epistemic space. If a subject did not know anything, all scenarios would be epistemically possible for the subject. When a subject knows something, some scenarios are excluded. Every piece of substantive knowledge corresponds to a division in epistemic space: some scenarios are excluded out as epistemically impossible for the subject, while others are left open. More specifically, it is natural to hold that for a given P, there may be scenarios in which P is the case, and scenarios in which P is not the case. Then when a subject knows that P, scenarios in which P is not the case are excluded, while others are left open. The scenarios that are epistemically possible for a subject are those that are not excluded by any knowledge of the subject.

One can naturally suppose that the space of scenarios is equally divided by belief, and perhaps that the division by belief underlies the division by knowledge. Every substantive belief, whether or not it qualifies as knowledge, corresponds to a division in the space of scenarios. When a subject believes that P, we might say that some scenarios (in particular, scenarios in which not-P) are ruled out as doxastically impossible, while others are left open. A scenario is doxastically possible for a subject if and only if it is not doxastically ruled out by any of the subject's beliefs. When a belief qualifies as knowledge, the scenarios ruled out as doxastically impossible are also ruled out as epistemically impossible.

A picture of this general sort is often present in philosophical discussions of knowledge and belief. Within epistemology, it is common to think of knowledge in terms of the "elimination of possibilities", with some sort of underlying space of possibilities presumed. In discussions of skepticism, for example, the fact that certain skeptical scenarios are not eliminated is used as evidence that certain knowledge claims are not true. And in epistemic logic and the theory of belief revision, it is common to model epistemic possibility using epistemic relations to an underlying space of possible worlds.

It is surprisingly difficult, however, to make the intuitive picture precise. What sort of possibilities we are dealing with here? In particular, what is a scenario? And what is the relationship between scenarios and items of knowledge and belief?

It is natural to think of scenarios as possible worlds, and to think of a scenario in which P as a world in which P. But it is immediately clear that this will not work, at least on the most common contemporary understanding of possible worlds. There are subjects for whom it is epistemically possible that Hesperus is not Phosphorus; but on the usual understanding, there is no possible world in which Hesperus is not Phosphorus. It is epistemically possible for me that my great-grandparents were cousins and it is epistemically possible that they were not; but on the usual understanding, my great-grandparents are cousins either in all worlds in which they exist or in none. In the Cartesian sense, it is epistemically possible for me that water is not H2O, but on the usual understanding (assuming that water really is H2O), there are no possible worlds in which water is not H2O. So if we are to maintain that it is epistemically possible that P iff there is an epistemically possible scenario in which P, we cannot identify a scenario in which P with a possible world in which P, at least on the usual understanding.

Some might react to this by denying the intuitions about what is epistemically possible (e.g. holding that it is never epistemically possible that Hesperus is not Phosphorus), and some might react by denying the coherence of the picture connecting epistemic possibility to epistemically possible scenarios. I think that both reactions would be premature: the first loses touch with the phenomenon we are trying to analyze, and the second assumes that possible worlds as currently understood are the only available tool.

Instead, we should try to understand epistemic possibility on its own terms. We are not dealing here with counterfactual space: the space of ways the world might have been. Here, we are dealing with epistemic space: the space of ways the world might be. This epistemic space calls for its own epistemic tools of analysis. Where the analysis of counterfactual space invokes possible worlds as maximally specific ways the world might have been, the analysis of epistemic space should invoke scenarios as maximally specific ways the world might be. The two notions are quite distinct, although they have a deep underlying relationship.

In this paper, I will explore some ways of understanding epistemic space. I hope to make at least a prima facie case that a version of the notion is coherent, and that there are no decisive problems with it. And I will argue that the notion, if coherent, has a large philosophical payoff in the analysis of the content of thought and the meaning of language. It promises an analysis of the dimensions of meaning and content that are tied constitutively to the epistemic dimension. In particular, it promises an account of a sort of Fregean sense for linguistic expressions, and it promises an account of the narrow content of thoughts.

2 Preliminaries

On the picture suggested above, we might say that the notion of strict epistemic possibility - ways the world might be, for all we know - is undergirded by a notion of deep epistemic possibility - ways the world might be, prior to what anyone knows. Unlike strict epistemic possibility, deep epistemic possibility does not depend on a particular state of knowledge, and is not relative to a subject. Whereas it is strictly epistemically possible (for a subject) that P when there is some epistemically possible scenario (for that subject) in which P, it is deeply epistemically possible that P when there is some deeply epistemically possible scenario in which P. Since all scenarios are deeply epistemically possible on this picture, we can put this more simply: it is deeply epistemically possible that P when there is some scenario in which P.

How can we understand deep epistemic possibility more precisely? A problem immediately presents itself. It might be held that for any proposition P, there is some subject for whom P is strictly epistemically possible. Perhaps there are subjects who are so limited or so confused that they know nothing at all: if so, then any P is epistemically possible for them. Or for any given P, there might be a subject who is so limited or so confused with respect to the domain of P that they know nothing in this domain: if so, it is plausible that P will be epistemically possible for the subject. This may apply even to elementary logical contradictions, and to the negation of trivialities. If so, then if strict epistemic possibility entails deep epistemic possibility, then for all P, it will be deeply epistemically possible that P.

This notion of deep epistemic possibility might be useful for some purposes, but it is not very useful for our purposes. If it is deeply epistemically possible that P for all P, so that there is a scenario in which P for all P, then the structure of epistemic space will be near-trivial. Under certain natural assumptions (of a kind outlined below), for any propositions P, Q, R, ..., there will be distinct scenarios in which P&Q&R&..., P&Q&~R&..., P&~Q&R&..., and so on (including all sorts of scenarios in which P&~P, and so on). So the space of scenarios will be as fine-grained as the power set of the space of propositions. This would rob the space of scenarios of interesting structure, and it would also trivialize the relationship between scenarios and belief. Beliefs and knowledge would divide the spacer of scenarios only in a trivial way, and there could no nontrivial inferences from uneliminated scenarios to contents of belief or knowledge, or vice versa.

For a more useful notion, we need a more constrained notion of epistemic possibility, one that builds in some degree of rationality, and precludes incoherent possibilities such as those in which P&~P. There are various ways that one might do this, but the most obvious way is to exploit the idea that these incoherent possibilities can be ruled out a priori. We can say that P is deeply epistemically possible when it is not a priori that ~P. On this understanding, if P is a priori false, there will be no scenario in which P. Scenarios will now correspond to maximally specific coherent ways the world might be. We might think of the resulting notion of epistemic space as ideal epistemic space, or rational space.

It is clear that the idealized notion diverges from the ordinary understanding of epistemic possibility, at least within the a priori domain. For example, on the ordinary understanding, negations of complex mathematical and logical theorems can be epistemically possible (I do not know enough to rule them out), but on the idealized understanding above, these negations will be deeply epistemically impossible. Ideal epistemic space is well-suited to analyzing much empirical knowledge and belief, however, and it also yields an elegant system with many useful properties. So I will focus on the idealized notion of deep epistemic possibility and of epistemic space, at least for now. Later, I will discuss ways in which the idealization might be relaxed.

Another question arises: what are the objects of deep epistemic possibility, and what sort of objects are true or false at scenarios? So far I have often talked as if these are propositions. I think that this view is ultimately correct, but the contested nature of propositions raises difficulties in developing the approach. For example, on some theoretical views of propositions (e.g. views involving singular propositions, and some views involving sets of possible worlds), "Hesperus is Hesperus" and "Hesperus is Phosphorus" express the same proposition. But on the face of it, it is deeply epistemically possible that Hesperus is not Phosphorus (we cannot know a priori that Hesperus is Phosphorus), but it is not deeply epistemically possible that Hesperus is not Hesperus (we can know a priori that Hesperus is Hesperus). So it is not easy for these views of propositions to capture this difference: they must either deny the epistemic possibility claim, or accept that the epistemic possibility that P is not to be analyzed as the epistemic possibility of the proposition that P. I think that there is an alternative understanding of propositions that avoids these problems, and that the approach developed in this paper can itself help support such an understanding, but it would be problematic to assume such an understanding at the outset. To avoid this sort of theory-dependent confusion, it is best to start without talking about propositions at all.

Instead, we can associate epistemic possibility, and truth or falsity at scenarios, with something in the vicinity of beliefs and sentences. I will take beliefs to be token mental states: occurrent beliefs will be the paradigm example. The sentences I will be concerned with are sentence tokens that are used to make assertions: call such a token a statement. I will take it that there is a relation of expression between statements and beliefs, and that when a statement expresses a belief, the statement and the belief have the same truth-value. Statements usually express a belief of the subject making the statement, and never express more than one belief of the subject. (We might think of expression here as direct expression: a given statement might indirectly express more than one belief, and indirect expression may allow variation in truth-value.)

When statements do not express a belief (e.g. because the subject is not confident in the truth of what is asserted), I will take it that they express a propositional attitude of a more general sort, one that we might call a thought. To think that P, as I will use the term here, is to entertain the hypothesis that P, and a thought that P is an entertaining of the hypothesis that P. Occurrently believing that P entails thinking that P, in much the same way that knowing that P entails believing that P. Like beliefs, thoughts can be true or false. A statement always expresses exactly one thought, and the thought and the statement will always have the same truth-value.

I will take it that there is a relation of continuance (or persistence) that can hold between thoughts of a subject at different times, and that continuance guarantees that two thoughts that stand in this relation have the same truth-value. I will take it that the thoughts (and derivatively, the statements) of a given subject can stand in a relation of negation to each other, and in a relation of conjunction (and other logical relations) to each other. And I will take it that a thought can come to be accepted, yielding a belief: in this case we can say that the belief is an acceptance of the thought.

We can say that a belief is a priori justified when it is justified independently of experience. A belief is a priori justifiable when it can be justified independent of experience, yielding a priori knowledge. More generally, a thought is a priori justifiable - or more simply, a priori - when it is possible for an acceptance of it to be justified independently of experience, yielding a priori knowledge. Here, apriority abstracts away from contingent cognitive limitations. If there is any possible mental life that starts from a thought and leads to an a priori justified acceptance of that thought, the thought is a priori. A statement is a priori when it expresses an a priori thought. A thought is epistemically possible when its negation is not a priori. A statement is epistemically possible when it expresses an epistemically possible thought. One thought (or statement) implies another when a conjunction of the first with a negation of the second is epistemically impossible.

Note that the uses of "epistemically possible" in the previous paragraph correspond to deep epistemic possibility, in the idealized sense. In the following discussion, I will be setting aside issues concerning strict epistemic possibility, and non-idealized notions of deep epistemic possibility. So from now on, unqualified talk of epistemic possibility should always be taken as talk of deep epistemic possibility in this sense.

