1. The Benacerraf-Mackie challenge

Lecture 4 of Being Realistic about Reasons (BRR) begins with a discussion of Mackie’s epistemological ‘Argument from Queerness’. In his (Reference Mackie1977), Mackie argues that normative properties would be ‘qualities or relations of a very strange sort, utterly different from anything else in the universe’, and that knowledge of them would require ‘some special faculty of [normative] perception or intuition, utterly different from our ordinary ways of knowing everything else’ (Reference Mackie1977, 38).Footnote ¹ The argument is akin to Benacerraf’s that ‘the connection between the truth conditions for the statements of [mathematics] and any … events connected with … people … cannot be made out’ (Reference Benacerraf1973, 673).

The challenge to explain our knowledge of truths of a kind, F, actually has two components – components which Scanlon does not distinguish.Footnote ² First, there is what I will call the justificatory challenge, or the challenge to explain the (defeasible) justification of our F-beliefs. Second, there is what I will call the reliability challenge, or the challenge to explain their reliability.Footnote ³ To see the difference, consider Godel’s analogy between mathematical intuition and sense perception. Godel claims that we ‘have a perception …of the objects of [mathematics], as is seen from the fact that the axioms force themselves upon us as being true’ (Reference Gödel, Benacerraf and Putnam1947, 483–484). As an answer to the justificatory challenge for mathematical realism, Godel’s remark seems apt. What explains the justification of our mathematical beliefs, Godel seems to suggest, is that they are ‘intuitively evident’ or are implied by the best systemization of what is. But Benacerraf was not satisfied with Godel’s remark, evidently because it did nothing to answer the reliability challenge for mathematical realism. He writes,

What is missing from Godel’s discussion, Benacerraf seems to suggest, is an account of why the ‘intuitive evidentness’ of mathematical propositions would be a reliable symptom of their truth.Footnote ⁴

2. Scanlon on the reliability challenge

Scanlon’s response to the ‘Benacerraf-Mackie Challenge’ is two-pronged. The first prong says that reasons and mathematical entities would not be a ‘special kind of entity which we could “get in touch with” only through a faculty analogous to sensory perception’ (BRR, 70). Why? Because ‘[n]othing in the content of normative or mathematical judgments suggests that they are about objects with any particular spatio-temporal location … hence … not one “outside of us”’ (BRR, 70). The second prong articulates an ‘account of the kind of thinking through which we can come to know [such] truths’ BRR, (70). In the normative and mathematical cases, Scanlon argues that the ultimate such account appeals to the method of reflective equilibrium.

It might be thought that the two prongs of Scanlon’s response correspond exactly to the reliability and justificatory challenges. The second does seem to be responsive to the justificatory challenge. What explains the justification of our normative and mathematical beliefs, Scanlon suggests, is that they have been arrived at via (a proper application of) the method of reflective equilibrium (BRR, 84). But how might Scanlon’s suggestion that ‘[n]othing in the content of normative or mathematical judgments suggests that they are about objects …“outside of us”’ be responsive to the reliability challenge? Scanlon appears to hold that if truths of a kind, F, do not involve commitment to ‘a special kind of entity’, then an answer to the justificatory challenge for F-realism suffices as an answer to the reliability challenge. For example, he writes ‘there is the question of how we could come to know [normative truths]. The method of reflective equilibrium is an adequate answer to this question unless the best understanding of the domain holds that facts about it are … inaccessible to us’ (BRR, 122).Footnote ⁵

