On The Two-Dimensionalist Reductio and The Response from Ideal Conceivability

According to an argument that was recently formulated by Robert J. Howell, there is a statement whose conceivable truth is enough to show that Chalmers’ two-dimensional approach to semantics is incorrect and directly self-refuting.[1] The statement Howell has in mind is this:

(SN) The space of metaphysically possible worlds is more limited than the space of conceivable worlds.

This statement is of course incompatible with Chalmers’ two-dimensional approach to semantics since, on that approach, the space of metaphysically possible worlds and the space of conceivable worlds coincide; they are in effect one and the same. Now, before I try to explain why Howell thinks that the conceivability of SN is enough to show that Chalmers’ two-dimensional approach to semantics is incorrect, I should introduce Howell’s epistemic notations ‘conceivable1’, ‘conceivable2’, and their modal counterparts ‘possible1’ and ‘possible2’.[2] Whenever a statement is said to be conceivable1 or possible1, what is meant is that it is conceivable or possible with regard to its primary intension. That is to say, the statement in question is conceivable when we treat other possible worlds as if they are actual and then evaluate the statement relative to those worlds. Conversely, when a statement is said to be conceivable2 or possible2, what is meant is that it is conceivable or possible with regard to its secondary intension. In other words, the statement in question is conceivable when we treat other possible worlds as if they are counterfactual and then evaluate the statement relative to those worlds. Clearly then, Howell’s distinction between conceivability1 and conceivability2 is equivalent to David J. Chalmers’ distinction between primary and secondary conceivability and his distinction between possibility1 and possibility2 is in turn equivalent to the distinction between a priori metaphysical possibility and Kripke’s a posteriori metaphysical possibility. With these clarifications firmly in mind, we can now turn to Howell’s two-dimensionalist reductio:

1. If metaphysical two-dimensionalism is true, the conceivability1 of a statement’s truth entails its possibility1.
2. SN is conceivable1.
3. If metaphysical two-dimensionalism is true, SN is possible1.
4. If SN’s primary and secondary intensions coincide, SN’s being possible1 entails that SN is possible2.
5. SN’s primary and secondary intensions coincide.
6. If SN is possible2, SN is true.
7. If SN is true, metaphysical two-dimensionalism is false.
8. If metaphysical two-dimensionalism is true, it is false.

Given our understanding of the two-dimensional approach to semantics, as well as our intuitions about conceivable truths, Howell assumes that premise 1 and 2 should seem rather reasonable to us. And since it follows straight-forwardly from premise 1 and 2, he assumes that premise 3 should seem rather reasonable as well. When it comes to premise 4, 5, and 6, however, things become a bit more complicated. Howell defends premise 4 by pointing out that, if a statement’s primary and secondary intensions were indeed to coincide, then one should obviously be able to infer from the fact that the statement is possible1 that it is also possible2.[3] The most pressing question, he admits, concerns premise 5. In other words, why do the primary and secondary intensions of SN have to coincide? Howell begins to answer this question by pointing out that SN is a statement about the whole of logical space, i.e. the set of all possible worlds. Since the set of all possible worlds remains the same no matter what particular world we treat as actual, SN or any statement concerning the whole of logical space must be necessary.[4] Howell further argues that, since SN is also knowable a priori, it follows that its primary and secondary intensions coincide. He explains:

If a statement is necessary a priori, its truth across possible worlds considered as counterfactual and as actual should coincide, because its truth requires no contribution from any particular world.[5]

To illustrate this point further, Howell compares SN with necessary statements that are only knowable a posteriori:

Primary and secondary intensions come apart in the water case, since which world is actual matters to the evaluation of necessary truths concerning water. That is why the necessities there are a posteriori. In the case of a priori truths, by hypothesis one doesn’t need to find out which world is actual in order to evaluate them. Given the way primary and secondary intensions are defined, therefore, they should coincide when it comes to the necessary a priori. So, premise five follows from the modal and epistemological nature of SN and the definitions of primary and secondary intensions.[6]

So statements that are necessary a priori differ from statements that are necessary a posteriori in that, when we evaluate the their respective truth-conditions, only the latter kind of statement require that we take into consideration what particular world we are treating as actual. This means that it is only statements that are necessary a posteriori that can have different primary and secondary intensions. If Howell is right then, the primary and secondary intensions of SN do indeed coincide. Now, given the truth of premise 5, Howell thinks that the conclusion cannot be far behind:

[F]rom the conceivability of SN, we can conclude its possibility in all relevant senses, and since it is a necessary truth, its possibility entails its truth. But if SN is true, metaphysical two-dimensionalism is false. Only if SN is false can we be guaranteed that there is a world corresponding to the truth of our primary intensions. If SN is true, therefore, there is no entailment from the conceivability of the primary intension to its possibility, and metaphysical two-dimensionalism is shown false under its own light.[7]

There are several ways in which one could object to the two-dimensionalist reductio. For example, it is not all that clear what justifies premise 6 and its inference from the possibility of SN to its truth. In the above-mentioned quote, Howell explains that the truth of SN is entailed by the fact that it is both a possible and necessary truth. This is problematic, since SN is merely possible and necessary if true. That is to say, SN is also possible and necessary if false — which would entail its falsehood if we followed Howell’s method of reasoning. I shall return to this point below. For now, I will turn my focus on the assumption that, on the two-dimensional approach to semantics, conceivability is always a good guide to modality. In light of this assumption, the two-dimensionalists might want to take a step back and claim that conceivability is only a good guide to modality provided certain epistemic or rational conditions are fulfilled. Such a response will be explored and defended below. This response, which will henceforth be referred to as the response from ideal conceivability, relies on a distinction between two different kinds of conceivability – only one of which tracks genuine metaphysical possibility. Indeed, Howell himself anticipates this kind of response. He writes:

[…] given our epistemic limitations, we can no doubt conceive of the falsity (or the truth) of Goldbach’s Conjecture. But no one wants to conclude from this level of conceivability that it is possibly false (or true) – if it is false it is necessarily false, so if it is possibly false it is false. It is open to the two-dimensionalist to say that conceiving of the truth or falsity of SN is like this – only weakly (or ‘prima facie’) conceivable.[8]

Specifically then, Howell admits that the two-dimensionalist could escape the conclusion of his argument by distinguishing ‘weak conceivability’ from what is commonly referred to as ‘ideal conceivability’. As opposed to weak conceivability, ideal conceivability is only available to us once we have overcome certain epistemic limitations. If a statement is weakly and prima facie conceivable then, it is only conceivable because we happen to lack a sufficient grasp of the terms in which it is formulated, or the key concepts it might be associated with. Perhaps the distinction between weak and ideal conceivability can be further illustrated by way of the following example: When I was a small child my mother often gave me clothes that were previously owned by relatives. As a result, some of my shirts were labeled with their names instead of my own. One day during school a teacher happened to recognize one of the names and claimed that it had belonged to a former student of hers. This eventually led her to ask me whether I was a relative to this student, whereby I responded that I was not but that my mother was! As a small child then, it seemed perfectly conceivable to me in a primary sense that my biological mother’s relatives were not my own. Obviously, the fact that this was primarily conceivable to me does not entail that it was metaphysically possible. Instead, I think we would be justified in saying that I lacked a sufficient grasp of the term ‘relative’ and the key concepts it might be associated with. Clearly, this is just another way of saying that when I was a small child, some statements involving the term ‘relative’ were not ideally conceivable to me.

So the response from ideal conceivability would save the two-dimensionalist from the conclusion of Howell’s argument as it would enable him claim that SN is not conceivable in a strong enough sense to entail its genuine metaphysical possibility. Howell has formulated the following two objections to this kind of response:

(O1) Ideal conceivability is difficult to define in a non-circular way. If, for example, one defines it as the kind of conceivability that manages to track genuine metaphysical possibility, then the two-dimensional approach to semantics has simply overcome the gap between conceivability and possibility by replacing it with a gap between weak conceivability and ideal conceivability.[9]

(O2) If the ideal conceivability of SN should be denied because we do not completely or sufficiently understand what it would mean for SN to be true, then the two-dimensional approach to semantics itself rests on a presupposition that we do not completely or sufficiently understand. After all, the two-dimensional approach to semantics presupposes that SN is false. So even though the response from ideal conceivability might help save the two-dimensionalist from the two-dimensionalist reductio, it does so at the price of giving us a new reason to reject the two-dimensional approach to semantics.[10]