3 Principles of epistemic space

On the picture developed at the start, items of knowledge and belief divide epistemic space. Any item of knowledge or belief excludes some scenarios, and endorses others. More generally, any thought can be held to excludes some scenarios and to endorse others. We can put this by saying that for any given thought (or belief, or item of knowledge), there is a class of scenarios that verify the thought and a class of scenarios that falsify the thought. Intuitively, a scenario verifies a thought when it corresponds to a way the world might be that the thought endorses, and a scenario falsifies a thought when it corresponds to a way the world might be that the thought excludes.

(Note that verification and falsification as defined here are not evidential notions. It may be epistemically possible, for example, that there is life in black holes for which we can have no evidence. Intuitively, this epistemic possibility is backed by various scenarios in which life exists unperceived in black holes. My thought that there is life in black holes will be verified by these scenarios, not because these scenarios could provide evidence for the truth of the thought, but because they correspond to ways the world might be that the thought endorses. In effect, verification and falsification are broadly representational notions. If one holds independently that truth is tied constitutively to evidence, as on some anti-realist views, it may be that verification will also be tied to evidence; but this tie is not compulsory.)

If we allow that thoughts can have multiple truth-values (such as true, false, and indeterminate), we should also allow that the verification relation can yield the same truth-values. We might say that there is a function that maps an arbitrary scenario S and thought T to a truth-value verifies(S,T). Then S verifies T iff verifies(S,T) is true, and S falsifies T iff verifies(S,T) is false. When S verifies T, we can say that T is true at S. When S falsifies T, we can say that T is false at S. In other cases, it may be that T is indeterminate at S, or perhaps that T has some other truth-value at S.

It is an important part of the picture I am developing that for any subject at a time, there is exactly one way the world is, relative to that subject and time. More specifically, relative to any thought, there is exactly one way the world is, relative to that thought. (Recall that thoughts are tokens, embedded in a world.) We can put this by saying that exactly one scenario is actualized at the thought. The thought will be true if it is verified by its actualized scenario, and false if it is falsified by its actualized scenario.

So we are postulating a space of scenarios, a function of verification from thoughts and scenarios to truth-values, and a relation of actualization between scenarios and thoughts. These must satisfy at least the following principles of epistemic space:

Plenitude: A thought T is epistemically possible iff there exists a scenario S such that S verifies T.

Actuality: For every thought T, there is a unique scenario S such that S is actualized at T.

Truth: For all S and T, if S is actualized at T, then T's truth-value is verifies(S,T).

Compositionality: For all S and T, if T is a logical composition comp(Ti), where comp is a truth-function, then verifies(T,S) = comp(verifies(Ti,S)).

Parsimony: If scenarios S1 and S2 are such that for all possible thoughts T, verifies(S1,T) = verifies(S2,T), then S1 and S2.

For any thought, there will be an associated set of scenarios that verify the thought. More generally, for any thought, there will be an associated function that maps a scenario to the truth-value of thought thought at the scenario. We can call this function the thought's epistemic intension. A thought's epistemic intension might be thought of as representing its epistemic content - the way that it divides epistemic space. Strictly speaking, we might think of an epistemic intension as defined here - relative to an idealized notion of epistemic possibility - as a thought's ideal epistemic content, since any two thoughts that are a priori equivalent will have the same epistemic content. (The latter usage would leave open the possibility of a less idealized notion of epistemic content such that thoughts that are nontrivially a priori equivalent can have distinct epistemic content.) We can define the epistemic intension of an statement in a similar way.

One can also postulate similar principles governing the relationship between scenarios and statements. We can say that a scenario verifies an statement if it verifies the thought expressed by that statement, and that scenario is actualized at an statement iff it is actualized at the thought expressed by the statement. Then analogs of Epistemic Plenitude, Actuality, and Truth for statements follow automatically. An analog of Compositionality will follow as long as we allow that statements have compositional relations that mirror the compositional relations of the thoughts expressed. An analog of Parsimony will follow as long as we allow that every possible thought could be expressed by a possible statement.

Of these, Plenitude is the core principle that captures the general picture set out at the beginning. Actualization and Truth capture the special role of the way the world actually is (relative to a thought), among all the ways the world might be. Compositionality ensures that negations and conjunctions, and other logical compositions of thoughts have the truth-values that one would expect. Parsimony says, in effect, that epistemic space contains no redundancy, so that it is a minimal space that accomplishes its epistemic purpose. Of these, it is arguable that Parsimony is not absolutely essential, but the other four principles seem crucial to the notion of epistemic space.

Of course it would be easy to simply postulate that there is a space of scenarios and relations of verification and actualization that satisfy these principles. But this would raise the question of whether the postulation is coherent. We can make a positive case for coherence by constructing epistemic space. That is, we can identify the space of scenarios with a space of abstract objects that we have prior reason to believe is coherent, and we can make the case that there are relations of verification and actualization over this space that satisfy the relevant principles. In the following sections, I explore two ways of making such a construction.

There is a further principle worth mentioning:

Maximal Thoughts: For every scenario S and every subject, there exists a possible thought T such that for all possible thoughts T' of the subject, S verifies T' iff T implies T'.

We might think of T as the thought that S is actual. Where S is a maximally specific epistemic possibility, T will be a maximally specific coherent thought. Intuitively, S will verify T' iff the thought that S is actual implies the thought that T' (that is, if the thought "S is actual and ~T" is epistemically impossible), and S will falsify T' iff the thought that S is actual is implies the thought that ~T (that is, if the thought that "S is actual and T" is epistemically impossible).

Let us say that a maximal thought is an epistemically possible thought T such that there is no thought T' such that T' implies T while T does not imply T'. (Intuitively, if both conjunctions were coherent, then T could not be a maximally specific coherent thought.) It is easy to see that if Maximal Thoughts is true, then for any S, the corresponding thought T is a maximal thought in this sense. (If it were not, then S would verify both T1 and ~T1 for some T1, so it would verify the epistemically impossible T1&~T1.) We can call T a maximal thought corresponding to S.

The Maximal Thought principle is not essential to the notion of epistemic space. It is not entirely obvious that maximal thoughts can exist, and the idea of epistemic space may still be coherent even if there is a problem with maximal thoughts. But if maximal thoughts can exist, the principle certainly provides a useful aid in the analysis of epistemic space.

4 Scenarios as centered worlds

The most natural way to think of scenarios, at least initially, is as possible worlds. In a way this is trivial - scenarios are defined as possible (in some sense) ways a world might be (in some sense). But the notion of possibility invoked here differs from the notion of possibility that is usually associated with possible worlds: it is a sort of epistemic possibility, whereas possible worlds are usually understood to be associated with a sort of "metaphysical" possibility. Still, the question arises as to whether possible worlds understood in the latter sense might serve to help us model the space of scenarios, at least indirectly. That is: can we use the space of metaphysically possible worlds to construct a space of scenarios, and to make the case for a verification relation between scenarios (so understood) and thoughts?

I think we might. The intuitive idea is simple: to every possible world W, there corresponds a very specific (deep) epistemic possibility: the epistemic possibility that W is actual. Given any specific epistemic possibility of this sort, it will be epistemically compatible with some thoughts, and incompatible with others. One might then proceed as follows: (i) identify the space of scenarios with the space of possible worlds; (ii) say that a world W verifies a thought T when the hypothesis that W is actual is epistemically compatible with T; and (iii) say that a world W is actualized at a thought T when W is the world containing T.

The idea here is intuitive, since I have invoked the intuitive idea of a hypothesis, and of a hypothesis being compatible with a thought. One could give more flesh to the idea by invoking the thought that W is actual instead, and holding that W verifies T iff the thought that W is actual is epistemically compatible with T (where two thoughts are epistemically compatible when their conjunction is epistemically possible). As with the maximal thought principle, it is not obvious that such specific thoughts can exist for all W, so we do not want to rest the framework on their existence. And even we we take this approach, there is still an intuitive element, in that we have have appealed to the intuitive idea of a thought that W is actual. But in any case, this way of thinking about things at least gives a sense of why one might try to model scenarios with possible worlds.

I think that this sort of approach is on the right track. But there are a few obstacles to understanding scenarios in terms of possible worlds, some of which we have already seen. The obstacles fall into four classes, which we can class under the four headings: indexicality, rigidity, strong necessity, and parsimony.

(1) Indexicality

The first obstacle is a problem for any view that identifies scenarios with objective states of affairs. As a number of philosophers have argued, there are certain questions that may be left open by any amount of objective information about the world. One can put this as follows. Let D be a full objective specification about the world. Let T be an indexical claim, such as "I am a philosopher" or "It is raining here now" or "Today is Friday". Then in each case, it may be that both D&T and D&~T is epistemically possible. That is, the information in D may not enable one to settle the status of T. To settle these indexical claims, one needs to be able to locate oneself and the current location and time with respect to an objective description of the world; and this locating information cannot be derived from objective information. So objective information is not maximal information.

The problem this poses is clear. In these cases, D&T and D&~T are epistemically possible, so both are verified by a scenario; and they are epistemically incompatible, so no scenario verifies both. But if D is a complete objective description of the world, then there will plausibly be only one world (objectively understood) in which D is the case. And even if there is more than one objectively indistinguishable D-world, it seems that there is no room in such a space of objective worlds for a division between worlds in which today is Saturday and worlds in which it is not.

The basic problem is in satisfying Plenitude. Maximal epistemic possibilities are differentiated not just by objective information but also by indexical information, and there are not enough possible worlds to go around. To model epistemic space, we need a more fine-grained space that allows indexical differentiation.

This suggests a natural solution: we need to supplement possible worlds with some further indexical structure. Specifically, we can identify scenarios with centered worlds: worlds marked with a "center" consisting of a subject and a time within the world. (Equivalently, centered worlds might be identified with ordered triples of a world, a subject in that world, and a time in that world.) We can think of the center as a sort of "you are here" marker, corresponding to a hypothesis about the subject's identity and location. If W' is a centered world, the hypothesis that W' is actual will be a sort of indexical hypothesis about the world and the subject's location within it. Putting things in terms of thoughts: if W' consists of an ordinary world W centered on a subject and a time, my thought that W' is actual will be the the thought that I am now at the center of W' (i.e. that W is actual, I am the subject indicated, and now is the time indicated).

When scenarios are modeled by centered worlds, the problems above can be handled straightforwardly. My thought that I am a philosopher will be verified by all centered worlds in which the subject at the center is a philosopher. My thought that it is raining here now will be verified by all centered worlds in which it is raining at the location of the marked subject at the time marked at the center. My thoughts T1 and T2 above will be verified by different classes of centered worlds, with an objective world in common but different locations for the center. And so on. Intuitively, this is just how we would expect these thoughts to divide the space of epistemic possibilities.

