Bob Hale. Essence and Existence: Selected Essays

Abstract

Essence and Existence: Selected Essays brings together fifteen essays by Bob Hale, mostly written between the publication of his last book, Necessary Beings, in 2013 and his death in 2017 from cancer, which he had been battling for some time. Six of the essays are published here for the first time, several of the remaining ones are not easily accessible, and there is a complete bibliography of Hale’s writings. The collection is carefully edited by Hale’s former student Jessica Leech, who has contributed some useful editorial notes detailing the editorial changes made to the manuscripts that Hale did not have time to complete entirely. The book contains an eight-page introduction by Kit Fine, which combines some touching personal reminiscences with a very useful critical discussion of two of the included essays, which are concerned with truthmaker semantics for universal generalizations and for modal statements. As will be no surprise to readers familiar with Hale’s work, the essays cover a wide range of topics: from truthmaker semantics, through modality and ontology, to logic and the philosophy of mathematics. My discussion will focus on the essays that deal with, or bear on, the philosophy of mathematics. Some of these essays — eight of the total fifteen — are useful further developments of themes that will be familiar to readers of Hale, while others introduce entirely new themes. A clear example of the latter is the new essay ‘What makes true universal statements true?’, which also benefits from a critical analysis in Fine’s introduction. Nearly all extant truthmaker semantics take a universal generalization to be made true by a state composed of states that verify each of the instances of the generalization. Hale concedes that this standard account seems correct of some generalizations, e.g., that all of his children live in England. But he insists that the standard account is incorrect of other generalizations, e.g., that every natural number has a successor. Perhaps most importantly, the standard account fails to do justice to an ‘obvious uniformity in the grounds for the truth of the instances’ (p. 107). It is far more plausible, Hale contends, that the generalization is made true by the essence or nature of the property of being a natural number, namely, that it lies in the nature of this property that every instance has a successor. On Hale’s view, then, universal generalizations can have two entirely different kinds of truthmakers. In addition to the instantial truthmakers invoked by the standard account, there are generic ones, where ‘a single state which has no proper parts’ (p. 114) verifies a universal generalization, along with each of its instances. These philosophical ideas are potentially very fertile in the philosophy of mathematics. In particular, generic truthmakers can be seen as a more abstract and general analogue of the intuitionistic conception of universal generality, thus making a version of this conception available even to philosophers who reject the more problematic aspects of intuitionism. To spell things out technically, Hale sketches a truthmaker semantics that permits a single non-composite state to verify a universal generalization directly, and where such generic verifiers are allowed to coexist with the usual composite verifiers invoked by the standard account. While this is a good start, the semantics has some undesirable features, as Fine points out. In particular, the semantics is unabashedly non-compositional, because the verifiers and falsifiers of a universal generalization need have no connection with those of its subformulas. Moreover, Hale’s motivating ideas are compatible with many different choices when setting up a truthmaker semantics. One must therefore be extremely wary of drawing strong consequences from the particular semantics that Hale sketches. In particular, it seems to me a wide open question whether the motivating ideas are best explicated by means of a semantics that validates classical logic (as Hale believes), as opposed to intuitionistic (as argued in unpublished work of mine that Hale criticizes in an appendix but regrettably conflates with a published article where the ideas are only briefly summarized). Another essay that breaks new ground is ‘Essence and definition by abstraction’ (2018), which is a valuable addition to the neo-Fregean program. Hume’s Principle, we recall, is the ‘abstraction principle’ stating that the number of |$F$|s is identical with the number of |$G$|s just in case the |$F$|s and the |$G$|s can be one-to-one correlated. This principle is often regarded as some form of definition of the number-of operator. What kind of definition might this be? As Hale observes, it would have to be an implicit definition, not an explicit one; for the principle affords no way to eliminate the definiendum. He also discusses where the definition might be located on the spectrum from stipulative to interpretative definitions — answering, in the spirit of Carnapian explications, ‘somewhere in the middle’. The heart of the essay, however, concerns a third axis of comparison, namely between what Locke called nomininal and real definitions. As Hale asks (p. 11), ‘what is the relation between Hume’s principle, taken as an implicit definition of the number operator […] and the essence or nature of the corresponding entities — the function from concepts or properties to objects, and those objects themselves, i.e., the cardinal numbers in general, and the natural numbers?’ The answer, Hale contends, is that abstraction involves a ‘coincidence of nominal with real essence’ (p. 15). If correct, this ‘coincidence’ promises a route to a priori knowledge of real definitions, namely that, in the relevant cases, ‘we fix essence by fixing meaning’ (p. 18). Why, though, should we believe in this happy match between nominal and real essence? Hale’s answer has two parts. First, he warns against a ‘false assimilation’ of abstraction principles ‘to cases in which we seek to capture the essence of natural kinds. Cardinal and natural numbers are not natural kinds — at least not in the same sense in which gold and tigers are. They are, as one might say, artificial kinds’ (p. 19). Second, Hale supports his view of definitions by appealing to the attractive neo-Fregean idea that numbers are ‘metaphysically lightweight’. The idea is that the existence of numbers ‘depends only upon there being properties which are one-one correlated’ (p. 19); and, as we shall see shortly, the needed properties too are taken to have a lightweight character. Clearly, this yields an ‘abundant conception’ of properties and abstracta (p. 20). Hale can now explain ‘how we achieve such a happy match between our definition and the essence’, namely because ‘given the abundant conception, we cannot very well miss the target’ (p. 21; italics original). Although the idea of metaphysically lightweight existence will no doubt strike many readers as elusive, Hale’s discussion thus reveals the potential for some entirely new theoretical benefits of the idea. ‘The problem of mathematical objects’ (2011) discusses a central question in the philosophy of mathematics, namely how we can know that there are infinitely many natural numbers. According to what Hale calls property-based approaches, ‘an infinite sequence of objects is seen as dependent on an underlying infinity of properties’. Unsurprisingly, he is particularly enamoured of the neo-Fregean version of this approach, where the existence of zero is derived from a property (such as non-self-identity) that necessarily has no instances, one is derived from the property of being identical with zero, and so on, following the famous Fregean bootstrapping argument. By contrast, object-based approaches argue directly that we have some form of access to at least a potential infinity of numbers (p. 213). Hale focuses on Charles Parsons’s [1980] account of mathematical intuition, which seeks to tie this form of intuition to ordinary perception and imagination. Parsons argues our ability to imagine an arbitrary sequence of strokes being extended gives us intuitive knowledge that every number has a successor, which in turn answers our central question. Hale is unconvinced. He takes Parsons’s account to require that we be able to imagine any given finite sequence of strokes. But this assumption seems questionable. Why can our imagination not enable us to grasp the operation of adding one stroke independently of our ability to imagine every argument of the operation, just as our perception enables us to grasp, say, the property of being red independently of our ability to perceive every instance of this property? To borrow terminology from Hale’s own discussion of universal generality, it seems more promising to understand the operation of adding one stroke in a generic, rather than an instantial, manner. Moreover, even if Hale’s criticism were upheld, his distinction between object-based and property-based approaches seems a red herring. For notice that the Fregean bootstrapping argument can be adapted to use pluralities of objects instead of properties. The only relevant difference between the two versions of the argument concerns our ability to obtain zero by abstraction. While there clearly are uninstantiated properties, the idea of an empty plurality is controversial. But even without an empty plurality, this wrinkle can be ironed out, say, by regarding the number-of operator as a ‘co-partial function’ (in the sense of [Oliver and Smiley, 2016, p. 88], which can have the value zero on an undefined argument. Thus, when classifying answers to the central question of how we can know that there are infinitely many numbers, what primarily matters is presumably whether an account relies on some form of Fregean bootstrapping, not whether we bootstrap on properties or (pluralities) of objects. In ‘Ordinals by abstraction’, published here for the first time, Hale discusses the thorny problem of ordinal abstraction — i.e., abstraction on well-ordered relations under the equivalence of isomorphism — whose naïve version falls prey to a version of the Burali-Forti paradox. First, he critically discusses a recent attempt by Ian Rumfitt [2018], based on restricting the second-order logic to so-called |$\Delta^1_1$|-comprehension. This solution would, however, have devastating consequences for other parts of the neo-Fregean program. Then, Hale attempts to develop a better solution, which is also more hospitable to the neo-Fregean agenda. In essence, his proposal is to restrict the abstraction to well-orderings that are in some sense ‘definite’. An interesting exploration of the desired notion of definiteness ensues, which opens promising avenues for further research, but stops short of presenting a worked-out theory of ordinals. A central theme of the collection to which three of its fifteen essays are devoted, is a broadly Fregean approach to properties. As I am the co-author of one of these essays, the previously unpublished ‘Ontological categories and the problem of expressibility’, I will only mention that its topic is how Fregeans can properly express their view that objects are denoted by singular terms, first-level properties by first-level predicates, and so on, without violating their own type-theoretic restrictions; and that the answer takes the form of a theory of nominalization, which allows higher-level properties to be denoted also by singular terms, albeit in a derivative manner. Thus, the essay builds on and extends the work of Hale and Wright [2012]. The other two essays in this category, ‘Second-order logic: Properties, semantics, and existential commitments’ (2015) and ‘Properties, predication, and arbitrary sets’ (previously unpublished), develop and defend the Fregean