1. Quine, Kaplan and De Re Modality

Quine’s objection arises from the observation that if expressions like ‘nine’ in (1) are genuinely referring expressions, then we should be able to quantify ‘into’ that position, i.e. (1) should entail that:

But to evaluate the truth of this sentence we need to know how to assign a truth-value to the following open sentence, relative to an assignment of a value to the variable x:

The example from the previous paragraph shows that whether (4) is true relative to some assignment of a value to the variable x depends not only on the object assigned but on how that object is designated. Relative to the assignment of the number 9 to x, (4) is true if we designate 9 by ‘7+2’ but false if we designate it by the description ‘the number of planets’. Either there is a metaphysically ‘preferred’ designation of each object (which Quine takes to be tantamount to Aristotelian essentialism) or the idea of quantifying into modal contexts like (1) is incoherent. Quine concluded the latter; expressions like ‘nine’ in (1) are not genuinely referring expressions. What look like ascriptions of modal properties de re to objects are really claims about de dicto necessity, to the effect that the sentence within the scope of the expression ‘necessary’ is necessarily (or possibly) true. So Quine would read (1) as meaning:

‘Nine is greater than five’ is a name for a sentence, a sentence that is mentioned, not used, in (5). Consequently, the object nine is no more referred to in (5) than it is in:

Quine’s point was not merely that de re modal contexts are referentially opaque, but that what look like de re modal contexts are really de dicto modal contexts like (5) in which necessity is a predicate of sentences.Footnote ³De re modality is either de dicto modality in disguise, or it is hopelessly confused.Footnote ⁴

In his widely influential 1969 paper ‘Quantifying In’ David Kaplan offered a Frege-inspired solution to Quine’s objection. His crucial innovation was to claim that variables within the scope of modal operators range over expressions for objects. Kaplan initially analyses (1) as:

where the variable α ranges over expressions and Δ(α, x) means that α denotes x and ^┌α is greater than five^┐ denotes the result of writing the expression asigned to α followed by the string ‘is greater than five’. The operator ‘N’ means that the expression following it is a necessarily true sentence. Note that the variable α, because it takes expressions as values, can occur outside and inside the modal context, and is referentially transparent. Intuitively, (7) says that there is an expression α that denotes nine such that ^┌α is greater than five^┐ is a necessarily true sentence. But if we give the same analysis to (2), we get:

which is true because ‘nine’ denotes the number of planets and ‘nine is greater than five’ is necessarily true. So to avoid Quine’s original problem Kaplan needs to restrict the range of admissible expressions to those that necessarily denote the objects they do, i.e. (7) becomes:

where Δ_N(α, x) means that α necessarily denotes x. (9) is now the ‘official’ paraphrase of (1). But notice that, while ‘nine’ appears to occur within the scope of the modal operator in (1), its paraphrase as (9) reveals this to be an artefact of the surface grammar; in (9), ‘nine’ occurs outside the scope of the modal operator. But the referential opacity of ‘nine’ in (1) remains: if we replace ‘nine’ with ‘the number of the planets’ in (9) we get:

which is false. So Kaplan shows us how to render consistent the referential opacity of ‘nine’ as it occurs in (1) with the fact that (1) is a genuinely de re modal sentence, contra Quine.

Kaplan also shows us how to distinguish (1) from (11):

Whereas the modal operator takes ‘wide’ scope in (1), it takes ‘narrow’ scope in (11). This is reflected in their underlying logical form as, respectively:

where λ stands for predicate abstraction.Footnote ⁵ (13) says that nine satisfies the open sentence obtained by replacing ‘nine’ with the variable x in (12); intuitively, it says of the nine that it is necessarily greater than five. This is why ‘nine’ as it appears in (11) (and its paraphrase, (13)) is referentially transparent: if we substitute ‘the number of planets’ for ‘nine’ in (11) we get something whose paraphrase is:

Since the number of planets is nine, (14) is equivalent in truth-value to (13) and (11). The principle of substitution holds for ‘nine’ in (11) because ‘nine’ occurs outside the modal operator; in (1) it occurs within it. So now we have a fully referentially transparent de re modal idiom, as exemplified by (11).