It is useful to stipulate that the marking of centered elements in a centered world is optional. This way, we can accommodate the (arguable but plausible) a posteriority of claims such as "Subjects exist" and "The universe is temporal". If we allow centered worlds without marked subjects or times, then there will be subjectless scenarios and timeless scenarios to falsify these claims. There can even be an empty scenario to verify "Nothing exists", which is arguably a deep epistemic possibility. (Here I assume that "I exist" is a posteriori, being justified by experience. If someone holds that "I exist" is a priori, then they can require that centered worlds contain marked subjects.) It may also be that we sometimes need additional optional marked information at the center of a world. For example, for the verification of certain demonstrative thoughts (e.g. Austin's "Two Tubes" puzzle - "that spot is red", in a symmetrical visual field with two spots), one needs to allow one or more marked experiences at the center of a world, to distinguish otherwise indistinguishable contents. (For more discussion, see...) But the general framework is much the same.

(2) Rigidity

The second problem has already been mentioned. There are subjects for whom it is epistemically possible that Hesperus is not Phosphorus: more specifically, there are subjects for whom the statement "Hesperus is not Phosphorus" expresses an epistemically possible thought. But on the usual understanding of possible worlds, "Hesperus" and "Phosphorus" are rigid designators: they pick out the same object (here the planet Venus) in all possible worlds. If so, then there is no possible world satisfying "Hesperus is Phosphorus". Something similar applies to the epistemic possibility that water is not H2O ("water" picks out H2O in all worlds, so no world satisfies "water is not H2O) and the epistemic possibility that my greatgrandparents were or were not cousins (I have the same ancestors in all worlds, so if my greatgrandparents are not cousins, no world satisfies "my greatgrandparents are cousins"). Adding centers to the possible worlds does not help with this. So it may seem that if scenarios are centered worlds, then Plenitude cannot hold: there are cases in which an statement (e.g. "water is not H2O") is epistemically possible, but there is no centered world satisfying the statement.

This conclusion would be premature, however. There will be a problem for Plenitude only on the assumption that a possible world verifies an statement precisely when it satisfies that statement. And this points immediately to a possible solution: we can hold that verification and satisfaction are quite different relations. On such a view, it may be that for epistemically possible statements such as the above (e.g. "water is not H2O"), then although no world satisfies the statement, there are nevertheless worlds that verify the statement.

In fact we have already seen reason to accept that verification and satisfaction are different relations. Above, we saw that "I am a philosopher" is verified by any centered world in which the being at the center is a philosopher. But it is false that any such world satisfies "I am a philosopher". For example, consider a counterfactual world in which David Chalmers is a mathematician and George Bush is a philosopher, and consider a centered version on this world, centered on George Bush. Then by the standard sort of evaluation, this is a world in which I am not a philosopher (instead I am a mathematician) - that is, the world satisfies "I am not a philosopher". But nevertheless, the world verifies "I am a philosopher".

The distinction is also suggested by the intuitive test for verification of a thought T given above: is the hypothesis that W is actual epistemically compatible with T? Take a "Twin Earth" world W that is superficially like our world, and in which the oceans and lakes around the center are filled with (clear, drinkable) XYZ. It is epistemically possible (i.e. not ruled out a priori) that W is actual. The epistemic possibility that W is actual is epistemically compatible with the thought that water is XYZ, and is epistemically incompatible with the thought that water is H2O. If so, then W verifies "water is not H2O". At the same time, Kripke and Putnam have argued that W is a counterfactual world in which XYZ is not water, and in which water is still H2O (if it exists). So W does not satisfy "water is H2O", even though W verifies "water is H2O".

What is going on here? The root of the difference is that satisfaction is a broadly subjunctive notion, concerning ways the world might have been. To determine whether the Twin Earth world satisfies "water is XYZ", we can ask: if the liquid in the oceans and lakes had been XYZ, would water have been XYZ? If Kripke and Putnam are correct, the answer is no. Verification, on the other hand, is a broadly epistemic notion, concerning ways the world might be. To determine whether the Twin Earth world verifies "water is XYZ", we can ask: if the liquid in the oceans and lakes is XYZ, is water XYZ? This indicative conditional behaves epistemically, turning on whether there is an appropriate epistemic relation between the antecedent and the conclusion. And in this case, the epistemic relation is present, so the answer is yes.

The notion of possibility that is operative in most contemporary discussions is what one might call subjunctive possibility: P is subjunctively possible if it might have been that P. (Kripke is quite explicit about this, for example.) The notion of possibility that is operative in the epistemic realm is epistemic possibility: P is epistemically possible if it might be that P. It is already a familiar point that these two notions of possibility behave differently. It is not surprising that there are correspondingly different relations between thoughts, statements, and possible worlds.

One can use verification and satisfaction to define parallel notions in the epistemic and subjunctive realms. For example, we defined the epistemic intension of an statement A or a thought T as the function mapping a scenario S to verifies(S,A) or verifies(S,T). If scenarios are centered worlds, then the epistemic intension is a function from centered worlds to truth-values. We can similarly define the subjunctive intension of A or T as the function mapping a possible world W to satisfies(W,A) or satisfies(W,T) (where we hold that a world satisfies a thought if it satisfies an statement that expresses that thought). If scenarios are centered world, these intensions will have closely related domains (the same up to centering), but for a given thought or statement, the intensions themselves will be very different.

In a similar way, one can postulate both epistemic intensions and subjunctive intensions for concepts and for tokens of expressions smaller than sentences, mapping. When an expression is a rigid designator - such as "I", "water", or "Aristotle" - its epistemic intension and subjunctive intension will usually behave quite differently. The subjunctive intension of a rigid designator will pick out the same object in all worlds, but its epistemic intension will not. For example, the subjunctive intension of my concept "I" will pick out David Chalmers in all worlds (mirroring the subjunctive necessity of "I am David Chalmers" - that is, I could not have failed to be David Chalmers). But the epistemic intension of "I" picks out whichever being is marked at the center of a world, irrespective of whether he is David Chalmers. Similarly, the subjunctive intension of "water" picks out H2O in all worlds (if Kripke and Putnam are correct), but its epistemic intension picks out different substances in different worlds, as we have seen.

The same applies to (tokens of) names such as "Aristotle". The subjunctive intension of a name picks out the same person across worlds, but the epistemic intension cannot. To see this, note that it is (deeply) epistemically possible for me that Plato was Aristotle: this is unlikely, but not ruled out a priori. So there must be a scenario verifying my statement "Plato is Aristotle"; and if scenarios are centered worlds, there must be a centered world verifying "Plato is Aristotle". One such is a world W in which an individual lived a double life, publishing one set of (familiar-looking) books as "Plato" and another as "Aristotle", and in which the community at the center has no knowledge of that distinctness. Intuitively, the hypothesis that W is actual is (deeply) epistemically possible, and it epistemically entails the hypothesis that Plato is Aristotle. If Kripke and Putnam are right, this is not a world satisfying my statement "Plato is Aristotle", but it may nevertheless be a world verifying "Plato is Aristotle".

This suggests a broadly two-dimensional approach to meaning and possibility, and as such may recall the two-dimensional approaches developed by Kaplan, Stalnaker, Evans, and others. I have explored the similarities and differences elsewhere. One important difference, however, is that an epistemic intension is not defined in terms of the context-dependence of a thought or statement's truth-value. The epistemic intension is a matter of epistemic dependence, turning on whether a thought T is epistemically compatible with the hypothesis that W is actual. Nothing here gives any special role for a copy of the thought T at the center of W, and the epistemic intension will usually be defined over worlds in which any such token is absent. The epistemic intension will even be defined over worlds in which language and thought are absent altogether. Witness "language does not exist", or "no-one exists", each of which is arguably deeply epistemically possible, and which will be correspondingly verified by languageless worlds and by subjectless worlds respectively.

Note that I have not given a formal definition of what it is for a centered world to verify a thought, but only an intuitive account: W verifies T when the hypothesis that W is actual is epistemically compatible with T. A formal definition would require two things. First, we would have to rigorize talk of the epistemic compatibility of hypothesis and thoughts, either by formally defining and characterizing epistemic possibilities, or by dispensing with hypotheses and appealing to the thought that W is actual instead. Second, we would have give an account of what is involved in the hypothesis that W is actual. One natural way to do this would be to distinguish a sort of canonical way of describing possible worlds, so that for any centered world W there will be a canonical description D (including indexical claims to capture the centering). Then the hypothesis or thought that W is actual might simply be the hypothesis or thought that D.

So one possible formal definition would be the following: a centered world W verifies a thought T when T is epistemically compatible with a thought expressed by D, where D is a canonical description of W. Of course this definition requires an account of canonical descriptions, and it requires that the relevant maximal thoughts can exist. I discuss the details of this and other formal definitions elsewhere. For present purposes, it is sufficient to note that the general approach to verification is prima facie coherent, and that the phenomena of rigidity pose no fatal problems.

(3) Strong necessities

We have seen that the existence of a posteriori necessities such as "water is H2O" poses no deep problem for the picture of scenarios as centered worlds, as long as we distinguish verification from satisfaction. It is not hard to see that all of the a posteriori necessities suggested by Kripke can be handled in a similar way: in these cases, when N is an a posteriori necessity there is plausibly a centered world verifying N. But on some philosophical views, there exist a posteriori necessary truths that do not have this property. We can call these strong necessities.

Consider a theist view on which it is necessary that an omniscient being exists (e.g. because it is necessary that God exists and is omniscient), but on which it is not a priori that an omniscient being exists (e.g. because it is not a priori that God exists). On such a view, "There is an omniscient being" is necessary but a posteriori, and "There is no omniscient being" is (deeply) epistemically possible. On such a view, it will not merely be the case that no possible world satisfies "There is no omniscient being". It seems clear that no world verifies "There is no omniscient being". That is, all possible worlds W are such that the hypothesis that W is actual is epistemically incompatible with the thought that an omniscient being exists. There are no relevant two-dimensional phenomena involving rigidity here: it simply seems that there are not enough worlds to go around. If this view is correct, then a model on which scenarios are centered worlds is incompatible with Plenitude.

There are some other views on which the same applies:

(i) A particular strong "strong laws" view on which the fundamental laws and properties instantiated in our world are the fundamental laws and properties of every possible world. Let say the view also holds (plausibly) that fundamental laws are a posteriori. On this view, a denial of the law of gravity (say) will be deeply epistemically possible, but there will be no possible world satisfying this denial, and there will also be no possible world verifying the denial.