conception of properties that Hale first advocated in [Hale, 2013], Chapters 1 and 8. This conception postulates an extremely tight connection between a property and a suitable ‘predicate’ (or open formula) that expresses or characterizes the property. Indeed, Hale writes that ‘a necessary as well as sufficient condition [for the existence of a property] would require only that there could be a suitable predicate’ (p. 62). The sufficient condition makes property existence very undemanding: ‘the existence of a suitable predicate is sufficient for that of a property’ (p. 191). The result is an abundant conception of properties, not a scarce one where properties are required, say, to figure in scientific laws. As Hale observes, however, the necessary condition — that there could be a suitable predicate — is ‘far from toothless’ (p. 63). For ‘[b]y a predicate here, we mean an expression of finite length — an expression which we could, at least in principle, understand and use to speak of the property’ (p. 192). Hale’s conception of properties thus involves a form of definabilism. This is a highly distinctive conception, wedged in between two more familiar and well-developed ones. On the one hand, Hale’s conception is far more restrictive than the truly abundant set-theoretic conception, which takes properties to be — or at least to be adequately represented by — arbitrary sets of objects from the first-order domain. On the other hand, Hale goes to great lengths to emphasize two respects in which his form of definabilism is more liberal than traditional forms thereof, such as that of Weyl [1918]. First, there is no fixed language in which the ‘suitable predicates’ must be available. In particular, to insist on definability in the relevant object language would be ‘needlessly crippling’ (p. 194). Second, the predicates in question are allowed to be impredicative (i.e., to involve bound second-order variables). Has Hale succeeded in articulating a stable middle ground between the yet more liberal, set-theoretic conception and the more restrictive, predicative form of definabilism? Like any defender of an intermediate view, Hale faces a battle on two sides. On one flank, he will be attacked by defenders of the set-theoretic conception. A powerful such attack can be found in [Shapiro, 2018]. As Shapiro emphasizes, contemporary mathematics freely uses the axiom of choice and would be completely crippled without at least a weak form of this axiom. But definabilists are not entitled to this axiom, for the simple reason that the ‘choice set’ asserted to exist need not be definable. Hale’s dismissive response in ‘Properties, predicates, and arbitray sets’ is unlikely to satisfy any reader who is unwilling to sacrifice vast tracts of actual mathematics for purely philosophical reasons. On the opposite flank, Hale will be attacked by traditional definabilists, who will challenge both of his desired liberalizations. Let us begin with Hale’s attempt to go beyond definability in any particular language. ‘[T]he sense of definable in which the only properties there are are definable’, he writes, ‘is what we might call an absolute sense — definable in some language, actual or possible’ (p. 195). But the idea of such an absolute notion of definability is problematic. The standard notion of a definable set (or property) relies on a Tarskian notion of satisfaction, which is defined only relative to a particular language. Presumably, the desired absolute notion of definability would require an absolute notion of satisfaction, which is applicable not only to all actual languages but also any possible one. However, there is no reason to believe that such a notion exists. Here the burden of proof falls entirely on Hale, who seeks to go beyond standard definabilism, but fails to explain how the desired extension is supposed to work.1 Next, let |$\textrm{Def}(\mathbb{N})$| be the set of all first-order definable subsets of |$\mathbb{N}$|⁠. Let us now consider the second-order model obtained from |$\mathcal{N}$| by letting |$\textrm{Def}(\mathbb{N})$| be the second-order domain. It is easy to verify that this is a model of second-order arithmetic with only predicative comprehension.2 The important thing to notice is that this model is obtained through a sequence of perfectly good mathematical definitions, without any reliance on the powerset operation. With this review of definability in place, let us now follow Hale and try to define the second-order domain as the set |$D$| of all and only the subsets of the natural numbers that are definable by means of any second-order formula, including impredicative ones. The circularity of this attempted definition is now apparent. Any talk about definability by means of a second-order formula presupposes a second-order domain, which is precisely what we are attempting to define. Hale has, as it were, given us an equation with an unknown |$D$|⁠. If this equation has a unique solution, we would at least have an implicit definition of the second-order domain. But does Hale’s equation have a unique solution, rather than none or more than one? Sam Roberts and Kameryn Williams have each proved that the answer turns out to be ‘more than one’.3 This is terrible news for Hale: not only is his attempted definition of the second-order domain circular, but it remains unsuccessful even when viewed as a implicit definition. I conclude that we lack any kind of worked-out model of Hale’s attempted conception of properties, leaving it doubtful whether there is a coherent conception here at all. To sum up, the book is full of interesting and often promising ideas. Some of these ideas need to be more fully developed, however, and at least one central idea, I have argued, is unlikely to admit of any such development. It is a shame that Hale is no longer with us to continue the discussion, which he would no doubt have done with insight as well as passion.