2. A Kantian Solution to Quine’s Objection

Greenberg begins the book with a short argument for a familiar ‘dual aspect’ interpretation of Kant’s transcendental idealism. On Greenberg’s reading, the very same objects that causally affect our minds are intuited in space and time. We thus have one domain of objects that play at least two distinct roles: the ‘noumenal role’ (things in themselves, which causally affect our senses, are not spatiotemporal in their own nature, and would exist even if no minds were affected by them) and the ‘phenomenal role’ (appearances, which are the objects of the resulting spatiotemporal intuitions).Footnote ⁶ ‘Appearances’ and ‘things in themselves’ refer to the same set of objects, but in virtue of playing two different roles. So we can say of one and the same object that it is an appearance (in virtue of being the object of a spatiotemporal intuition) and that it is a thing in itself (in virtue of being capable of existing whether or not it causally affects a spatiotemporally intuiting mind). This is a simplification of Greenberg’s complex reading of the Kantian notion of appearances and things in themselves (as well as his own technical notion of an ‘initiator’) but it will suffice to understand his analysis of Kant’s modal theory.Footnote ⁷

This resolute ‘one object’ reading of Kant’s idealism gives rise to a modal problem, and it is to Greenberg’s credit that he notices it, although he never states it as clearly as he might. Of the very same set of objects we seem able to make two incompatible modal judgements:

The grounds for attributing (I) to Kant are his repeated remarks that, for all we know, it might be possible to have a different form of spatial intuition or even a non-spatial form of intuition; if we had a non-spatial form of intuition, these objects would still exist but would not be intuited in space. Strictly speaking, this only supports attributing (I) to Kant if we preface it with ‘for all we know’ but I will ignore this complication. Greenberg wants to interpret (I) and (A) as de re claims:

Although I agree that Kant makes de re modal claims like (A*), Greenberg never really argues for this. He points to Kant’s claim at A38/B55 that space and time ‘necessarily … belong to the appearances of this object’ but this is a thin basis on which to attribute a view about de re modality to Kant. Greenberg’s thought, I take it, is that if (A*) is not true then in knowing the necessary Euclidean properties of outer objects I really only know the de dicto necessity that all objects of my outer intuition are Euclidean and spatial, but I do not know any necessary properties of any given object. So I may know that necessarily any object of outer intuition is spatial and obeys the axioms of Euclidean geometry but, for all I know, any such object is only contingently Euclidean. That very object could have been non-Euclidean or even non-spatial, for all I know.

Perhaps further evidence that Kant attributes de re modal properties to appearances comes from various passages (not cited by Greenberg) in which he claims that if there were no cognitive subjects with our spatiotemporal form of intuition, objects in space and time would not exist, for instance:

Kant might be read as making the uninteresting de dicto claim that:

However, I think it is more natural to read this passage as making a de re claim about spatiotemporal appearances: if appropriately constituted minds did not exist to intuit them, these very objects would not exist. This might be support for Greenberg’s claim that appearances are de re necessarily spatial, while noumena are not: if cognitive subjects like us (with spatial intuition) did not exist, things in themselves (the mind-independent things that affect our minds) would presumably still exist. Going forward, I will assume that Greenberg is right and that Kant wants to hold onto (A*) and (I*).

The obvious problem is that things in themselves and appearances are the same domain of objects. Let a be an appearance. From (A*) it follows that:

Since, by hypothesis, all appearances are also things in themselves, it follows from (I*) that:

To make (I*) and (A*) compatible with the fact that things in themselves and appearances are the same domain of objects, we need to deny that modal contexts are referentially transparent. So we need to do precisely what Quine argued we could not: we need to restrict the principle of substitution within modal contexts in (I*) and (A*) while maintaining the idea that they are genuinely de re. Thus Kaplan is one natural place to turn to to understand how, on a ‘dual aspect’ reading of transcendental idealism, there can be genuine de re modality in Kant’s system.Footnote ¹⁰

Let a be a Kantian appearance. (A*) entails that:

Recall that Kaplan analysed ‘necessarily, nine is greater than five’ ((1) from earlier) as

If we just apply this flat-footedly to Kant’s (3), we get:

Clearly, α cannot range over linguistic expressions, for whatever Kant’s theory of necessity is, it is not linguistic. So let α range over mental representations instead. This raises the question of how to understand the corner quotes in a Kantian context. On this point, Greenberg is notably unhelpful, writing: ‘let the quotes be [corner] quotes and let the representations inside the quotes, except for the variable, denote themselves’ (119–120). On the usual way of understanding corner quotes (or ‘quasi-quotation’) ^┌α is intuited in space^┐ denotes the result of concatenating the expression designated by ‘α’ followed by ‘is intuited in space’. Inside the corner quotes, standardly conceived, are not ‘representations’, but a string of symbols. As far as I can tell, Greenberg seems to have in mind the idea that ^┌α is intuited in space^┐ designates the judgement that consists in combining the representation α, the subject of the judgement, with the representation <is intuited by space>, the predicate of the judgement. (I use angle brackets <. . .> to denote the representation that is the meaning of the linguistic expression.) It is crucial to point out, though, that the predicate <is intuited in space> is not being predicated of α, but of the object (or objects, if α is a concept), designated by α. In a Kantian judgement <A is B>, the predicate concept <B> is not predicated of the concept <A> (i.e. the judgement does not say that the concept <A> is within the extension of <B>); the predicate is applied to the objects represented by <A>. Objects are represented in Kantian judgements by concepts, but Kantian judgements are about objects. This will be important in what follows.

To return to Greenberg’s analysis, (5) says that there is a representation of a that necessarily denotes it and that necessarily represents it spatiotemporally. But this allows us to turn questions about the de re modal properties of objects into questions about the modal properties of representations of those objects. (5) will be true just in case there are ways of representing a that could not but represent a as spatial. So we need to restrict the condition on α to include only representations that are spatiotemporal intuitions:

where Δ_{S&T, N}(α, a) means that Δ is a spatiotemporal intuition (S&T) that necessarily (N) denotes a. So (6) says, intuitively, that there is a spatiotemporal intuition of a such that the judgement obtained by concatenating that intuition with the predicate <is intuited in space> is a necessarily true judgement.Footnote ¹¹

Now we can see how (1) is compatible with (2), in other words, how one and the same object can have different de re modal properties depending on whether it is designated as a thing in itself or as an appearance. For applying the same strategy to (2) we get:

We need only to determine which conditions to impose on the representations that designate a. Although Greenberg’s view on this is somewhat unclear, I take it that the concept of a thing in itself is the concept of a mind-independent thing that causally affects our mind. In other words, the concept of a thing in itself is the concept of an object of a discursive intellect in general, a purely intellectual representation. Our representations of objects as things in themselves are conceptual, not intuitive. So the representations that designate an object as a thing in itself are conceptual and hence not intuitive (and thus not spatiotemporal). So (7) should be refined to

where Δ_{C&D, N}(α, a) means that Δ is a conceptual (C) representation of an object of a discursive (D) intellect in general that necessarily (N) denotes a. But this shows that (8) is false: the concept of an object of a discursive intellect in general does not necessarily represent it as spatial. However, if we want to account for the truth-value of

where ‘t’ is the name of a thing in itself, we would analyse it as

which is true, because a conceptual representation of an object as a thing in itself represents it (by definition) as the object of knowledge for a discursive intellect, hence as an object that causally affects the subject’s mind.Footnote ¹²

However, a problem remains, namely, that the object-names that appear in (7) and (10) do so in referentially opaque contexts. The easiest way to see this is that the different conditions on the admissible representations in (7) and (10) depend upon the means by which the object-names were introduced. Since t was introduced as a name for a thing in itself, the representations are limited by the condition to conceptual representations of an object of a discursive intellect; because a was introduced as the name of an appearance, the representations are limited to spatiotemporal intuitions. Aside from the mere problem of opacity, this means that we cannot quantify into those positions, which means that we do not get an analysis of (I*) and (A*) on this account. (I*) and (A*) are de re modal claims whose variables are bound by a universal quantifier

Greenberg’s analysis so far gives us no account of their truth-conditions because we do not know how to evaluate open sentences like ‘N(x is spatial)’. The truth of this sentence relative to assignment of an object to the variable x will depend upon how that object is picked out.