Such a suggestion would not be unprecedented.Footnote ⁶ Mackie strains to ‘reify’ objective values, comparing them to Plato’s Forms (Reference Mackie1977, 24), and an influential line of response to Benacerraf says that, while mathematical truths are relevantly independent of minds and languages, they are not really about special entities, like numbers, sets, and tensors.Footnote ⁷ But note, first, that it would be very surprising if one could dismiss the reliability challenge in this way. That challenge is widely supposed to appear pressing for varieties of realism which do not involve commitment ‘to a special kind of entity’. For example, logical (as opposed to metalogical) realism is typically supposed not to involve such commitment. ‘If there are dogs, then there are dogs’ seems to be true, and true relevantly independent of minds and languages (even of necessity). But few would add that this is thanks to the existence of ‘a special kind of entity’, The Conditional. Similarly, in the context of nominalism about universals, traditional formulations of moral realism involve commitment to no new ontology. While the realist holds that, e.g. ‘Hitler is wicked’ is true relevantly independent of human minds and languages, and while she accepts that this is thanks in part to the existence of Hitler, she does not accept that its truth owes anything to the existence of yet another entity, The Good. (What about sentences like ‘Generosity is a virtue’? The question of whether any sentence of the form ‘F-ness is G’ is true on a face-value construal is just the problem of universals.Footnote ⁸) Finally, one who takes modal operators as primitive merely admits new ‘ideology’, whose meaning may be given by introduction and elimination rules, on analogy with negation.Footnote ⁹ Nevertheless, there is widely supposed to be a pressing reliability challenge for realism about logic, morality, and modality.Footnote ¹⁰

Of course, this supposition could be mistaken. What, Scanlon might ask, could the reliability challenge for F-realism amount to if there are no special F-entities with which to ‘get in touch’? The answer is that it could amount to the challenge to explain the correlation between our F-beliefs and the F-truths. This challenge appears pressing whenever the F-truths are relevantly independent of human minds and languages, whether or not they are about ‘a special kind of entity’. But would not F-truths themselves be ‘a special kind of entity’? Even if they would, the challenge can be stated so as not to presuppose them. As Field (Reference Field1989, 25–30) notes, it can be stated as the challenge to explain a significant array of instances of the following schema:

Instances of Correlation use instances of p. They do not mention them.

Nevertheless, Scanlon might protest that, absent an account of what it takes to explain a significant array of instances of Correlation, the challenge is still without force. Benacerraf notoriously demanded the specification of a causal connection between our beliefs and the truths (Reference Benacerraf1973, 671–673). But not even the author of the theory to which Benacerraf appealed regarded that as legitimate.Footnote ¹¹ Another suggestion is that it takes specifying an explanatory – even if not causal – connection between our beliefs and the truths.Footnote ¹² But this seems both too easy and too hard. It seems too hard because sufficiently ‘pluralist’ accounts of the truths (to be discussed) seem to ‘solve the problem by articulating views on which though [the truths] are mind independent, any view we had had of them would have been correct …’ (Field Reference Field, Gendler and Hawthorne2005, 78). Such views do not, in general, imply that there is an explanatory connection between our beliefs and the truths. It seems too easy because the contents of our true logical beliefs are implied by the best explanation of everything. In particular, they are implied by the best explanation of our having the true logical beliefs that we have (Clarke-Doane (Reference Clarke-Doane and Pataut2017, Sec. 5)). But surely this truism does not suffice to explain the correlation between our logical beliefs and the truths.Footnote ¹³

A better suggestion is that the reliability challenge requires the specification of a counterfactual (even if not causal) dependence between our beliefs and the truths.Footnote ¹⁴ Such a challenge seems prima facie legitimate but also impossible to meet in the normative and mathematical cases. As Bedke writes, ‘[w]hatever form the [normative] facts or properties take, one would have the very same [normative] seemings and beliefs because such things are causally determined, and the causal order has not changed’ (Reference Bedke2009, 196). Bedke’s point is naturally taken to be that had the ‘explanatorily basic’ normative or – we may add – mathematical truths been different, our corresponding beliefs would have been the same.Footnote ¹⁵ The causal forces that shaped our normative and mathematical beliefs would still have led us to have the same such beliefs (Reference Bedke2009, Sec. 1.2).