(ii) A materialist view on which truths Q about experience are necessitated by the conjunction P of physical truths, but on which Q is not a priori derivable from P. Here, a psychophysical conditional "P and not-Q" will be epistemically possible. It is not hard to show that if there is even a possible world verifying this conditional (as in the Kripkean cases), problems for materialism ensue. So some materialists deny that even a verifying world exists. If so, the conditional "If P, then Q" is a strong necessity.

(iii) A view on which there are mathematical claims M - perhaps the Continuum Hypothesis? - that are true and are necessary, but are not knowable a priori by any possible being. On such a view, it seems that M will be a strong necessity: ~M will be epistemically possible, but verified by no possible world.

Other such views could be developed: e.g. one on which moral claims can be true and necessitated by natural truths, without being a priori derivable from natural truths; or a similar view about vague claims. In each of these cases, the distinction between verification and satisfaction does not seem to help. If the views in question are correct, there are simply not enough possible worlds to verify all epistemically possible thoughts and statements.

The simplest response to this problem, and the response that I believe is correct, is to deny that strong necessities exist. In each case, the claim in question about necessity is at least controversial. In some cases, proponents claim support from the Kripkean cases, but these cases give no reason to believe in this much stronger phenomenon. In fact, one can argue in reverse: the fact that the link between epistemic possibility and verification by possible worlds is so strong elsewhere gives reason to believe that these claims are incorrect. One can also argue that there are deeper problems with these views. But I have argued for these claims elsewhere, and will not repeat those arguments here.

It is at least clear that these views provide no clear reason to reject the model of scenarios as centered worlds, since in no case is the view in question clearly true. Still, the existence of these views entails that the claim that scenarios can be modeled by centered worlds will be at least as controversial as the denial of the views. And it would be desirable to give an account of scenarios that even holders of these views could accept. If so, that provides at least some reason to look at other models of scenarios.

(4) Parsimony

So far, we have looked at problems of the sort: there are not enough possible worlds to act as scenarios. But there are also potential problems of the sort: there are too many possible worlds to act as scenarios. That is, while the problems above are mostly problems for Plenitude, one can also raise problems for Parsimony. In particular, it seems that there exist groups of centered worlds such that any possible thought is equally verified or falsified by any world in the group. If so, it seems that each world in the group corresponds to the same scenario.

One way this can happen is with symmetrical worlds. Say that a world is mirror-symmetrical, and consider centered worlds W1 and W2 centered on corresponding subjects on each side, at the same time. Then it seems that there is no thought T such that the thought that I am in W1 is epistemically compatible with T, while the thought that I am in W2 is epistemically incompatible with T. The same goes for a world with a cyclic Nietzschean eternal recurrence of indistinguishable cycles, extending indefinitely into the past and the future. If we take a group of centered worlds Wi centered on corresponding subjects and times in different cycles, then it seems that for any T, if one world Wi verifies T, then all worlds Wi verify T. In these cases, it seems that the different centered worlds all correspond to the same epistemic possibility, violating Parsimony.

Parsimony might also be violated if possible worlds can contain inconceivable features. Say that there are two possible worlds W1 and W2 that are otherwise indistinguishable, except that at a certain point they contain different features F1 and F2. And say that F1 and F2 are inconceivable, in the sense that there is no possible concept that refers to F1, or to F2. Then it seems that there will be no possible thought T such that T is verified by W1 but not W2.

Finally, suppose that (as some believe) there are qualitatively indistinguishable possible worlds. Take two identical twins Bill and Bob in the actual world. Some argue that there can be qualitatively indistinguishable worlds W1 and W2 such that only Bill exists in W1 and only Bob exists in W2. Then it is plausible that W1 and W2 to not correspond to distinct epistemic possibilities: any thought verified by W1 is also verified by W2. If so, Parsimony is violated.

Of these three cases, the last two are controversial and might be denied. But the first is relatively uncontroversial, and the other two at least raise problems. So it seems that the space of centered worlds and the verification relation, as we understand them, do not satisfy Parsimony.

One could respond in different ways. One might simply jettison Parsimony, holding that it is an inessential principle: certainly it seems less essential than Plenitude. One might also modify the picture slightly, by identifying scenarios with equivalence classes of centered worlds, where the worlds in groups such as the above will all fall into the same equivalence class. Either response will still allow a serviceable construction. Still, both responses suggest that there is at least a mild mismatch between scenarios and centered possible worlds.

What is the upshot of the four obstacles to identifying scenarios with possible worlds that we have discussed? The obstacles due to indexicality and rigidity can be overcome relatively easily, by invoking centered worlds and distinguishing verification from satisfaction. The obstacle due to strong necessity can be denied, and the obstacle due to parsimony can be dealt with as above.

Still, the last two obstacles suggest that while centered worlds may do a good job of modeling scenarios, the match is not perfect. The existence of philosophical views on which there are strong necessities suggests that even if these views are misguided, an analysis of scenarios as centered worlds will be at least mildly controversial. Because it makes a substantive (if plausible) claim about the relationship between possible worlds and epistemic possibility, it goes beyond a surface analysis of epistemic possibility itself. The problems with parsimony also suggest a slight conceptual mismatch between the notions. So while centered worlds may provide a very useful way of thinking about scenarios, it may also be useful to look at other notions.

5 The epistemic construction of scenarios

The obstacles in the previous section all have a common source. They arise because we are taking a class of entities - the possible worlds -- developed in the service of a different notion of possibility (what might have been the case), and adapting it to help analyze the notion of epistemic possibility (what might be the case). It is inevitable that this adaptation will lead to certain complications. An alternative strategy suggests itself. Instead of adapting a different modal space, we might construct the space of scenarios directly, by a construction grounded in epistemic notions. In particular, we might take (deep) epistemic possibility as basic, and proceed from there. In this way, we can give an account of epistemic space in its own right.

The natural way to proceed is to identify scenarios with constructions out of thoughts or statements. We already have a notion of epistemic possibility that applies to these entities, and this notion can be exploited to construct scenarios directly. Of course we will need to appeal to possible thoughts and statements, since it is unlikely that all the thoughts and statements we need will exist in the actual world. For present purposes, I will understand these as subjunctively possible thoughts and statements - thoughts and statements that might have existed. Because of this, our construction of epistemic space will not be entirely independent of subjunctive modal notions, but the relationship will be much more indirect than in the previous constructions, so that the resulting epistemic space may be quite different from subjunctive modal space. An alternative strategy would be to appeal to the epistemic possibility that certain thoughts and statements exist, and bootstrap the construction from there, but I will not pursue that strategy here. Henceforth (to avoid clutter and also to avoid confusion with the existing and distinct notion of an epistemically possible thought), reference to thoughts and statements should be understood as reference to possible thoughts and statements.

I will work here with thoughts rather than statements, but everything here could straightforwardly adapted to statements. The approaches pursued here will start with the possible thoughts of a subject at a time, and so will first define a space of scenarios relative to a given subject and time. After that, we can look at defining a common space across subjects and times.

The simplest strategy is to identify scenarios with equivalence classes of thoughts. In particular, we can appeal to the class of maximal thoughts. As defined previously, a maximal thought is an epistemically possible thought T such that there is no epistemically possible thought T' such that T' implies T while T does not imply T1. We can say that two maximal thoughts T1 and T2 are equivalent if T1 implies T2 and T2 implies T1. Then we can say that a scenario is an equivalence class (under this equivalence relation) of maximal thoughts, and that a scenario verifies a thought T if a maximal thought in its equivalence class implies T.

It is easy to see that this construction will satisfy Compositionality: this follows from the analogous principle about implication. The construction will also satisfy Parsimony: if two maximal thoughts imply the same thoughts, then they will imply each other, and will be members of the same equivalence class.

In order to satisfy Plenitude, we need the following principle: for every epistemically possible thought T, there is a maximal thought that verifies T. It is not absolutely obvious that this principle is correct: it is not absolutely obvious that maximal thoughts exist (perhaps for every possible thought, there will be a more specific thought?), and it is not absolutely obvious that there will be one for every epistemic possibility. Still, there is no obvious problem with the idea. And it is as least intuitively reasonable to suppose that from a given thought, a maximal thought could be obtained as a sort of maximal conjunction, by conjoining other thoughts or their negations until the limits of epistemic possibility are reached.

It is not obvious how to define the actualization of a thought: the maximal thought corresponding to how things are from the point of view of a given subject at a given time, thinking a given thought. It is tempting to say: a maximal thought M is actualized at a thought T if, were the subject to think M at the same time as T, M would be true. But this cannot work: any M that implies "I am not thinking a maximal thought" will fail this test, but intuitively, an M like this is required to capture how things are for most subjects. In effect, we want M to correctly describe how things were before M was thought. I will not pursue the project of defining actualization here. Instead I will take it that we have a reasonably good intuitive grip on the notion, and I will simply assume a relation of actualization between maximal thoughts and thoughts, such that a thought if true iff it is implied by its actualization.

If one has worries about maximal thoughts, one might also construct scenarios via maximal classes of thoughts. Let us that a class of thoughts is epistemically possible when, intuitively, the conjunction of thoughts in the class would be epistemically possible. Let us say that a complex is an epistemically possible class of thoughts. We can say that a complex C implies a thought T when the union of C with {~T} is not epistemically possible. A complex C implies a complex D when C implies every thought in D. We can say that a maximal complex is a complex C such that there is no complex C1 such that C1 implies C while C does not imply C1. We can then identify a scenarios with equivalence classes of complexes under mutual implication.

The definition of a complex appeals to the notion of an epistemically possible class of thoughts, which in turn appeals to the notion of the epistemic possibility of an arbitrary conjunction of thoughts. If we allow that for every class of thoughts (that is, of possible thoughts of a subject at a time), there is some possible thought that is their conjunction, then we can define this notion wholly in terms of the epistemic possibility of thoughts. But even if we deny this, perhaps because of in-principle cognitive limitations, it is arguable that the notion of an epistemically possible class of thoughts still makes sense. One could try to define it in various ways (e.g. via the claim that for every thought T that is a conjunction of thoughts in the class, T is epistemically possible), or one could simply assume the notion. In any case, the claim that every thought is implied by some complex is weaker than the claim that every thought is implied by some maximal thought, so it is useful to have this notion on the table.

In any case, given the assumptions above, we can define the space of scenarios (relative to a subject) out of the possible thoughts of that subject, and the five central principles will be satisfied.

A question arises: will the space of scenarios (so defined) have interesting structure? Or will it have the structure simply of arbitrary classes of thoughts? This question depends on the extent to which thoughts are epistemically related, and to which there is epistemic entailment between different classes of thoughts. The question is too large to say much about here, but we can at least say a little.

