The purpose of this essay is to provide some arguments as to why conceivability can’t entail possibility. The debate is broad, so I will only enter into a particular, recent part of the dialectic; I will try to show that Chalmers’ distinction between positive and negative conceivability will not save the anti-Materialist from the counterexamples to the conceivability-possibility entailment (the "C-P thesis") which its implementation might otherwise serve to avoid. I will not neglect giving a brief summary of the issues involved which motivate the issue, so this paper will have three parts: In the first, I will expound Chalmers’ view and terminology. In the second, I will discuss some of the difficult cases for Chalmers’ view and the corresponding application of the positive/negative conceivability distinction. The last part of this paper will expound Hill’s alternative model for explaining the apparent contingency of a posteriori necessities. I will conclude that invoking positive/negative conceivability to escape counterexamples to the C-P thesis doesn’t work, and that Hill’s model can carry the burden assumed by anyone—i.e., the Materialist—who classifies mind/brain identities as a posteriori necessities.

Chalmers is well-known for his attempt to repudiate (philosophical) Materialism, so I can be relatively brief here. In caricature, the argument goes as follows. Premise 1: Zombies—functional, microphysical duplicates of ourselves which have no consciousness—are conceivable. Premise 2: Conceivability entails possibility. Conclusion: Zombies are possible, and therefore Materialism is false. In order to flesh out this argument, I will now briefly elucidate Chalmers’ semantics and his analyses of conceivability and possibility.

Chalmers takes the import of Kripke’s distinction between epistemic and metaphysical possibility, and Putnam’s Twin-Earth thought-experiments, to show that concepts and words have two meanings: in his terminology, a "primary intension" and a "secondary intension". The primary intension is a function from centered possible worlds to extensions, determined by what the concept picks out at a world considered as actual; a "centered" possible world is a world with a specified individual and time, or ‘point of view’. The secondary intension is a function from worlds to extensions which depends on what the concept picks out in a world considered as counterfactual. So, thought-experiments reveal that the primary intension of ‘water’ is something like ‘the drinkable clear liquid that comes from the faucets and fills the lakes’, whereas the secondary intension of water is ‘H2O’. The former corresponds to the narrow content of the concept and is typically invoked in stating a priori truths; the latter corresponds to wide content and is typically invoked in stating necessary truths. Most of this paper won’t be too concerned with semantic issues like these, since conceivability and possibility with respect to primary intension is all that really matters; pain states, arguably, have identical primary and secondary intensions. But Chalmers does make important use of them to explain the apparent contingency of a posteriori necessities, and so they will come up when Hill’s alternate explanation is considered.

Conceivability is, as Chalmers has done a significant amount of work to show, a widely variegated notion. He distinguishes modes of conceivability along three dimensions: primary vs. secondary conceivability; prima facie vs. ideal conceivability; and positive vs. negative conceivability. The first dimension corresponds to conceivability claims which employ the primary and secondary intensions of words or concepts, respectively. The distinction between prima facie and ideal conceivability is a little less clear. Prima facie conceivability, for Chalmers, is conceivability on a first pass, and is avowedly an imperfect guide to possibility. Secunda facie conceivability—a term defined more recently—is represents conceivability after a far more thoroughgoing process of reflection, and will be defeated only in extreme cases, if at all. Ideal conceivability, finally, represents conceivability with all contingent psychological limitations removed. However, exactly what Chalmers counts as a ‘contingent’ psychological limitation not clear, and this is an issue to which I briefly return later in the paper.

I turn now to the distinction between positive and negative conceivability. For any statement S, positive conceivability of S is "clear and distinct conceivability of a situation verifying S", and negative conceivability of S is "the absence of any apparent contradiction in S". (MM, 7) (Note: ‘MM’ and all other abbreviations used here are defined in the bibliography.) Some sections of TCM seem to indicate that Chalmers has negative conceivability in mind—he "can discern no contradiction in the description" of a zombie world, and that is grounds enough for its conceivability. (96ff.) But in his later works he clarifies his view, and insists upon the positive conceivability of zombies; as we shall see, his argument relies upon it. For now, it is important to bear in mind that in order for negative and positive conceivability to be distinct notions, positive conceivability has to amount to more than merely intuiting that a particular description of a scenario is consistent with S. (I.e., that it not a priori entail not-S.) That’s just what negative conceivability is. In order to underwrite a positive conceivability claim, it seems evident that the scenario in question—specified in some non-question-begging way—must in some sense entail S. What sense this could be will be considered below.