How should the two-dimensionalist respond to these objections? Let us start out by looking at O1. I think Howell is right in saying that, when we try to define ideal conceivability, we should not simply say that it is the kind of conceivability that manages to track genuine metaphysical possibility. This is indeed problematic. Instead, I think we should say something to the effect that ideal conceivability is the kind of conceivability that tends to persist even after some considerable effort has been made to completely or sufficiently understand the terms with which a statement is formulated, and the key concepts it might be associated with. Interestingly, David J. Chalmers himself makes a similar claim in his article ‘Does Conceivability Entail Possibility?’ where he attempts to describe different methods of conceivability and clarify the thesis that some of these methods are good guides to modality. He writes that,

S is ideally conceivable when S is conceivable on ideal rational reflection. It sometimes happens that S is prima facie conceivable to a subject, but that this prima facie conceivability is undermined by further reflection showing that the tests that are criteria for conceivability are not in fact passed.[11]

Admittedly, this kind of argument begs the question as to how a ‘complete or sufficient understanding’ or an ‘ideal rational reflection’ should be defined. Chalmers briefly explores the possibility of defining the latter notion in terms of an ideal conceiver, i.e. an ideal agent who is “free of all contingent cognitive limitations.”[12] On this understanding, a statement would be ideally conceivable just in case such an ideal conceiver would judge it to pass the relevant tests on ideal conceivability. When applied to my characterization of ideal conceivability above, the very same understanding would entail that a statement is ideally conceivable if and only if someone with a complete and sufficient understanding of the terms with which the statement is formulated – or the key concepts it might be associated with – determines it to be ideally conceivable. Chalmers quickly discounts this kind of view, however, as he finds it doubtful whether the notion of an ideal conceiver is actually coherent. One of the worries he brings to our attention is that for every possible conceiver there could probably be an even better conceiver. Though there might be something to this observation, I’m not sure it would necessarily be all that damaging for the account of ideal conceivability just outlined. Nonetheless, it seems intuitively preferable to avoid appeals to ideal agents in philosophy – at least in contexts where our hope is to explain the everyday talk and practices of ordinary people. To a certain extent then, I still agree with Chalmers intuitions.

After discounting the understanding of ideal conceivability in terms of an ideal conceiver, Chalmers then turns his focus on a related but apparently less severe idealization according to which a statement is ideally conceivable if and only if it is conceived with what he calls ‘undefeatable justification’. He writes:

Alternatively, one can dispense with the notion of an ideal reasoner, and simply invoke the notion of undefeatability be [sic] better reasoning. Given this notion, we can say that S is ideally conceivable when there is a possible subject for whom S is prima facie conceivable, with justification that is undefeatable by better reasoning. The idea is that when prima facie conceivability falls short of ideal conceivability, then the claim that the relevant tests are passed will either be unjustified, or the justification will be defeatable by further reasoning. For ideal conceivability, one needs justification that cannot be rationally defeated.[13]

When applied to my characterization of ideal conceivability above, this view would entail that a statement is ideally conceivable just in case someone with a complete understanding of the terms in which it is formulated – and the key concepts it might be associated with – judged it to be so, and this person’s judgment would not defeatable by better reasoning. To some philosophers, this characterization of ideal conceivability will seem hopelessly vague. In particular, the sceptic will want to press the two-dimensionalist on exactly how the notion of ‘better reasoning’ should be defined. Regretfully, Chalmers does not provide us with any such definition. Instead, he simply settles for our intuitive grasp of such rational notions and choses to regard them as primitive and unanalyzable. This in my opinion is a bit too quick. Instead of regarding the notion of better reasoning as a primitive, Chalmers could after all define it directly in terms of better reasons and then chose among the many analyses of reasons that have been offered up in the philosophical literature. In any case, since he does regard the notion of better reasoning as a primitive, Chalmers runs the risk of casting the concept of ideal conceivability in a mysterious light. And because of this, he is also careful to point out that – besides playing a crucial role in the explanation of ideal conceivability – the notion of better reasoning is also involved in some key epistemic concepts:

… it is generally held that if one’s justification for a belief that P is defeatable by better reasoning, then one does not know that P. So the notion of conceivability is not obviously worse off than the concept of knowledge.[14]