Earlier we noted that at least some thoughts (perhaps all) are composed of concepts. Let us say that a thought T is composed from a class C of concepts if T is composed only of concepts in C. Then we can say that a class C of concepts is a basis class if all thoughts are implied by some thought composed from C. (Note that we do not require that all thoughts are equivalent to some thought composed from C.)

It is plausible that basis classes exist. The class of all concepts is plausibly a basis class (unless there are thoughts that are neither composed of concepts nor verified by any thoughts that are composed of concepts). Given any basis class C, then given any maximal thought M, there will be a thought M' composed from C such that M' implies M. Then M implies M' (by maximality of M), and M and M' are equivalent maximal thoughts. It follows that for any basis class C, we can construct arbitrary scenarios using only concepts in C.

The question then arises: how small can a basis class of concepts be? This is one of the deepest questions in philosophy. But here, I will simply note that it is plausible that many concepts are eliminable from a basic class. I have argued elsewhere that for many thoughts (e.g. "water is H2O", "Oswald killed Kennedy") there are statements about the world that are nontrivially sufficient to settle the truth of the thought. The exact form of these statements is not important here (plausible candidates include qualitative descriptions of the world in physical, mental, and indexical terms). What matters is there exist such nontrivially sufficient statements: sufficient in that they imply the thought or its negation, and nontrivially sufficient in that the statements in question do not invoke the main concepts from the original thought (e.g. "water") or their cognates. This phenomenon is not restricted to the actual world: in general, for an arbitrary epistemic possibility, there are statements about that possibility that are nontrivially sufficient to settle the status of the thought with respect to that possibility. If so, it follows that the main concepts in these thoughts are eliminable from a basis set of concepts.

Just how far this can be taken is an open question. I think it is plausible that a basis set can be stripped down quite a long way. I have argued that at least for possibilities in the vicinity of the actual world, a basis set consisting of microphysical concepts, phenomenal concepts, and indexical concepts (plus logical and mathematical concepts and other "framework" concepts) suffices. One might argue that microphysical concepts are themselves eliminable in favor of general spatial, temporal, causal, and experiential concepts, and perhaps ultimately in favor of phenomenal concepts plus the concept of causation. But I will not defend any such strong claim here. (Note again that the claim is not quite as strong as it may sound, as we do not require that every thought be equivalent to a basis thought.) What matters is that at least some concepts are eliminable from a basis set. To the extent that there are eliminable concepts, there will be nontrivial structure in the relation between scenarios and thoughts.

Having defined a space of scenarios for each subject at a time, we must now ask: how can we define a common space of scenarios for all subjects? (For ease of discussion, I will talk simply of subjects instead of subjects-at-times.) This requires some principle of translation between the maximal thoughts (or the complexes) of one subject, and those of another. We could simply stipulate that there is a relationship of translation among maximal thoughts, such that for any two subjects, a maximal thought of one subject is translated by exactly one maximal thought of another (at least up to a priori equivalence). This would not be very informative, however. Instead, it is useful to ground translation in some more basic notion of translatability.

To translate maximal thoughts, it suffices to define a translation relation across basis classes of concepts in all subjects. In more detail, let us say that a basic translation relation is an equivalence relation (translation) among a class C of translatable concepts across all subjects - yielding a derivative class of translatable thoughts composed from C, with a translation relation among them - such that:

(i) For any two subjects, any translatable concept of the first is translated by some concept of the second.

(ii) If a thought T1 translates a thought T2, then T1 is epistemically possible iff T2 is epistemically possible.

(iii) For any subject, the translatable concepts of that subject are a basis class.

It is easy to see that any basic translation relation yields a mapping between scenarios across subjects. Given scenarios S1 and S2 of subjects, we can stipulate that S1 translates S2 iff for any pair of translatable thoughts of the two respective subjects, S1 verifies T1 iff S2 verifies T2. Then for any two subjects and any scenarios S1 of the first subject, there will be a unique S2 that translates S1.

(To see this, note first as a lemma that if T1' implies T1 and if these two thoughts are respectively translated by T2' and T2, then T2' implies T2: the epistemically impossible T1'&~T1 will be translated by T2'&~T2, which will be epistemically impossible by (ii). Now take any two subjects and a scenario S1 of the first subject. By (iv), there will be a translatable maximal thought T1 associated with S1. By (i), this will be translatable by a thought T2 of the second subject. T2 is maximal: if it were not, it would be implied by a translatable maximal thought it does not imply, so (by the lemma) T1 would be implied by a (translated) thought it does not imply, so T1 would not be maximal. Given two any two translatable thoughts T1' and T2' of the subjects, the lemma entails that T1 implies T1' iff T2 implies T2', so S1 implies T1' iff S2 implies T2'. So S2 translates S1. If any scenario S3 translates S1, then S3 will verify T2 (since S1 verifies T1 and T2 translates T1), so S3 verifies T2, and S3=S2. So S2 is the unique scenario that translates S1.)

So for a common space of scenarios, we need not start with a translation relation over all concepts: a translation relation on a class of basis concepts will suffice. That is, we need a notion of epistemic translatability that clearly applies to some concepts, and is such that the concepts to which it applies forms a basis class. I will not try to define this relation here, but I will say a few words about it.

For two thoughts to be epistemically translatable, then intuitively, they must divide epistemic space in the same way. We can get an initial grip on the notion from the discussion of centered worlds above. Two thoughts "I am a philosopher" by me and a friend will each be verified by all centered worlds in which the being at the center is a philosopher (at least, if there are no relevant differences in our concepts of "philosopher"); so these thoughts will be epistemically translatable. The same goes, plausibly, for two thoughts "my greatgrandparents are first cousins", and the like. In a similar way, for two concepts to be epistemically translatable, they must at least have the same epistemic intension across centered worlds. It seems that any two "I" concepts will be epistemically translatable, and that two "first cousin" concepts are epistemically translatable, and so on.

Clearly, epistemic translatability is not a notion that requires sameness of extension. This is illustrated by the case of "I", and by the more general point that two concepts can have the same epistemic intension across centered worlds but quite different extensions, due to embedding in different centered worlds. The crucial property of epistemic translatability is requirement (ii) above: for this requirement, sameness of extension is irrelevant (and most extension-dependent notions of translatability will not meet it). Rather, what matters is something like sameness of inferential role, and in particular isomorphism of a priori connections.

I will not try to formally define this notion of epistemic translatability here. But even without a formal definition, I think that we have a clear grasp of an epistemic translatability relation over a wide class of concepts. Translatability intuitions become difficult where extension-dependent (or otherwise environment-dependent) concept types are involved, as with the concepts expressed by names and natural kind terms, and also with concepts that are expressed by terms used with semantic deference. But even when these concepts are eliminated, a robust class of concepts remains (such as those above), with a relatively uncontroversial translation relation among them. And it can be plausibly argued that these concepts form a basis class (or something close to it), so that we have a basis translation relation. If so, this yields a common space of scenarios, and a construction of epistemic space.

(There is one obstacle to a basic translation relation here, involving demonstrative concepts in the sort of case mentioned before. Say I demonstrate one of two spots in a symmetrical visual field and think "this is red" (T); and say another possible subject in the same situation thinks two analogous "this is red" thoughts (T1 and T2), one for each spot. Then T plausibly is no better or worse translated by T1 or T2; and T1 or T2 do not translate each other, as they do not imply each other. So T is translated by neither T1 nor T2. More generally, it seems that no possible thought of the other subject is appropriate to translate T. For the same reason, no possible concept of his is appropriate to translate my "this" concept, and there seems to be no underlying translatable basis concept either. I think the moral is that experiential demonstratives are not strictly translatable, but are sui generis.

If this is right, then the space of scenarios for a subject will involve a common translatable basis plus a further basis of untranslatable demonstratives. We can either have a mildly different space of scenarios for each subject (slightly modifying the original picture), or an extended common space with all possible demonstratives (such that some subjects cannot be related to some scenarios). Either way, demonstrative thoughts will never have identical epistemic content across subjects. But we can define an epistemic isomorphism relation between thoughts and between scenarios, such that T stands in this relation to both T1 and T2, which will allow good enough cross-subject comparisons for most purposes.)

It is worth noting that insofar as we have an independent grasp on a notion of epistemic translatability of (at least some) thoughts, this imposes a further constraint on epistemic space that might be worth listing alongside the original principles of epistemic space:

Translatability: If thoughts T1 and T2 are epistemically translatable, then for all scenarios, S verifies T1 iff S verifies T2.

To be made precise, this principle would require a precise and independently grounded definition of epistemic translatability across some class of concepts and thoughts. But even in the absence of a precise definition, insofar as we have an intuitive grasp of epistemic translatability (at least in clear cases), the Translatability principle yields a substantial constraint that epistemic space should satisfy.

Instead of further pursuing the strategy based directly on epistemic translatability of thoughts, I will instead pursue a closely related linguistic strategy for constructing scenarios instead. This strategy starts with sentences, where these are understood as types within a language rather than as tokens.

It is natural to try to construct scenarios by invoking an epistemic possibility operator over sentences, and proceeding from there. But there is an obstacle: many words and sentences exhibit a sort of epistemic variation, so that one token of a sentence may be a priori and another not. This happens most obviously with ambiguous terms, and terms with context-dependent criteria of application. It can also happen with names and natural kind terms. For example, if Leverrier uses "Neptune" as a name for whatever perturbs the orbit of Uranus, and if his wife picks up the term from him without knowing this, then their two corresponding concepts may have different epistemic intensions. To see this, note that "Neptune perturbs the orbit of Uranus, if it exists" expresses an a priori thought for Leverrier, but not for his wife. So the two statements will have different epistemic intensions, arising from difference in the subjects' concepts of "Neptune". So sentences like this are not well-suited to constructing epistemic space.

To deal with this obstacle, we can restrict attention to a special class of epistemically invariant expressions. This class should at least have the property that if a sentence S contains only epistemically invariant expressions, then if one token of S is epistemically possible, all tokens of S will be epistemically possible (setting aside tokens of S that are used deferentially.) And more generally, these expressions should be such that (intuitively) their epistemic intensions across centered worlds cannot differ.

To obtain such a class, we can exclude names and natural kind terms, which allow phenomena such as the above. We can regiment language to eliminate ambiguity. And we can regiment language to eliminate terms with context-dependent criteria of application (especially criteria that vary with speaker's intentions or with conversational context; simple dependence of extension on the environment is allowable). The terms that remain are epistemically invariant. These terms will include pure indexicals such as "I", "now", and "here", and logical and mathematical terms. Also included will be pure descriptive terms: perhaps terms such as "philosopher", "circle", "justice", "object", "cause", "perceive", and "conscious", at least if these are regimented to avoid contextual variation.