Finally, the way ‘possibility’ is being used must be elucidated. It is taken, plausibly, as a metaphysical primitive. A state of affairs S is possible iff there is a possible world where S occurs. The particular metaphysics of modality one ultimately adopts—Lewisian modal realism, Actualism, ersatz modal realism of one sort or another—will not really matter, since the modal claims in question in this discussion are ones that (at least on a cursory examination) will be truth-value-invariant with respect to the different metaphysical theories. The last point to make is that possibility is wholly objective. Although primary possibility is relativized to an observer and time, this is just done in order to specify the semantics of his concepts and words; once fixed, the space of possibilities is the same no matter who or what that individual turns out to be. Whether or not the same holds of conceivability is, of course, a different and substantial question.

To return now briefly to the anti-Materialist argument that began this section: It is clear that the relevant sense of conceivability being used in the argument must be spelled out. Crucially, the sense of conceivability in which zombies are conceivable, and the sense of conceivability which entails possibility (if there is such a sense), must be the same sense. There must not be a gap between the respective conceivabilities; or, to put it another way, the sort of conceivability that we have epistemic access to must be possibility-entailing. Finding such a sense presents a difficult challenge; so it is not surprising that Chalmers, as I will argue, fails to meet it.

There are a number of cases which have been brought as counterexamples against the C-P principle; Chalmers’ initial strategy has been either to deny that the purported conceivable state of affairs really is (ideally) conceivable after all, or (less frequently) to assert that what was presumed conceivable but impossible really is possible, after all. Into the latter category fall zombie worlds, and the denials of identity statements of the Kripke/Putnam variety when conceived with respect to primary intension; into the former fall denials of the identities when conceived with respect to secondary intension, certain mathematical truths like the false member of the pair {Goldbach, ~Goldbach}, theistic claims, and metaphysical claims. That is, Chalmers has argued that Fermat’s last theorem, for example, is not conceivably false, in any possibility-entailing sense of conceivability; and the apparent falsity of a posteriori necessities arises only through making a semantic mistake—which amounts to mistaking one’s intuitions about the way the actual world might have turned out for one’s intuitions about the nature of counterfactual worlds.

Chalmers’ diagnosis, and verdict of ‘not properly conceivable after all’, has been accepted for many of these cases, most notably the a posteriori necessities, with respect to which he is considered to have given a valuable codification of the insights into modality and semantics provided by Kripke’s Naming and Necessity. (McLaughlin, Yablo) But some cases resist such easy subsumption into the rubric of ‘not properly conceivable’, and Chalmers has elaborated his theory accordingly, by progressively refining the senses of conceivability in question and introducing corresponding versions of the C-P principle. I’ll consider the hard cases first, and argue that on their account the only sense of conceivability which could entail possibility is ideal positive conceivability; I will not question this entailment, but rather the claim that zombies are conceivable in this sense. If I succeed in these tasks then Chalmers’ anti-Materialist argument fails.