One way that Scanlon might respond to this objection is by complaining that it makes no sense to say what would have happened had the explanatorily basic normative or mathematical truths been different. Such truths are widely supposed to be ‘metaphysically’ necessary, and counterfactuals conditionalizing on metaphysically necessary truths are vacuous on a standard semantics.Footnote ¹⁶ But as Bedke notes, ‘it is at least conceptually possible’ to vary ‘the [normative] facts and properties … however one wishes’ (Reference Bedke2009, 196). Moreover, it surely appears false that had the explanatorily basic normative or mathematical truths been different, our explanatorily basic normative and mathematical beliefs would have been correspondingly different.

In my view, whether the explanatorily basic normative and mathematical truths count as ‘metaphysically’ necessary is of little metaphysical consequence.Footnote ¹⁷ The important point is that there is no principled reason to regard a counterfactual conditionalizing on alternative normative or mathematical ‘laws’ as vacuous which is not equally a reason to regard counterfactuals conditionalizing on the likes of alternative physical or biological laws as vacuous too.Footnote ¹⁸

The real problem with the present objection is not that it involves a ‘counterpossible’. The problem is that it threatens to overgeneralize. For virtually any alleged ‘metaphysically’ necessary truth, p, it seems had it been the case that ~p, we still would have believed that p. In particular, it seems that had the bridge principles which link subvenient to supervenient properties – such as atoms arranged chairwise compose a chair – been different, our beliefs would have been the same. Of course, such truths are intuitively ‘theoretical’, unlike the truth that I am sitting in a chair. But, as Sturgeon (Reference Sturgeon, Copp and Zimmerman1985) pointed out, even the normative realist can establish the counterfactual dependence of our atomic normative beliefs, such as that ‘Hitler is wicked’ – assuming, what we must assume in the ordinary object case too, that the closest worlds in which Hitler is not wicked are worlds in which the bridge principles are the same.Footnote ¹⁹

3. Safety and pluralism

If the reliability challenge cannot be understood as the challenge to establish a causal, explanatory, or even non-causal counterfactual dependence, between our beliefs and the truths, then one might be tempted to conclude, with Scanlon, that there really is no distinctive reliability challenge for normative and mathematical realism. But this would be too quick. For any area, F, there are two ways of having false F-beliefs. First, it could happen that the F-truths are different while our F-beliefs fail to be correspondingly different. Second, it could happen that our F-beliefs are different while the F-truths fail to be correspondingly different. Even if the first possibility is inapt when the F-truths are metaphysically necessary, the second possibility remains. Moreover, while the second possibility per se just says that the F-truths do not counterfactually depend on our beliefs (which any ‘non-pluralist’ F-realist should concede), there is a possibility in the neighborhood which is genuinely worrisome. This is that our F-beliefs are not safe – that we could have easily had false F-beliefs (using the method that we actually used to form them), because we could have easily had different F-beliefs.Footnote ²⁰ The reliability challenge for F-realism can be understood as the challenge to show that our F-beliefs are safe.Footnote ²¹

This interpretation finally allows us to understand why Scanlon might claim that ‘there is the question of how we could come to know [normative truths] … [but t]he method of reflective equilibrium is an adequate answer to this question…’ (BRR, 122). In Lecture 3, Scanlon introduces a kind of ‘metaphysical pluralism’ according to which all truths about the world can be non-arbitrarily segregated into kinds, which he calls ‘domains’, and ‘as long as some way of talking [is] well defined, internally coherent, and [does] not have any presuppositions or implications that might conflict with those of other domains, such as science’, such talk is true (Reference Scanlon2014, 27, emphasis in original).Footnote ²² Such a pluralism arguably affords a reduction of the reliability challenge to the justificatory challenge – understood as the challenge to show that our beliefs are safe – because it bridges the gap between ‘coherence’ and truth. It shows that, as long as we could not have easily had ‘incoherent’ beliefs of a kind, F, we could not have easily had false F-beliefs – assuming that our F-beliefs do not conflict with truths about other domains, and that the F-truths could not have easily been different. The last claim is trivial when the F-truths are metaphysically necessary, as the (explanatorily basic) normative and mathematical truths are widely supposed to be, and Scanlon claims – and I will not dispute – that neither normative nor mathematical truths conflict with truths from other domains. Modulo an argument that we could not have easily had ‘incoherent’ normative and mathematical beliefs (using the method that we actually used to form ours – i.e. according to Scanlon, the method of reflective equilibrium), it follows that that our normative and mathematical beliefs are safe.