Let us say that a language is epistemically invariant if all terms in it are epistemically invariant. English is not epistemically invariant, although it plausibly contains epistemically invariant sublanguages (by restricting the lexicon). But there are certainly many possible languages that are epistemically invariant, and some of them will have significant expressive power.

If L is an epistemically invariant language and S is a sentence in L, then if one token of S is epistemically possible, all tokens of S are epistemically possible (again, setting aside tokens used deferentially). So here we can associate epistemic possibility with types: a sentence S is epistemically possible if any (nondeferential) token of S is epistemically possible. It is reasonable to hold that any sentence of L can be used in principle by an arbitrary subject, expressing a thought. We can define an implication relation between sentences of L and arbitrary thoughts: a sentence S implies a thought T if the thought that the subject would express with S implies T. The epistemic invariance of L ensures that there is a fact of the matter here. Then the following thesis is attractive:

Invariant Basis Thesis: There is an epistemically invariant language L such that every thought is implied by some sentence of L.

In fact, an apparently weaker thesis would suffice: every thought is implied by some epistemically invariant sentence. We need not require that all the sentences be part of the same language. But given the plausible thesis that for any class of languages, their union constitutes a language, the weaker principle entails the stronger.

I will not defend this thesis at length here. I think, however. that it (or something like it) is plausible. The central intuition behind it is that there is nothing about the variability of language that increases our ability to entertain maximal thoughts: variability simply allows for a certain sort of coarse-graining. And where there is coarse-graining, there are more specific fine-grained states, which can be expressed in a fine-grained language. If so, it is plausible that even if a specific thought cannot be expressed in an epistemically invariant sentence, it will be implied by a more fine-grained thought that can.

(As earlier, the thesis needs qualification to handle demonstrative thoughts. To handle such thoughts, the language can be supplemented by an arbitrary number of experiential demonstratives, different demonstratives for each subject.)

In any case, if the thesis is true, we can use it to straightforwardly construct epistemic space. The natural approach is to identify scenarios with equivalence classes of maximal sentences (epistemically possible sentences of L that that are not implied by any sentences that they do not imply), and to say that a scenario verifies a thought when one of the associated maximal sentences implies the thought. As for the actualization relation, formal definitions are subject to the same complications as before, but we given that we have a reasonably firm intuitive grasp on the notion, we can simply assume it.

Is Plenitude satisfied? If every thought T is implied by a maximal thought M, then M will be implied by a sentence of L, which will be maximal (given that every sentence expresses a thought), so there will be a scenario verifying T. Even if there is a problem with maximal thoughts, one could construct scenarios from maximal complexes of sentences, assuming we have an epistemic possibility operator over classes of sentences (yielding notions of implication and maximality over complexes), and that any sentence is implied by some maximal complex. Any thought will be implied by some sentence of L, which will be implied by a maximal complex, so the thought will be verified by a scenario.

As for the other principles: Compositionality will be satisfied straightforwardly. Parsimony will be satisfied if every sentence of L can express a thought; if not, it may be satisfied under other minimal assumptions. Actuality and Truth will be unproblematic, given an appropriate actualization relation. Translatability will be guaranteed, if we take our initial notion of translatability to hold that two thoughts are translatable when they are expressible by the same sentence of L. So if the thesis is true, minor assumptions yield a space of scenarios and a verification relation satisfying the central principles of epistemic space.

It is worth noting that this general discussion of epistemic translatability gives us a general constraint on epistemic space that may be worth noting. The discussion here gives us one way of independently grounding a notion of epistemic translatability among at least a subclass of thoughts: two thoughts are epistemically translatable when they are expressible by the same sentence of an epistemically invariant language.

6 Applications

We have seen that there are various ways in which we might construct a space of scenarios satisfying the principles of epistemic space. As with many such constructions, certain assumptions and/or idealizations are required for the constructions to work and I have not demonstrated the truth of the assumptions and the reasonableness of the idealizations. But the assumptions and idealizations do not look obviously problematic. And as with other constructions of abstract objects, the notion of a scenario should not be seen as hostage to a construction. Rather, the constructions provide a prima facie reason to think that the notion of epistemic space is coherent, and give a sense of how it should behave.

If the notion of epistemic space is coherent, it has a large payoff. One obvious application is in the analysis of knowledge and of epistemic possibility. But there are also applications in the analysis of the content of thought and language. I will briefly outline three such application in the following: to the analysis of Fregean sense, narrow content, and indicative conditionals. Each of these topics deserves a much extended treatment (see "On Sense and Intension", "The Components of Content", and "The Tyranny of the Subjunctive" for this treatment). But even a brief examination shows that epistemic space has much to offer here.

6.1 Fregean sense

Frege held that every expression was associated with a referent, and also with a sense. The sense of an expression mirrors the cognitive value of the expression: for example, since "Hesperus is Phosphorus" is cognitively significant, "Hesperus" and "Phosphorus" have different senses, although they have the same referent. Frege also held that sense determines reference. In a little more detail, sense was supposed to obey at least the following four principles:

(1) Every expression with a referent has a sense.

(2) Sense reflects cognitive significance: singular terms 'a' and 'b' have different senses if 'a=b' is cognitively significant; sentences 'S' and 'T' have different senses iff 'S iff T' is cognitively significant.

(3) The sense of a complex expression is determined by the senses of its parts.

(4) Sense determines reference.

In contrast to Frege, many contemporary philosophers hold that there is no such thing as sense: nothing satisfies theses (1)-(4).

Epistemic space yields a notion that is closely akin to sense. We have seen that every statement can be associated with an epistemic intension, a function from scenarios to truth-values. Further, every expression token within a statement, can be associated with an epistemic intension, a function from scenarios to extensions. To unify these, let us say that the extension of a statement is its truth-value. Let us say that a statement is epistemically contingent if its negation is epistemically possible. Then epistemic intensions has the following properties:

(1) Every expression token with an extension has an epistemic intension.

(2) Epistemic intension reflects (deep) epistemic contingency: singular terms 'a' and 'b' have different epistemic intensions iff 'a=b' is epistemically contingent; statements 'S' and 'T' have different epistemic intensions iff 'S iff T' is epistemically contingent.

(3) The epistemic intension of a complex expression is (usually) determined by the epistemic intension of its parts.

(4) An expression's epistemic intension determines its extension, in conjunction with the actual world.

Principle (1) arises from the general framework. Principle (2) follows from Plenitude and Compositionality. Principle (3) follows from Compositionality. Principle (4) follows from Actuality and Truth.

These four principles are closely related to Frege's, with epistemic intensions playing the role of sense. In (1), senses are associated with expression tokens rather than types, but this seems also to be Frege's view at least of natural languages. In (3), we may need to put minor restrictions on compositionality of epistemic intensions (for reasons below), but it is close to correct. In (4), we must understand determination as determination in conjunction with the world: roughly, that sense provides a condition on extension, which entities in the world may satisfy. This may or may not have been Frege's view of determination, but in any case his view is often understood this way.

The key principle (2) will correspond to Frege's principle if cognitive significance is understood as epistemic contingency. Epistemic contingency behaves very much like cognitive significance: for example, my statement "Hesperus is Phosphorus" is epistemically contingent while "Hesperus is Hesperus is not. More generally, any a posteriori statement will be both cognitively significant and epistemically contingent, while any trivial statement will be neither. There is a difference, however, due to the idealization in the notion of epistemic possibility. Frege held that many a priori statements, such as statements of mathematics, are cognitively significant. As I have defined (ideal deep) epistemic possibility, they are not epistemically contingent but epistemically necessary. It follows that this notion of epistemic contingency departs from Frege's notion of cognitive significance, as it is idealized and less fine-grained. It is nevertheless closely related, and can do much of the work done by cognitive significance. (It is also possible that we can define more fine-grained notions of epistemic possibility and epistemic intension, as we will see.)

In recent years, it has been argued that no broadly Fregean account of the content of linguistic expressions can work. There have been at least three prominent arguments here: the argument from indexicality (Perry), the modal argument (Kripke), and the epistemic argument (Kripke). It can easily be seen that these objections do not apply to the present account.

The argument from indexicality holds that there is no coherent way for a Fregean view to handle the sense indexicals such as "I" and "now". Frege held that these vary between speakers and/or occasions, but this yields a mismatch between sense and cognitive significance. On my approach, these are handled straightforwardly: each token of such an indexical has the same (indexical) epistemic intension, mapping a scenario to the individual or time at the center. This way, sense mirrors cognitive significance. This view departs from Frege's own view: because the sense of an indexical sentence is itself indexical, it will not have an absolute truth-value (as Frege held), but rather a truth-value relative to a subject and time. But the departure ensures consistency, and is still compatible with the spirit of a broadly Fregean view.

The modal argument suggests that a name is not semantically equivalent to any description because of differences in modal contexts. For example, it might have been that Hesperus was not the evening star (e.g. it it had been knocked off course by a comet), so 'Hesperus' is not equivalent to 'the evening star'. And it seems that these cannot have the same sense, since they have different referents in possible worlds. On my view: once we distinguish epistemic from subjunctive notions, all this is compatible with 'Hesperus' and 'the evening star' having the same sense. 'Hesperus is not the evening star' is subjunctively possible but (roughly) epistemically impossible. Correspondingly, 'Hesperus' and 'the evening star' have the same epistemic intension (sense), but different subjunctive intensions. The modal context 'it might have been that ...' is governed by the subjunctive intensions of the expressions, as is the referent of these expressions in alternative possible worlds (considered as ways the world might have been). But all this is compatible with sameness of sense, where sense reflects cognitive significance and determines actual extension.

The epistemic argument suggests that many names are not equivalent to descriptions, because of epistemic differences. For example, 'Gödel' is not equivalent to 'the man who proved the incompleteness of arithmetic': it might turn out that Gödel did not prove incompleteness (perhaps he stole the proof), so it is not a priori that Gödel proved incompleteness. So the name and the description are inequivalent and have different senses. In response: nothing here can show that the name lacks an epistemic intension. In fact, Kripke's argument proceeds by considering an epistemically possible scenario (one in which the man called 'Gödel' steals the proof from a man called 'Schmidt'), and evaluating it as an instance of the epistemic possibility that Gödel did not prove incompleteness. All this is entirely compatible with the epistemic model. At most, it shows that the epistemic intension of a name is not equivalent to that of any associated description; it does not show that names do not have epistemic intensions.