There are two hard cases I have in mind: The Generalized Continuum Hypothesis (GCH), and metaphysical claims, such as claims about the metaphysics of modality. (Chalmers 1999 considers some others, but the others are too easy I think.) What makes the GCH hard—what distinguishes it from the Goldbach conjecture—is that mathematicians have (so I am told) already proved all there is to know about it relative to the axioms of Zermelo-Fraenkel set theory plus the Axiom of Choice (ZFC): namely, it is provably consistent with ZFC, and its negation is provably consistent with ZFC. (These results belong to Kurt Gödel and Paul Cohen, respectively.) Nearly every other mathematical truth on the books can be derived from the axioms of ZFC, but not GCH or ~GCH. However, the GCH seems conceivably true; I’m told by one set theorist that the majority of set theorists assume it, although some don’t. More importantly, it seems perverse to demand—as Chalmers presumably must—that the criterion on conceivability of a mathematical claim be the conceiving of a proof of truth, rather than a proof of consistency. (For surely having a proof of some sort, derived from intuitively obvious axioms, is the only plausible sufficient condition on the ideal conceivability of a mathematical proposition.) Having either proof in hand represents a process of conceiving carried out to its ideal completion, and a guarantee that employing the claim in question will not lead you to provably false statements. It’s hard to see how there could be a stronger, still-plausible sufficient condition on conceivability, if ‘conceivability’ is to retain any of its pretheoretical content. In addition, it’s also hard to see how the situation is helped by an appeal to an improved set of axioms—ones which better capture our mathematical concepts—with which the mathematical proposition in question is inconsistent. (This is the other way around the GCH case, and one which I believe Chalmers favors.) If the axioms of ZFC themselves are called into question, then the problems generated by mathematical truths can be assimilated to the case of metaphysical claims; for there can not of course be a proof, or demonstrative argument, that any particular consistent set of axioms is the ‘correct’ one, since the notion of proof (or demonstrative argument) already presupposes the existence of an axiomatic system. The problem then becomes justifying the claim that a particular candidate axiom is ideally inconceivable, or inconceivably false; and as I will discuss again in the metaphysics case, there seems to be no such justification available.

If the members of {GCH, ~GCH} are indeed ideally conceivable, then one path which is suggested by Chalmers 1999, and which in fact entails the inconceivability of GCH, ~GCH—denying that the GCH and its negation even have truth values—is eliminated. That is, it is eliminated if one makes the assumption that mathematical truths are necessary truths; in that case the conceivable truth of GCH would entail its necessary truth, and obviously that would give it a truth value (both truth values, that is). So it seems that the only way to go is to break the C-P link. The GCH is conceivably true and conceivably false, but not both possibly true and possibly false; although God knows whether it’s true or false, we don’t. I’ll discuss how the link is to be broken after running through the second hard case.

The second hard case concerns certain metaphysical claims, over which there is large disagreement among philosophers yet all of which constitute necessary truths or falsehoods. The issues concerning the metaphysics of modality mentioned earlier—modal realism, Actualism (there is only one world), and modal ersatzism—are examples par excellence of this. They concern the nature of possible worlds, which Chalmers explicitly takes as a primitive notion. So they seem prior to the conceivability-possibility debate; yet, at most one of these positions can be conceivably true, and therefore on Chalmers view the others are ideally inconceivable.

One answer Chalmers (personal correspondence) provides to these cases is to accept the inconceivability of all but the true member of the set of different views given above; he claims that they are a priori true or false. I think it is easily seen that this does too much violence to our concept of conceivability to be an acceptable answer, however. Unlike virtually all mathematical truths, there is no axiomatic basis from which metaphysical truths can be derived; it is impossible to explain away our intuitions that e.g. modal realism is conceivably true and conceivably false by appealing to an ‘incomplete proof’ or a pronouncement by metaphysicians which is what we’re really conceiving and which is in fact compatible with the falsity of what we purport to conceive. (As mentioned above, the axioms of ZFC themselves fall into this category; Chalmers suggests that they too—even the disputed ones—are a priori.) Rather, it seems that all that is required in order to be able to conceive of a metaphysical proposition—or of a particular mathematical axiom—being true is simply to entertain it and to have discerned that it coheres with or is consistent with one’s other views. This will involve, in the ideal case, grasping all the consequences of the proposition, and the logical relationships between that proposition and others. But that is all it will involve; for there is simply nothing else that conceivability of a metaphysical claim could be. There is surely no ‘direct metaphysical intuition’; at the very least, there is no introspective evidence for such intuitions, and that’s the only sort of evidence there could be. So, there is no other means of epistemic access to such claims than through assessing their consistency. I don’t think this view commits us to the claim that there is no fact of the matter, or that we can’t come to have overwhelming evidence in favor of one view and not the others. Denying the a priority of some metaphysical truths merely amounts to denying that we will ever have a demonstrative argument for or against any one of them, but a demonstrative argument is what Chalmers needs if he is to justify his claim that any of them are inconceivable. If this argument is correct, then as before—unless we’re metaphysical irrealists, which is somewhat more plausible than mathematical irrealism yet still extreme—ideal conceivability can’t entail possibility.