The solution to this problem in Greenberg’s Kantian modal semantics is to iterate Kaplan’s strategy one more time. We need to understand the antecedent in I* not as restricting the object-quantifier to things in themselves but restricting admissible representations of objects to ‘thing in itself’-representations of them. In other words, (I*) should really be saying that, for every object, there is a ‘thing in itself’-representation of that object (a way of representing it as a thing in itself) that does not necessarily represent the object as being spatial. Consequently, the true logical form of (I*) is:

Likewise, (A*) will be analysed as:

In other words (A*) says that for every object there is a spatiotemporal intuitive representation of that object that necessarily represents it as being in space. The same basic move is retained: we move from talking about the de re modal properties of objects to talking about the de re modal properties of representations of those objects.

3. Contingent Necessities and Presupposition Failure

The second main objection to Kant’s modal theory that Greenberg discusses is the problem originally raised by Russell and Moore, but later sharpened by James Van Cleve.Footnote ¹³ I present here Van Cleve’s version of the problem.

Let p be any proposition of Euclidean geometry and let ‘Fp’ mean that p is made true by the forms of experience. Let → stand for the material conditional and ⇒ for the ‘strict’ or necessitated conditional.Footnote ¹⁴

The idea behind (2) is that if it is necessary that p, then p is made true by the forms of experience. If p is made true by the forms of experience, then necessarily, if p is true, it is made true by the forms of experience; necessarily, if we have different forms of experience, p would not be true. (3) states that it is contingent that we have the forms of experience we actually do. The proof shows that this is incompatible with there being any such necessary truths. The moral that Van Cleve draws is that necessities cannot be grounded in contingent facts; if the contingent facts were different, the necessary truths would be false, contra hypothesis.

One might doubt that Kant holds the Contingency premise. However, Van Cleve points to texts in which Kant appears to assert it, for example:

Again, strictly speaking, this only supports the Contingency premise with the qualification: as far as we know. As for the Dependency premise, it seems doubtful that it is true for all truths of Euclidean geometry, much less for all Kantian necessary truths. For Kant might hold the Contingency premise while denying that every truth of Euclidean geometry could be falsified if we had different forms of experience (as (2) requires). There might be very general truths (e.g. that there is more than one possible object) that would be true no matter what form of intuition we had. But this does not affect the real point of the argument. The argument shows that for any truth of Euclidean geometry for which (2) holds, this truth is not necessary, and this will presumably be the case for many fundamental axioms of Euclidean geometry, for example, the parallel postulate. So Van Cleve’s argument poses a deep problem for Kant’s modal theory.

Greenberg’s strategy for solving this problem involves distinguishing the logical relation of material implication from that of presupposition. On the standard definition, p materially implies q just in case, if q is false, then p is false. By contrast, p presupposes q just in case, if q is false, p lacks a truth-value. Greenberg traces this distinction back to Frege, Russell and Strawson and the debate about empty names. According to Frege and Strawson, if a denoting expression is empty (denotes no object) then sentences involving that expression lack a truth-value. According to one common misreading of Russell,Footnote ¹⁵ he disagrees; because he analyses names as definite descriptions, a sentence involving a non-referring name is simply false. ‘The present King of France’ is false, according to Russell, but lacks a truth-value according to Strawson and Frege. So, on the Strawson/Frege view, ‘the present King of France is bald’ presupposes but does not materially imply that there is a present king of France (that ‘present King of France’ refers).

Greenberg adapts the Frege/Strawson view about names to the relation between Kantian appearances and the forms of experience. If we had different forms of experience, our actual judgements about appearances would lack a truth-value, rather than be false. So in Van Cleve’s original argument he wants to replace (2) with:

where ‘∝’ means the relation of presupposition. So Kantian synthetic a priori necessities have a special modal status: necessarily, they are true if they have a truth-value at all.