So it seems that the framework of epistemic space provides the basis for a workable Fregean approach to the content of linguistic expressions. These arguments simply suggest that senses must be indexicals, that sense and extension do not exhaust semantic value, and that senses are intensions rather than descriptions. The result differs in some respects from Frege's own view (particularly with regard to the indexicality of senses, and the cognitive significance of a priori statements), but it clearly has much in common.

It should be noted that because epistemic intensions are associated in the first instance with expression tokens rather than types, this does not constitute an account of senses as linguistic meanings: the sort common to all tokens of an expression type. It may be that in some cases (but not all), it could be extended into such an account. But in any case, Frege himself allows that senses could vary between tokens of a type in natural language; and there is much work that a token-relative notion of linguistic content can do.

6.2 Narrow Content

Let us say that the content of a thought is wide if it depends on the environment: that is, if subjects who are internal (physical and phenomenal) duplicates have corresponding thoughts with different content. The content of a thought is narrow if it does not depend on the environment, and is the same between duplicates. In contemporary philosophy of mind, it is commonly accepted that many thoughts have only wide content. In response, some have argued that thoughts also have narrow content, and that even when a thought has wide content, it also has narrow content underlying. But no account of narrow content has received widespread support.

I think that the epistemic content of a thought, as I have defined it, is a sort of narrow content. The corresponding thoughts of two internal duplicates will have the same epistemic content. This can be seen initially though examining the cases that re commonly used as arguments for wide content.

(1) Twin Earth (Putnam 1975). Oscar lives on Earth, surrounded by water (i.e. H2O). Twin Oscar lives on Twin Earth, which is much like Earth except that H2O is replaced by superficially identical XYZ, which is not water but twater. Oscar and Twin Oscar are (near) duplicates and are chemically ignorant. Both utter "water is expensive", expressing a belief. Oscar believes that water is expensive, but Twin Oscar does not: he believes that twater is expensive. So their beliefs have different content.

Let B1 and B2 be the respective beliefs of Oscar and Twin Oscar. What is their epistemic content? We can see this by examining four relevant scenarios:

S1: Watery stuff in the environment is H2O, H2O is expensive. S2: Watery stuff in the environment is H2O, H2O is not expensive. S3: Watery stuff in the environment is XYZ, XYZ is expensive. S4: Watery stuff in the environment is XYZ, XYZ is not expensive.

Oscar's belief B1 is clearly verified by S1, and falsified by S2. It is also verified by S3: the thought that S3 is actual implies B1. That is, for Oscar, the hypothesis that S3 is actual and water is not expensive is epistemically impossible: if he accepts that S3 is actual, he will rationally accept that water is expensive. For similar reasons, B1 is falsified by S4.

For symmetrical reasons, Twin Oscar's belief B2 is verified by S3 and falsified by S4, and verified by S1 and falsified by S2. That is, both B1 and B2 are verified by S1 and S3, and falsified by S2 and S4. So at least as far as these four scenarios are concerned, B1 and B2 have the same epistemic intension. But these four scenarios are the crucial scenarios here: if B1 and B2 have the same epistemic intensions across these scenarios, they plausibly have the same epistemic intension overall. So B1 and B2 plausibly have the same epistemic content.

Note that B1 and B2 plausibly have different subjunctive content. B1 is plausibly satisfied only by worlds in which H2O is expensive and not by worlds in which XYZ is expensive, while for B2, the situation is reversed. So B1 and B2 have different subjunctive intensions. The subjunctive content here can be seen as the wide content of B1 and B2, while the epistemic content can be seen as the narrow content.

(2) Arthritis (Burge 1979). Bert and Twin Bert both have a term "arthritis", which they use for a disease whose nature they do not fully understand. Bert lives on Earth, where experts in his community use "arthritis" for disease D1 (arthritis), Twin Bert lives on Twin Earth, where experts in his community use "arthritis" for disease D2 (twarthritis). Bert and Twin Bert are duplicates: both utter "I have arthritis in my thigh". Bert believes that he has arthritis in his thigh. Twin Bert believes that he has twarthritis in his thigh. So it seems that their beliefs have different content.

Let B1 and B2 be the respective beliefs of Oscar and Twin Oscar. What is their epistemic content? We can see this by examining four relevant scenarios:

S1: Experts use "arthritis" for D1, D1 is in my (center subject's) thigh. S2: Experts use "arthritis" for D1, D1 is not in my thigh. S3: Experts use "arthritis" for D2, D2 is in my thigh. S4: Experts use "arthritis" for D2, D2 is not in my thigh.

(Scenario S1 may be impossible due to the nature of D1, but set this inessential worry aside.) Oscar's belief B1 is clearly verified by S1 (the actualized scenario) and falsified by S2. It is also verified by S3: the thought that S3 is actual implies B1. That is, for Oscar the hypothesis "S3 is actual and arthritis is not in my thigh" is epistemically impossible: if Oscar accepts that S3 is actual, he should rationally accept "I have arthritis in my thigh". For similar reasons, B1 is falsified by S4.

For symmetrical reasons, Twin Bert's belief B2 is verified by S3 (his actualized scenario) and falsified by S4, and verified by S1 and falsified by S2. That is, both B1 and B2 are verified by S1 and S3, and falsified by S2 and S4. So at least as far as these four scenarios are concerned, B1 and B2 have the same epistemic intension. But these four scenarios are the crucial scenarios here: if B1 and B2 have the same epistemic intensions across these scenarios, they plausibly have the same epistemic intension overall. So B1 and B2 plausibly have the same epistemic content.

Again, B1 and B2 may have different subjunctive intensions, so their subjunctive content may constitute wide content. But their epistemic content constitutes narrow content.

All this enables us to reconcile the facts that (a) the truth of a belief ascription such as "Oscar believes that water is expensive" or "Bert believes that he has arthritis in his thigh" depends on the subject's environment, and that (b) the two corresponding beliefs have the same narrow (epistemic) content. To reconcile these, we need only accept that the truth of belief ascription can depend on both the subjunctive content and the epistemic content of a subject's beliefs. Given this dual dependence, the wideness of subjunctive content explains the wideness of belief ascriptions, in a manner that is entirely compatible with epistemic content being narrow. (See "The Components of Content" for more on how this dual dependence works.)

In addition to examining cases, we can also give a principled argument that epistemic content is narrow. All we need are the following two theses:

Duplication Thesis 1: Given any two duplicates A1 and A2 and corresponding thoughts T1 and T2, then T1 is epistemically possible iff T2 is epistemically possible.

Duplication Thesis 2: Given any two duplicates A1 and A2 and corresponding concepts C1 and C2, then if C1 is an epistemically translatable basis concept, C2 translates C1.

The first thesis is extremely plausible, and nothing in arguments for externalism gives reason to deny it. In effect, the claim is that the apriority of beliefs is internally determined. This is what we would expect: whether a belief is a priori depends only on whether it can be nonexperientially justified, and whether a belief can be nonexperientially justified is independent of the environment.

The second thesis is also plausible, for the epistemic translation relation invoked earlier. The sorts of concepts over which translation is invoked are concepts to which externalist arguments do not apply: extension-dependent concept types such as names and natural kind concepts are excluded, and deferential uses of concepts are excluded. For the indexical and purely descriptive concepts that remain, the translation mapping between duplicate individuals will be relatively uncontroversial. And if these concepts form a basis class, this mapping can be extended into a mapping of arbitrary scenarios.

Given these two theses, then take any two duplicates with corresponding beliefs B1 and B2, and consider an arbitrary scenario S. Given that the translatable concepts are a basis class, S will be associated with a translatable maximal thought M1 in the first subject. If S verifies B1, then M will imply B1: that is, the thought "M1&~B1" will be epistemically impossible. The thought corresponding to "M1&~B1" will be "M2&~B2". By the first thesis, M2&~B2 will be epistemically impossible. By the second thesis, M2 will translate M1, so it will also be associated with S. So we can see that S verifies B1 if and only if S verifies B2. So B1 and B2 have the same epistemic content. This holds for arbitrary beliefs, so epistemic content is narrow content.

(As before, there is a mild exception for demonstrative beliefs, whose content is sui generis between individuals. When I attend to a spot in a symmetrical visual field and believe "This is red", and when a duplicate of me does the same, our thoughts will not have exactly the same epistemic content, since there is no canonical translation between experiential demonstratives. Instead, we can say that the thoughts are epistemically isomorphic, and have isomorphic epistemic content. Given the general thesis that corresponding demonstrative concepts are epistemically isomorphic concepts, as is plausible, we can still maintain the thesis that duplicates always have isomorphic narrow contents.)

6.3 Indicative and subjunctive conditionals

There is a striking, although infrequently noted asymmetry between indicative and subjunctive conditionals. Consider the following pairs:

(1a) If Prince Albert Victor murdered those women, he is Jack the Ripper.

(1b) If Prince Albert Victor had murdered those women, he would have been Jack the Ripper.

and

(2a) If XYZ is (and has been) the clear liquid in the oceans and lakes, water is XYZ.

(2b) If XYZ had been the clear liquid in the oceans and lakes, water would have been XYZ.

In these pairs, the first member seems intuitively correct. (1a) might reasonably be accepted by a detective investigating the murders. (2a) might reasonably be accepted by someone who is ignorant of the chemical composition of water, or even by someone who knows that water is H2O but is not certain of this. These conditionals pass the Ramsey test: if the antecedent is accepted as a hypothetical premise, the consequent should be accepted as a hypothetical conclusion. Let us say that in these conditions, the indicative conditional is correct. (I will be neutral on whether correctness corresponds to truth, assertibility, or something else.) Clearly, (1a) and (2b) are correct indicative conditionals.

In these pairs, the second member seems intuitively incorrect, at least if Kripke's intuitions are accepted. Given that Prince Albert Victor is not actually Jack the Ripper (as seems likely), then even if he had committed the murders, he would not have been Jack the Ripper. Rather, he would have committed the murders instead of Jack the Ripper. Similarly, given that water is actually H2O, then even if XYZ had been the clear liquid in the oceans and lakes, XYZ would not have been water. Rather, XYZ would have been in the oceans and lakes instead of water.

(I should note that my own intuitions about these subjunctive conditionals are not quite as clear as Kripke's. But for the sake of discussion I will accept the Kripkean intuitions.)

There is one especially striking feature of the asymmetry: it seems that the correct indicative conditionals have a "metaphysically impossible" consequent. How can this be accommodated in a semantic theory? More generally, how can the asymmetry be explained?

Many philosophers hold that one can give a possible worlds account of subjunctive conditionals, along something like the following lines:

"If it had been that S, it would have been that T" is true <-> the closest S-world is a T-world.