At least, the sense of ideal conceivability in which the GCH and metaphysical truths are conceivable can’t entail possibility. Chalmers allows for this possibility—although he doesn’t think it’s the right one—by invoking his distinction between positive and negative conceivability. Recall that p is negatively conceivable iff p is a consistent description of the world. Positive conceivability of p, on the other hand, consists of conceiving of a situation which verifies p. Clearly, the GCH and its negation as well as (some) metaphysical truths and their negations are ideally negatively and not positively conceivable; they are both consistent with everything else we know yet are not entailed by those facts. So Chalmers distinguishes the C-P principle into strong and weak versions, corresponding to negative and positive conceivability, respectively; and although he does not repudiate the principle that negative conceivability entails possibility, he puts the burden of his anti-Materialist argument onto the principle that positive conceivability entails possibility. (Cf. MM, 1999: 24) (Of course, if the above arguments for the ideal conceivability of inconsistent metaphysical and mathematical claims are incorrect, he will not need to do this.) That is, he claims explicitly that the zombie world is positively conceivable; and since ideal positive conceivability is not impugned by the foregoing arguments, the possibility of zombies is preserved. So there is a sense of conceivability which is strong enough to exclude the ‘hard cases’; I now turn to Chalmers’ claim that that zombies are in fact conceivable in this sense.

Chalmers does not, so far as I am aware, really provide any arguments for this claim; rather he just assumes that it is obvious. (1999) I shall not assume this. Let me observe now that one does not have to be an Analytic Functionalist or a Logical Behaviorist—the typical type-A Materialists—to deny the positive conceivability of zombies; for these type-A Materialists will deny even the negative conceivability of zombies, something which for the purposes of this paper I am willing to assume is true. Materialism and concept Dualism are perfectly compatible.

The reason that the positive conceivability of zombies needs substantial argumentation is that there appears to be no non-question-begging sense of positive conceivability which makes zombies come out as positively conceivable. ‘Verifies’ is the crucial word in that definition; in service of its elucidation Chalmers writes the following:

Being able to positively conceive of S is also equated with having a Cartesian "clear and distinct" conception of S, although Chalmers admits that these characterizations are intuitive and not reductive. (1999: 5-6) In this case, the stage is set for the observation and corresponding inference which I believe doom Chalmers’ project, and so upon which this entire paper turns. The observation is simply that in order for positive conceivability of S to be distinct from negative conceivability, S can not be included as part of the description of the scenario. (Of course, there are many ways to describe a scenario. S can’t be included in the description which underwrites a positive conceivability claim.) Suppose not; suppose that S could be included as part of the description of the scenario. Now suppose that for some S, S is ideally negatively conceivable; then any complete world-description D is consistent with S. Now consider the description D&S; by supposition, this description describes a coherent scenario and so one which could potentially verify S. Of course, the scenario described by D&S does verify S, since D&S trivially entails S. So S is positively conceivable. But all we assumed about S at the outset was that it was negatively conceivable. I take this to be a rather straightforward demonstration that if negative and positive conceivability are distinct, then S can not be included as part of the description of the scenario.

I note that one may object that positive conceivability of S requires a clear and distinct conception of a scenario which verifies S, and nothing in the above was said to ensure that that was the case for D&S. It’s not clear, however, what work ‘clear and distinct’ (in Chalmers’ definition of positive conceivability) is supposed to do. It can’t mean ‘positively conceivable’; in that case the definition collapses into circularity. It can’t mean anything like ‘introspectively vivid’, for as Chalmers is aware, that is not sufficient for ideal positive conceivability. It’s hard to see how it could mean anything other than ‘possessing a clearly consistent description’, a property which the scenario described by D&S has.