What does Scanlon mean by ‘coherent’? There are two possibilities. According to the first, coherence just requires (first-order) consistency. According to the second, coherence requires consistency with some privileged set of claims (and perhaps more).Footnote ²³ Scanlon seems to take coherence to require the latter (BRR, 79). While both PA + Con(PA) and PA + ~Con(PA) are consistent if PA is, Scanlon suggests that only one gives a coherent account of the natural numbers (BRR, 72). The truth of Con(PA) is ‘settled by the standards of the domain it is about’, even if not implied by its axioms (BRR, 19).Footnote ²⁴ In the case of set theory, Scanlon even suggests that such recondite postulates as the Axiom of Replacement are settled in this way (BRR, 73). Apparently, Scanlon takes PA + ~Con(PA) and ZF – Replacement + ~Replacement to be ‘incoherent’, and takes typical axioms of arithmetic and set theory, plus consistency statements, to be ‘privileged’.Footnote ²⁵

Let us call the pluralism resulting from the stipulation that consistency with some privileged set of (non-logical) statements is necessary for coherence, moderate pluralism.Footnote ²⁶ This form of pluralism is moderate in the sense that it is not obviously too anti-objectivist. In the case of set theory, the view allows that there is a serious question as to whether typical axioms are true (not just a serious question as to what follows from them). Consider the Axiom of Replacement again. The moderate pluralist says that one can be competent, consistent, attentive, and wrong about this. Indeed, a logician no less eminent than Boolos (Reference Boolos and Jeffrey1999, 121) doubted that there was an ordinal greater than all f(x), where f(0) = Aleph_0 and f(x + 1) = Aleph_f(x) for all natural numbers, x, and, thus, had ‘incoherent’, and so false, set-theoretic beliefs according to the moderate pluralist (since Replacement implies the existence of such an ordinal in the context of the other axioms).Footnote ²⁷

On the other hand, moderate pluralism is still pluralist because if T is the set of privileged statements, and A and ~A are both consistent with T, then, according to the moderate pluralist, T + A and T + ~A may both be true. For instance, in the case of set theory, the Continuum Hypothesis (CH) might be such an A (BRR, 76). Of course, CH and ~CH cannot be true of the same domain. But the moderate pluralist can hold that claims which are independent of the privileged ones lead to ‘bifurcations’ in our relevant concepts, so CH and ~CH do not really conflict (BRR, 80).

However, while moderate pluralism affords a level of objectivity, it threatens to preclude answering the reliability challenge. If it is this difficult to have ‘coherent’ beliefs, then it is hard to see why we could not have easily had incoherent ones. For instance, it is hard to see why we could not have easily doubted, with Boolos, the existence of the aforementioned ordinal – especially given that such enormous (albeit tiny for set theory!) sets seem superfluous to ‘ordinary’ mathematics and empirical science. It is not as if Boolos’s skepticism turned on a inconsistency in his beliefs, for instance. If Scanlon takes beliefs to be coherent only if they are consistent with, e.g. the standard axioms of set theory, then it is doubtful that his pluralism does afford a reduction of the reliability challenge to the justificatory challenge.