The problem is that Van Cleve’s original problem, Greenberg’s presentation of it and his presentation of his solution are all formulated in terms of de dicto necessity and Greenberg’s proposal is highly implausible in the de dicto case. Take a particular Kantian de dicto necessity like ‘necessarily, the internal angles of every triangle sum to 180 degrees’. On Greenberg’s proposal, the truth of ‘the internal angles of every triangle sum to 180 degrees’ presupposes that we have a Euclidean form of spatial intuition, so that if we did not, the proposition expressed by the sentence ‘the internal angles of every triangle sum to 180 degrees’ would be devoid of a truth-value. This is implausible. For if we had a non-Euclidean form of spatial intuition, we might have had spatial intuition whose form is given by Riemannian geometry. Greenberg’s view entails that if we had a Riemannian form of spatial intuition then the proposition expressed by the sentence ‘the internal angles of triangles sum to 180 degrees’ would be truth-valueless, but clearly it would be simply false. The only ground I can see for adopting Greenberg’s analysis is if we thought that mathematical concepts like ‘triangle’ are essentially concepts of mathematical objects of our actual form of intuition (making them so-called ‘rigid designators’). On this view, although there are three-sided objects in Riemannian geometry that closely resemble triangles, they are not triangles because ‘triangle’ refers only to Euclidean triangles (perhaps because it was first tokened by beings, us, with a Euclidean form of spatial intuition). This does not strike me as plausible.

However, Greenberg’s solution in terms of the presupposition relation is much more plausible in the case he was originally interested in: de re necessities. Consider Van Cleve’s argument, but where p is some de re claim about a particular Kantian appearance a, say, ‘a is Euclidean’:

The very same argument purports to show that it is not de re necessary that a is Euclidean. Since we know from the first half of the book that Greenberg thinks that a is de re is necessarily Euclidean, this is a consequence he should want to avert. And his presupposition move allows him to do just that. For consider (2) as reinterpreted in terms of presupposition:

The consequent of this conditional says that if we had different forms of experience, then the proposition that a is Euclidean would lack a truth-value. Interpreting (2) as (2*) suffices to block Van Cleve’s argument; the argument is no longer valid with (2*).

The question then becomes whether Greenberg has good reasons to read (2) as (2*) and in fact the theory of de re modality given in the first half of his book provides him with excellent grounds. For on that theory, the antecedent of the conditional in (2*) means that:

This means that there is a spatiotemporal intuition α of a that necessarily designates a such that ‘α is Euclidean’ is necessarily true. But very plausibly this is true only if ‘a’ is itself a spatiotemporal way of designating the object a – otherwise, why would it be that there is a representation that necessarily co-designates a and satisfies the rest of (7)? But if ‘a’ is itself a spatiotemporal way of designating an object, and thus tied to our actual forms of intuition, if we had different forms of intuition, ‘a’ would cease to be a meaningful expression. This delivers Greenberg a general way of resolving Van Cleve’s problem, but only in the case of de re necessities. De re necessary truths about appearances would not be false if we had different forms of experience, because our actual forms of experience are built into the semantic devices we use to refer to appearances. If we had different forms of experience, those semantic devices would cease to be significant; they would not refer to objects with different modal properties (e.g. objects that are necessarily Riemannian).

4. Concluding Remarks

I would like to conclude with some remarks about the project and the methodology of this book. While I think the general project of the book – to defend Kant’s theory of necessity from Quine’s and Van Cleve’s objections – is eminently worthwhile, I am not sure that Greenberg has gone about it in the best way. At various points in the book, he seems to have jumped too quickly to putting Kant in dialogue with contemporary philosophy, without first trying to get a solid grasp on what Kant’s own view is. First of all, he simply assumes at the outset that Kant shares Quine’s rejection of Aristotelian essentialism. As Quine himself points out, his objections to de re modality would be moot if there were a metaphysically ‘preferred’ way of denoting an object, say, by describing it in terms of its essential properties. But there is significant evidence that Kant believes in ‘real essences’ and thus would not find the Quinean problem about de re modality as troubling as Greenberg does.Footnote ¹⁶ Secondly, there is the problem that Greenberg never substantiates one of the basic assumptions of the book: that Kant even has a theory of de re modality, or accepts such a notion in the first place. While I agree that he does, and I have attempted in this essay to substantiate that somewhat, Greenberg could have done more in this regard. Thirdly, Greenberg has a tendency to translate some technical piece of analytic philosophy into Kantian terms without considering how appropriate it is within Kant’s theory. The clearest example of this is Kaplan’s linguistic device of corner quotes, which Greenberg transfers into Kant’s theory without explaining how to understand ‘corner’ quotes functioning on mental representations. Thus, while I am entirely in solidarity with the spirit of this quite interesting book, I have reservations about some of the details.Footnote ¹⁷