In that case, I think it’s clear that this observation demolishes Chalmers’ prospects for plausibly claiming that zombies are positively conceivable (this is the ‘corresponding inference’). Because precisely what creates the "hard" problem of consciousness is the fact that it does not connect in any epistemically accessible way to the physical properties that are tractable to verificationist means. So, suppose for the moment that ‘verify’ in Chalmers’ definition is construed very weakly as ‘provides good evidence for’. Then it is clear that the putative zombie scenario, call it ‘Z’—a scenario in which there are people who are just like us according to every observational criterion imaginable—does not verify the claim that there are zombies. It verifies the claim that there are conscious people—us. Moreover, making the verification criterion stronger will only make the problem more difficult; Z does not provide undefeatable evidence for the existence of zombies, and it certainly does not entail that there are zombies. The problem, to recap, is that there is no logical or evidential intermediary between the observational facts and the facts of consciousness: there is no fact which entails any consciousness fact which is not itself a consciousness fact; and there is no fact which verifies, in any sense, a negative consciousness fact contra our conclusions drawn from the evidence aside from a consciousness fact. So the only way to derive zombies from Z is to stipulate them in; but to do that is to forfeit any claim that zombies are positively conceivable (and surely doesn’t constitute having a ‘clear and distinct’ conception of zombies in any case). I conclude that zombies are not positively conceivable.

The only possible response I can see to this is to claim that we can directly conceive a zombie scenario, unmediated by any appeal to descriptions. In the same way that imagining (‘picturing in the head’) a mile-high unicycle, to use Chalmers’ example, is all there is to conceiving a mile-high unicycle, one might argue that zombies can be similarly directly represented. However, to argue this way would seem as unsubstantiated and desperate as positing direct metaphysical intuitions. Although a scenario where there is a mile-high unicycle may well be given to us directly, this is only because the existence of such can be verified perceptually, and our imagination is perceptual in nature. (It may allow for other kinds of direct representation as well, like statements concerning natural numbers, but these won’t be relevant for the zombie case.) But zombie scenarios can not be verified perceptually. As with the metaphysics cases, there is no direct zombie intuition nor ‘zombie faculty’ which allow us to directly represent zombie scenarios; therefore zombies are not directly conceivable and so descriptions are necessary in order to specify scenarios in which they exist.

The reader may observe that I have not given much attention to ideal conceivability in the above discussion, and may wonder if the appropriate idealizations may save Chalmers’ arguments. I think it is fairly clear that they do not, as I will now demonstrate. Chalmers (1999: 2, 16) thinks that ideal conceivability should be conceivability unconstrained by any contingent psychological limitations on the part of the agent, although he is far less clear about what he thinks those limitations should be. Plausibly, they include specific limitations on memory, processing speed, and (to some extent) the ability to perform abstraction. Such constraints do not play a role in the inability to positively conceive of zombies; all information was assumed to be had by the conceiver at the outset. In fact, it is interesting to observe that taking idealized conceivability seriously seems to hurt Chalmers rather than the Materialist. That’s because, assuming the necessity of identity (as a Kripkean like Chalmers presumably does) and assuming that there is consciousness at all, it follows that Chalmers must claim that Materialism is inconceivable. (I owe this point to Brian McLaughlin.) Suppose not; then for any mental state there is a conceivable world where it is identical to a physical state. Therefore the corresponding possible worlds exist by Chalmers’ C-P principle; and then by necessity of identity, at every possible world where those physical states exist they are identical to the corresponding mental states. In that case, there can be no zombie worlds; so if zombies are conceivable, then on Chalmers’ argument Materialism is inconceivable. The inconceivability of Materialism surely becomes less plausible as constraints on conceivability are lifted, so it seems clear that no appeal to ideal conceivability could help Chalmers with the positive conceivability of zombies—quite the opposite, in fact.