It might be thought that this worry is less pressing in the case of normative beliefs.Footnote ²⁸ If banalities like that we have reason to avoid pain and pursue pleasure exhaust the (non-logical) constraints on normative coherence, then prima facie we could not have easily had incoherent normative beliefs. However, these cannot exhaust the constraints on coherence if normative inquiry is to be remotely objective. Such truisms are analogous to the set-theoretic axioms of Extensionality or Pairing, and do not begin to ‘decide’ most questions characteristic of the domain in question. In particular, they leave open how much reason we have to avoid pain and pursue pleasure, and so how these considerations are to be weighed against one another.

4. Pluralism and objectivity

Perhaps, then, Scanlon takes (or ought to take) coherence to just require consistency. Again, the view cannot be that set theory with, e.g. Replacement and set theory with its negation are both true of the same domain on pain of inconsistency. However, Scanlon could claim that apparently contradictory axioms are actually about different domains, on analogy with CH and ~CH.Footnote ²⁹ Indeed, Scanlon makes just this move in connection to the Axiom of Foundation (BRR, 76, fn. 9 and 88). Set theories with Foundation and with its negation are in the same language (syntactically individuated), but Scanlon suggests that they do not contradict one another.Footnote ³⁰ Since it is arguable (though certainly not trivial) that we could not have easily had (first-order) inconsistent normative or mathematical beliefs, this understanding of ‘coherence’ arguably affords an answer the reliability challenge.

Let us call the pluralism resulting from the stipulation that consistency suffices for coherence, radical pluralism.Footnote ³¹ Then, while radical pluralism appears to afford an answer to the reliability challenge,Footnote ³² it is too anti-objectivist. To see what I mean by ‘objective’, consider the question of whether the Parallel Postulate is true – not as a hypothesis about physical spacetime, but as a pure mathematical conjecture. As is well known, the Parallel Postulate is true of Euclidean space, false of, e.g. hyperbolic space, and that is all there is to it. To be sure, an eccentric metaphysician could declare that some one geometry is somehow ‘metaphysically privileged’.Footnote ³³ However, such a view is entirely unprincipled (and, as far as I know, unprecedented – even in philosophy!). In an obvious sense, the Parallel Postulate question has no objective answer.

Note that the problem is not that it lacks a mind-and-language independent answer. Given a determinate use of ‘point’, ‘line’, and so on, the Parallel Postulate question has an answer, and for all that has been said, it depends entirely on the way the mind and language-independent geometrical facts are. Nor is the problem that the question lacks a unique answer, given such a use of primitive terms. (One need not be a paraconsistent logician in order to accept the truth of different geometries!) The problem is that all we would learn in answering that question (as a pure mathematical question) is something about us. We would just learn what geometrical structures we were talking about, rather than learning what geometrical structures there were.

On the present interpretation of Scanlon’s pluralism, he is suggesting that set theoretic, and presumably normative, inquiry is a lot like geometrical inquiry. The question of whether, e.g. Foundation is true is like the question of whether the Parallel Postulate is true. If logic is objective, then the question of what follows from set theory with and without Foundation remains genuine. So does the question of whether Foundation is ‘packed into’ our concept of set. But the peculiarly mathematical question of what the set-like universe (better: pluriverse) contains is trivialized. Assuming that we already know that set theory with Foundation and set theory with its negation are each consistent, then we already know that each is true. The more Scanlon views different set-theoretic and normative theories as being like different geometries, the more like the Parallel Postulate question fundamental questions of normativity and set theory become.

Note that the worry that normative and mathematical inquiry is not objective is distinct from the worry which preoccupies Scanlon throughout much of BRR, that ‘statements within these domains [lack] determinate truth values’ (BRR, 122). It is commonly conceded that if our concept of set is (appropriately) indeterminate, then the ‘search for new axioms’ to settle undecidables like the Continuum Hypothesis (CH), as traditionally conceived, is misguided (Martin Reference Martin and Browder1976, 90–91). However, if radical pluralism is true, then that search appears to be misguided even assuming that our concept of set is determinate. It is like the ‘search for the truth-value of the Parallel Postulate’, given a determinate use of ‘point’, ‘line’, and so on (although the answer is presumably less transparent). Again, such a search may tell us what is ‘packed into’ our concept of set.Footnote ³⁴ But it will not tell us anything that we did not know about which such concept is satisfied – i.e. about what the universe of set-like things contains.Footnote ³⁵