In other words, Chalmers claims that it is generally recognized that a set of beliefs do not count as knowledge if our reasons for holding those beliefs are weaker than the reasons someone else has for not holding those beliefs. If he is right, this might explain why we would regard someone with suspicion if they said that they know P, even though they also know that their justification for believing P is somewhat weaker than someone else’s justification for not believing P. So then if the explanation of ideal conceivability outlined above is mysterious just because of its reliance on the notion of better reasoning, perhaps Chalmers is right in believing that the same should be true of our concept of knowledge. Indeed, Chalmers goes on to argue that the notion of better reasoning is also implicit in our concept of a priority:

If I cannot know that P independent of experience, but another less limited being could do so, then it is a priori that P. And if I believe that P, but the justification for my belief is defeatable by better reasoning, then it is not a priori that P (unless there is another undefeatable justification). So the notion of apriority idealizes away from cognitive limitations in much the same way as the notion of ideal conceivability.[15]

Chalmers concludes that, although idealizations in terms of better reasoning are not perfectly clear or unproblematic, they should seem intuitively attractive as they can prove themselves useful in the analysis of some very difficult epistemic concepts. I think this is right. In fact, I suspect that the notion of better reasoning might turn out to serve an important role in the elucidation of a great number of difficult concepts – the most obvious examples being the kind of concepts that implicitly or explicitly involve a reference to various rational notions such as reasons and justification. The fact that these further notions are difficult to analyze and perhaps should be taken as primitive then, should not deter us from their use in our philosophical investigations.

At this point, it should be admitted that the challenge of providing a clear definition of ideal conceivability is a difficult one. And although I believe the above-mentioned views to be on the right lines, it will seem obvious to some that this challenge has yet to be met. For example, this will be true of anyone who denies that our epistemic concepts necessarily involve a reference to rational notions such as better reasoning.[16] Even if this is right, however, it seems to me that the two-dimensionalist can still rule out certain types of statements from the category. That is to say, certain statements seem so “queer” on a two-dimensionalist’s account, that any clear definition of ideal conceivability would have it that they are not ideally conceivable. SN I will now argue might be such a statement. As we have seen, the two-dimensionalist recognizes a close link between conceivability and modality. Hence, if he also accepts that SN is meaningful and should be analyzed the same way as other statements of its kind, he might have to accept that there are true contradictions. To see that this is true, reconsider Howell’s arguments above. Howell has suggested that the statement expressed by SN must be true because it is conceivably true. One of the problems with this suggestion is that the negation of SN is also conceivably false. In other words, it is also conceivable that the space of metaphysically possible worlds is not more limited than the space of conceivable worlds. Now, if this is taken to mean that the statement in question is also possibly true and therefore actually true, then it appears as if we should also accept the truth of the following contradiction:

(SNC) The space of metaphysically possible worlds is more limited than the space of conceivable worlds, and the space of metaphysically possible worlds is not more limited than the space of conceivable worlds.

The truth of SNC follows from the fact that both SN and its negation are conceivably true. In the light of this contradiction, it seems as if the two-dimensionalist should be skeptical of Howell’s claim that SN and its negation are conceivable in an a strong enough sense to entail the truth of SNC. In other words, the fact that the conceivability of SN and its negation seem to imply a true contradiction gives the two-dimensionalist a good reason to be skeptical about the ideal conceivability of those statements. Even if we have yet to formulate a sufficiently clear definition of what it means for a statement to be ideally conceivable then, it seems reasonable from the two-dimensionalist standpoint to exclude these statements from the category and claim that they cannot be conceivable in a strong enough sense to entail their possibility. In fact, if what has been said so far is true, the two-dimensionalist might be justified in saying that necessary statements such as SN are ideally conceivable only if their negations are not conceivable as well. This is obviously not a sufficient condition for the ideal conceivability of such statements, but it does seem reasonable to insist that it is a necessary one.