The foregoing considerations were intended to show that sense in which zombies are conceivable does not entail their possibility. The last thing left to do is to explain why our conceivability intuitions in such a case are misguided; why the negative conceivability of zombies is not a guide to their possibility, and so why mind/brain identities plausibly count as posteriori necessities. It turns out that there is a very good explanation of this, which is provided by Chris Hill (1997). That is, there is an explanatory model under which all a posteriori necessities, including mind/brain identities, can be subsumed. The apparent contingency of a posteriori necessities can be explained by virtue of the fact that the simultaneous employment of two different faculties of imagination or conception are required to imagine or conceive of the given necessity. This contrasts with the ‘textbook kripkean’ approach, whereby the apparent contingency is explained by the fact that the reference-fixing property and the property in the identity are distinct. With respect to imaginability, the different faculties can be different sense modalities, or sympathetic versus perceptual imagination; when conceivability is involved, different classes of concepts, such as sensory and neurophysiological, may be in play. In the water/H2O case the analysis goes something like this: The apparent contingency of the identity is due to the fact that our concept (primary intension) of water is largely a perceptual concept—we associate it with perceptual qualities like clearness, tastelessness, and dispositions to perceive like the disposition for thirst to disappear after one drinks it. H2O, on the other hand, is a microphysical concept; it is acquired through a process of theoretical abstraction and postulation of observable entities. There are no "substantive a priori ties" between perceptual and microphysical concepts, so one can "splice" together the concept of water with the concept of the absence of H2O. This process of image- or concept-splicing generates the intuition that water ¹ H2O, as well as a host of other bogus non-identity claims, and is therefore quite unreliable.

Image- or concept-splicing of this sort is also responsible, then, for the intuition that pain ¹ pyramidal cell activity (p.c.a.) Pain is a sensory concept, the justification for the application of which comes from experiencing the sensation. P.c.a. is a neurophysiological concept—not quite a microphysical concept, but still clearly a theoretical one rather than an observational one. To conceive of zombies is to splice together the p.c.a. concept with the concept of the absence of pain; since, as before, there are no a priori connections between the two, the intuition that there is no real identity between them is unreliable. This is not, as Hill argues, to impugn conceivability intuitions (as they lead to modal judgments) generally; rather, only those conceivability intuitions which are a consequence of not having access to all the facts—facts like ‘pain = p.c.a.’ something for which there are good methodological grounds for believing.

Chalmers responds to this argument in MMM, claiming that although such an analysis explains the apparent contingency of a posteriori necessities, it doesn’t explain the "co-presence" of apparent contingency and necessity in such statements. (10ff.) Such an explanation is given by the Kripkean model, Chalmers claims; on that model the co-presence is due to the fact that a counterfactual situation (watery stuff = XYZ) is considered as actual (water = XYZ). This is equivalent to explaining the co-presence as a confusion between the reference-fixing property (being watery stuff) and the property in question (being water). But, it seems, this sort of confusion is just the sort that occurs on Hill’s model, as well; as a result of not knowing (or not attending to) all of the facts, we think that the two concepts can be spliced together. But they can’t, and the reason for this is that one of the concepts (water) has as its constituents reference-fixing properties (potability, etc.), which are distinct from the property (being H2O) it picks out. Chalmers complains that "What we need is an explanation of how the two states could be necessarily one." (MMM, 10) But Hill’s model gives the same explanation Kripke’s does: if we can use two different imaginative faculties for, or have two different concepts of, the same state of affairs, we should expect that in some cases necessary truths will appear contingent—again, because as imaginers and conceivers we are in most cases not apprised of all the facts, facts which would make the appearance of contingency disappear. (Facts such as pain = p.c.a., perhaps.) The fact that water = H2O, after all, is a datum, not an explanandum. So, to splice concepts is (in some cases) to treat a counterfactual situation as though it were actual; Hill’s model possesses just as many explanatory resources as Kripke’s. So it seems clear that Hill’s model is certainly no worse at explaining co-presence than Kripke’s is, and therefore provides an adequate alternate explanation of a posteriori necessities.

To summarize: Some mathematical and metaphysical claims seem to violate the conceivability/possibility entailment; they are necessary truths yet conceivably false and conceivably true. When conceivability is differentiated into positive and negative versions, such that only positive conceivability and not negative conceivability entails possibility, then it can be successfully argued that the claims in question are only negatively conceivable. But it is also clear that zombies are not positively conceivable. Moreover, there is an adequate alternate model of the apparent contingency of a posteriori necessities, potentially including mind/brain identities; so Chalmers’ arguments against Materialism fail.