5. Objections and replies

I have argued that Scanlon’s metaphysical pluralism, according to which any ‘coherent’ set of beliefs meeting certain conditions is true, can be understood as a response to the reliability challenge, qua the challenge to show that our normative and mathematical beliefs are safe. But his pluralism admits of a radical and a moderate interpretation. Under the radical interpretation, it apparently answers the reliability challenge, but undercuts the objectivity of mathematical and normative inquiry. Under the moderate interpretation, it seems to afford a degree of objectivity in these areas, but no longer affords an answer to the reliability challenge. The twin desiderata of answering the reliability challenge and vindicating the objectivity of the areas may be mutually exclusive.Footnote ³⁶

How might Scanlon respond to this dilemma? I can think of three ways. First, grasping the second horn – and defending moderate pluralism – Scanlon could deny that it matters whether we could have easily had ‘incoherent’, and so false, normative and mathematical beliefs. Epistemic luck is pervasive. Why think that this particular kind is malignant? But while the issue certainly deserves discussion, the answer is straightforward. Evidence that we could have easily had a false belief as to whether p (using the method that we actually used to form it) is a paradigm undermining (as opposed to rebutting) defeater of our belief that p. It gives us reason to give up our belief, but not by giving us ‘direct’ reason to believe that its content is false.Footnote ³⁷ If Scanlon’s epistemology required that such undermining evidence was impossible, then its plausibility would be greatly diminished.

Second, grasping the first horn – and defending radical pluralism – Scanlon could argue that the existence of apparently conflicting normative or mathematical truths does not really undercut the objectivity of the areas, because they are about different domains. Scanlon might be thought to make just this point when he considers the challenge from ‘counter-reasons’ – the correlates of another ‘coherent’ practice superficially similar to our practice with reasons, but intuitively flipped. We might, say, have a counter-reason to hurt people just for the fun of it, even if we do not have any reason to do so. Scanlon, apparently allowing that there are counter-reasons in addition to reasons, suggests that they generate no challenge to the objectivity of normative inquiry because ‘[c]onclusions about “counter-reasons’ conflict with our conclusions about reasons only insofar as they are interpreted as conclusions about reasons” (BRR, 29). However, as indicated above, it does not matter whether reasons ‘conflict’ with counter-reasons.Footnote ³⁸ The Parallel Postulate question has no objective answer in the relevant sense, despite the fact that, given a determinate use of ‘point’, ‘line’ and other primitive terms, it has exactly one answer.

Finally, again grasping the first horn, Scanlon could argue that while his (radical) pluralism does come at the cost of the objectivity of normative and mathematical inquiry, this is as it should be. In the mathematical case, this response has some plausibility. Mathematics (as opposed to the philosophy of mathematics) seems to be overwhelmingly concerned with what follows from such and such axioms, not with what axioms are true. But no self-respecting normative realist should be satisfied with an analogous view. Normative inquiry is concerned with categorical questions, rather than with merely logical questions about what follows from such and such normative principles. Practically, however, radical pluralism is indistinguishable from the most uncompromising ‘if-thenism’.Footnote ³⁹ The question of whether we have most reason to give to charity is like the question of whether the Parallel Postulate is true. We speak a language, and, in that language, the question may have a mind-and-language independent answer. But this answer stands to the Parallel Postulate as the negation of that answer stands to the negation of that postulate. Euclidean and hyperbolic geometries are on a metaphysical par, and there is no useful sense in which the first gives the ‘right’ account of geometrical reality, even if we mean Eucidean line by ‘line’, Eucidean point by ‘point’, and so on. To give up on the objectivity of normative inquiry is to give up what ought to be the central commitment of normative realism.