Let us now move on to O2. With this objection Howell seems to suggest that we should be skeptical of any theory that presupposes the negation of a statement that it also states that we do not completely or sufficiently understand. To find out whether this suggestion is on the right lines, it would be helpful to consider an example from other areas of philosophy. Consider e.g. meta-ethics. Would it be unreasonable for a moral nihilist to base his meta-ethical stance on the fact that we do not completely or sufficiently understand what it would mean for moral values to be part of the furniture of the world? I find this very doubtful. In fact, it seems quite absurd to me that the moral realist’s trouble in explaining what it would mean for moral values to be part of the furniture of the world should itself constitute a problem for any theory that presupposes the falsity of his position. Indeed, judging by the many queerness and supervenience arguments that have taken the center stage of discussions around moral realism over the past twenty-five years, it appears as if there are many philosophers who would share my intuitions. After all, many or most of these arguments against moral realism seem to rely on the observation that the position in question has peculiar metaphysical implications that are difficult (if not impossible) to make sense of given our current linguistic intuitions or metaphysical commitments.[17]

Perhaps the objection could be made that the above-mentioned example misses its mark since, unlike the moral nihilist say, two-dimensionalists do not assume their approach to be the correct one for the simple reason that we do not understand what SN says. Even if this is true, however, what I have tried to show is that it would not be all that unreasonable for a two-dimensionalist to do exactly that. That is to say, if my reasoning above is sound, it would perhaps follow that someone could plausibly be tempted towards conceivability-based accounts of modal knowledge and the two-dimensional approach to semantics because he was having difficulties understanding what it would mean for the space of metaphysically possible worlds to be more limited than the space of conceivable worlds.9 Furthermore, even if we do judge this to be unreasonable, I suspect that the above-mentioned example could be reformulated to get around this objection. Consider quasi-realism. In meta-ethics, the quasi-realist is someone who accepts a nihilistic stance on moral values and thinks that the primary purpose of our first-order moral statements is to express attitudes, desires, preferences or some other set of non-cognitive mental states. Despite her anti-realism, however, the quasi-realist also thinks it legitimate to conduct our moral talk and practices as if there were some truth to be held on moral matters. In other words, it is the quasi-realist’s ambition to explain why and how it can be legitimate to say things such as ‘it is true that murder is wrong’, ‘I believe that murder is wrong’, ‘if murder is wrong, then so is rape’, even though the primary purpose of our first-order moral statement is not to express beliefs or propositions. In fact, quasi-realists such as Simon Blackburn have argued quite extensively that their analysis of such statements can be far more precise and intuitive than that of other meta-ethical theories such as moral realism.

The more specific example just mentioned might be better formulated than the one with which I began, since it clearly illustrates how one need not be a moral nihilist simply because one does not understand what it would mean for moral values to be part of the furniture of the world. Instead, a moral nihilist that is also a quasi-realist say, might be tempted towards her position because it affords her with the best explanation of our everyday moral talk and practices. What we should ask ourselves then, whether it would be unreasonable for this kind of moral nihilist to claim that, besides being able to provide a plausible account of our everyday moral talk and practices, her meta-ethical theory also has the benefit of presupposing the negation of a position that we do not completely or sufficiently understand. Again, if we find that Howell’s suggestion is correct, then our first intuition should be to answer this question in the affirmative. That is not, however, my intuition. Indeed, I think examples of this kind illustrate quite clearly that if we were to accept Howell’s suggestion, we might have to abandon a very useful tool in our philosophical investigations.[18]

1. See Robert J. Howell’s The Two-Dimensionalist Reductio 2008, p. 352.
2. Ibid, p. 350.
3. Ibid, p. 352.
4. Ibid. pp. 352-353.
5. Ibid.
6. Ibid, p. 353.
7. Ibid.
8. Ibid, p. 353.
9. Ibid, 354.
10. Ibid, pp. 354-355.
11. See David J. Chalmers’ Does Conceivability Entail Possibility? 2002, p. 147.
12. Ibid, p. 148.
13. Ibid.
14. Ibid.
15. Ibid.
16 This is probably true of Crispin Sartwell, for example, who is famous for arguing that knowledge can be adequately analysed without mention of justification or any related rational notions; see his Knowledge is Merely True Belief 1991.
17. See e.g. Blackburn’s Essays on Quasi-Realism 1993, pp. 111-129, 130-148.
18. This should also become quite clear whenever one considers the various positions that have been taken with regard to the existence of God. After all, it seems safe to assume that many of the people that have been tempted towards atheism or theological non-cognitivism have been so tempted because it is exceedingly difficult to make sense of the proposition that God exist. It might turn out then that this general way of deciding between conflicting theories in philosophy is quite common wherever there is some sort of deep ontological issue at stake.

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