Here, closeness is a similarity metric on the space of possible worlds, and the closest S-world is the world such that of all worlds in which S in the case, it is the most similar to the actual world. Note that the closeness metric may vary between contexts, and that one can adjust the definition in various ways, e.g. to allow for multiple closest worlds.

Many philosophers hold that by contrast, one cannot give a possible-worlds account of indicative conditionals, holding that they are deeply epistemic rather than modal. The considerations above are not usually invoked here, but they might be thought to strengthen the case: if correct indicative conditionals can have metaphysically possible antecedents and impossible consequences, then possible worlds are an inappropriate tool.

However, the framework of epistemic space allow us to give an analysis of indicative conditionals that is parallel to the analysis of subjunctive conditionals, except that we invoke the space of epistemic rather than subjunctive possibility. In particular, one can hold the following

"If S is the case, then T is the case" is correct <-> the epistemically closest S-scenario is a T-scenario.

I will not attempt an exact analysis of the notion of epistemic closeness here. But it should be understood as something like the following: one scenario is epistemically closer than another (for a subject) to the extent that it is more compatible with the subject's knowledge (or beliefs), or to the extent that it is compatible with a smaller revision of the subject's knowledge-constituting beliefs. Note that epistemic closeness is subject-relative, and depends on what a subject knows (or believes). Whether knowledge or belief should be invoked here is a tricky issue that I will not go into here.

Then in the cases above: there is a wide variety of scenarios in that verify "Prince Albert Victor committed the murders", and all (or almost all) verify "Prince Albert Victor is Jack the Ripper". Similarly, there is a wide variety of scenarios in which XYZ is and has been the liquid in the oceans and lakes. Those most compatible with a subject's beliefs are plausibly those in which things are superficially identical except for the substitution for XYZ for H2O, and these all verify "water is XYZ". So in both cases, the epistemically closest scenarios verifying the antecedent also verify the consequent.

Note that all this is very much compatible with the Ramsey test for the assertibility of indicative conditionals. When we hypothetically accept S, we hypothetically exclude all scenarios verifying ~S, while making minimal revisions to our beliefs. The scenarios we will then be led to hypothetically endorse are those that verify ~S and that are compatible with a minimal revision of our beliefs. That is, we are led to hypothetically endorse the epistemically closest S-scenarios. We will conclude that T precisely to the extent to which the endorsed S-scenarios are also T-scenarios. So the Ramsey test will be satisfied precisely when the conditions above are met.

(Note that the Ramsey test corresponds to the belief-relative notion of epistemic closeness. Does this mean that our correctness conditions should also appeal to the belief-relative notion? Not necessarily: or only to the extent that correctness and assertibility converge. It is arguable that when an assertible indicative conditional is grounded in false beliefs, the conditional is intuitively incorrect. If so, then it is plausible that correctness conditions should not depend on a subject's false beliefs, and perhaps should depend only on a subject's knowledge.)

The subject-relativity in the notion of epistemic closeness mirrors the often-noted subject-relativity in the correctness conditions of indicative conditionals. To use a version of the "Sly Pete" example: say that Pete holds one card, that I know that Pete has a 3 or a 5, and that you know that Pete has a 5 or a 7 (of course he actually has a 5). Then I can correctly say: if Pete does not have a 5, he has a 3. And you can correctly say: if Pete does not have a 5, he has a 7. It is plausible that I could not correctly assert the second, and you could not correctly assert the first. The difference results from the difference in our epistemic closeness measure, resulting from the difference in our knowledge. For me, the closest scenarios in which Pete does not have a 5 are scenarios in which he has a 31. For you, the closest scenarios in which Pete does not have a 5 are those in which he has a 7.

Note that the analyses of subjunctive and indicative conditionals are almost exactly parallel, except for the differences between (i) S-worlds vs. S-scenarios, and (ii) counterfactual closeness vs. epistemic closeness. It is natural to hope that many of the tools that have been used to analyze subjunctive conditionals can also be adapted to analyze indicative conditionals.

7 Hyperintensionality

The notion of (deep) epistemic possibility that we have been dealing with is an idealized one: if S is knowable a priori, then it is not epistemically possible that ~S, even when S is far from obvious, and even when no-one in the world knows that S. This has the result that any two sentences that are a priori equivalent will have the same epistemic intension, and the same epistemic content. For example, all a priori sentences, including complex sentences of logic and mathematics, have the same epistemic intension. And all a priori thoughts, including complex logical and mathematical thoughts, have the same epistemic intension.

It is natural to wonder if there is a less idealized notion of epistemic possibility that will a more fine-grained notion of epistemic content. This is a complex issue, but I think there can be.

To develop this notion, we must start with a nonideal notion of deep epistemic possibility. Instead of saying that it is epistemically possible that P when ~P cannot possibly be ruled out a priori, we might say that it is epistemically possible that P when ~P cannot be ruled out through reasoning of a certain sort. Here, there are various possibilities.

We might hold that it is epistemically possible that P when:

(i) it is not obvious a priori that ~P; (ii) P cannot be ruled out by such-and-such amount of a priori reasoning; (iii) P cannot be ruled out through logical reasoning alone; (iv) P cannot be ruled out through non-moral a priori reasoning; (v) it is not obvious that ~P (vi) it is not trivial that ~P (vii) it is not certain that ~P (viii) it is not known that ~P

and so on. Here, the definitions work have different properties. The first four definitions all appeal to varieties of a priori reasoning. These guarantee that if it is epistemically possible that P in the previous sense, it will be epistemically possible that P in the new sense. The last four definitions do not appeal only to a priori reasoning; so it is possible that some P will be epistemically possible in the original sense but not in the new sense (e.g. obvious empirical falsehoods, such as "I do not exist" or "I do not have hands"). Because we are mostly concerned with more fine-grained notions of epistemic possibility here, I will concentrate on definitions in the first class.

Given a notion of non-ideal epistemic possibility, we can attempt to set up a corresponding non-ideal epistemic space, made up of non-ideal scenarios. The principles governing this space will be much as before. The key principle, once again, will be Plenitude: there is a scenario verifying S iff S is epistemically possible. Because many more thoughts and statements will be epistemically possible for non-ideal notions of epistemic possibility, it follows that there will be many more corresponding non-ideal scenarios.

It seems reasonable that the principles of Actuality, Truth, Parsimony, and Translatability should all hold on this model. There is a question about whether Compositionality should be endorsed. If a compositional principle is itself nonobvious, it may be that each Ti can be epistemically possible without comp(Ti) being epistemically possible. If compositionality holds, it is likely that if a scenario verifies some statements, it will verify all logical consequences of those statements, which may be undesirable. If so, we may wish to do without Compositionality, or restrict it in some fashion to obvious compositions and the like.

The processing of constructing scenarios will be more complex where non-ideal epistemic possibility is concerned. It is clear that centered worlds will be inappropriate here. On natural models of verification, a priori falsehoods will be verified by no centered world; and on any reasonable model of verification, there will not be enough centered worlds to satisfy Plenitude. As for the direct epistemic construction, we will probably need to avoid appeal to maximal statements and thoughts. Maximal statements and thoughts are likely to be indiscriminately epistemically possible, because of their complexity, and they may have no interesting implication relations to ordinary thoughts.

Instead, it may be best to appeal to classes of statements of thoughts: perhaps classes such that no statement or combination of statements is epistemically impossible. Then we can say that a class implies a statement T if some conjunction of ~T with a set of statements in the class is epistemically impossible, or perhaps if there is some reasoning process of the relevant sort that takes us from a statement or group of statements in the class to T. We can say that one class implies another class if it implies every statement in that class. We can say that a class is maximal if it is implied by no class that it does not imply. A maximal class will verify a thought when it implies that thought. There will be difficulties in setting up equivalence relations on maximal classes, due to failures of transitivity in implication, but this problem might be dealt with in a variety of ways. It seems that this sort of approach is at least promising.

If we can set up a non-ideal epistemic space corresponding to a non-ideal notion of epistemic possibility, we will then have a corresponding notion of non-ideal epistemic content. We can say that the non-ideal epistemic content of a thought is the thought's intension over non-ideal scenarios, according to whether those scenarios verify the thought. Then for any two thoughts T1 and T2 such that it is epistemically possible that T1 holds without T2 and vice versa, T1 and T2 will have different non-ideal epistemic content. The same goes for the non-ideal epistemic content of statements.

When this way of thinking is applied to different notions of epistemic possibility, it will yield various different applications. For example, if we are concerned with Frege's notion of cognitive significance, we can say that T is epistemically possible when ~T is cognitively significant (perhaps this will be whenever ~T is nontrivial), and we can set up a corresponding non-ideal epistemic space. This will yield a variety of non-ideal epistemic content that behaves very much like Fregean sense.

Another application: For the ordinary notion of epistemic possibility (strict epistemic possibility) with which we started this paper, it is plausible that P is strictly epistemically possible when one could not easily come to know that ~P given what one already knows. The corresponding notion of deep epistemic possibility is something like the following: it is deeply epistemically possible that P when one cannot easily know a priori that ~P. From this notion, we will be able to set up a corresponding non-ideal epistemic space. For this space, it is plausible that P is strictly epistemically possible iff there is a P-scenario that is not excluded by any item of one's knowledge. So this space is perhaps the closest to delivering the intuitive picture of strict epistemic possibility as discussed at the start of the paper.

Another application: say that we are concerned with hypotheses about the relationship between the nonmoral and the moral. We may think that the connection is ultimately a priori, or we may think that moral beliefs are ultimately not truth-evaluable, but as long as the connection and the non-truth-evaluability is not obvious, there will be an interesting hypothesis space to investigate. To do this, we can invoke notion (iv) above: it is epistemically possible that P when P cannot be ruled out through nonmoral a priori reasoning. This will plausible yield a space of "moral scenarios" which is much like the space of ideal scenarios, except that it may have an additional dimension of variation in the way that it associates moral claims with nonmoral claims. Such a space may be very useful for analyzing the moral domain without presupposing moral views. (This will be closely related to Gibbard's notion of a factual-normative world.)

It may be that there is no canonical notion of nonideal epistemic possibility. If so, there will be no canonical notion of nonideal epistemic content. Instead, we might have a spectrum of notions of deep epistemic possibility, from the ideal to the nonideal, perhaps ending at the notion on which anything is epistemically possible, and on which there are no interesting relations among epistemic content. There will be a corresponding spectrum of epistemic spaces. Every statement might then be associated with a spectrum of epistemic intensions, each of which is an intensions across scenarios within a given epistemic space. For different purposes, different intensions from within this spectrum may be relevant. Between these intensions and these epistemic spaces, there will be enough material to do significant explanatory work in many different epistemic domains.