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.
Reviewed by: Therapeutic Revolutions: Pharmaceuticals and Social Change in the Twentieth Century ed. by Jeremy A. Greene, Flurin Condrau, Elizabeth Siegel Watkins Dominique Tobbell Jeremy A. Greene, Flurin Condrau, and Elizabeth Siegel Watkins, eds. Therapeutic Revolutions: Pharmaceuticals and Social Change in the Twentieth Century. Chicago: University of Chicago Press, 2016. 320 pp. $110.00 (978-0-226-39073-4). “The practice of medicine in 1980 looked quite a bit different than it had in 1930,” write Jeremy Greene, Elizabeth Watkins, and Flurin Condrau, in the introduction to Therapeutic Revolutions (p. 8). A major component of the transformation of medicine in these decades was the massive expansion of pharmaceutical research and development that led to the introduction of scores of new prescription drugs, including antibiotics, corticosteroids, minor and major tranquilizers, oral contraceptives, and the first drugs to treat hypertension. This period of transformation has been termed by physicians, the pharmaceutical industry, policy makers, patients, and some medical historians alike as a therapeutic revolution. But as [End Page 224] Charles Rosenberg noted forty years ago in his seminal essay, “The Therapeutic Revolution,” claims of therapeutic efficacy—and thus the basis of therapeutic revolution—are both historically contingent and locally specific.1 This impressive collection of essays by leading scholars in the history and anthropology of pharmaceuticals builds on Rosenberg’s earlier analysis to problematize the concept of a twentieth-century pharmacotherapeutic revolution. By examining the intersections of pharmaceutical innovation and social transformation in North America, Western Europe, and Africa, the essays underscore the historical contingency and geographic specificity of any such revolution, and reveal what was at stake politically, economically, and culturally for the various historical actors that have mobilized the language of therapeutic revolution. The volume can be broken into four sections. The first section, with chapters by Scott Podolsky and Anne Kveim Lie, Elizabeth Watkins, and Nicolas Henckes, considers three classes of drugs held as emblematic of the mid-twentieth-century therapeutic revolution—antibiotics, oral contraceptives, and antipsychotics, respectively—and examines the degree to which these drugs were revolutionary and the political, economic, and social gains made by those who mobilized the rhetoric of revolution. The second section interrogates the measures by which pharmaceuticals introduced in the mid-twentieth century were considered revolutionary: pharmaceutical consumption and statistical measurements of therapeutic efficacy. Using pharmaceutical sales data from IMS Health, Nils Kessel and Christian Bonah show that it was older drug products like analgesics, hypnotics, and sedatives, rather than “revolutionary medicines” like antibiotics and cardiovascular drugs, that dominated the West German pharmaceutical market in the 1960s and 1970s. In their history and ethnography of tuberculosis treatment, Janina Kehr and Flurin Condrau show the ways in which clinical trials data documenting the efficacy of streptomycin to treat tuberculosis in the 1950s was mobilized as “proof of the universal effectiveness of antibiotics regardless of social situation,” even as antibiotic resistance undermined treatment efforts and tuberculosis persisted as a global health concern (p. 135). This “statistical modeling of success” (p. 135), Kehr and Condrau argue, “transformed tuberculosis from a subject of cutting-edge biomedical research into a disease that physicians considered ‘boring’ . . . until it reemerged in the late twentieth century as multidrug resistant tuberculosis and extremely drug-resistant tuberculosis, and thus a subject of renewed biomedical excitement” (p. 12). The third section shifts focus from the United States and Western Europe to the Global South and examines the intersections of pharmaceutical innovation, international development, and global public health, highlighting the spatial-temporal character and political economy of the therapeutic revolution. Jeremy Greene analyzes the ways in which American lawmakers, physicians, patients, regulators, and pharmaceutical companies mobilized arguments about the [End Page 225] cause of and potential solutions to geographic disparities in pharmaceutical access during the 1960s and 1970s in an attempt to shape both domestic health policy and international development agendas. Paul Farmer, Matthew Basilico, and Luke Messac revisit the debate, ignited by Thomas McKeown in the 1960s, over the role of medicine in public health. Their analysis of recent economic, ethnographic, and epidemiological data makes clear that despite the persistent problem of uneven access, pharmaceuticals have contributed to a significant population-level mortality decline in the Global South. Julie Livingston’s history and ethnography...
Quine (1966) offered his classic characterization of the notion of paradox, a taxonomy for paradoxical arguments and some vocabulary for discussing them. In this article, I shall generalize Quine’s taxonomy and defend a simpler characterization. The simpler characterization will have the virtue or the flaw (as might be) of making paradox a matter of degree. For Quine, a paradox is an apparently successful argument having as its conclusion a statement or proposition that seems obviously false or absurd. That conclusion he calls ‘the proposition of’ the paradox in question. What is paradoxical is of course that, if the argument is indeed successful as it seems to be, its conclusion must be true. On this view, to resolve the paradox is (i) to show either that (and why) despite appearances the conclusion is true after all, or that the argument is fallacious, and (ii) if the former, to explain away the deceptive appearances. Quine divides paradoxes into three groups. A ‘veridical’ paradox is one whose ‘proposition’ or conclusion is in fact true despite its air of absurdity. We decide that a paradox is veridical when we look carefully at the argument and it convinces us, i.e. it manages to show us how it is that the conclusion is true after all and appearances to the contrary were misleading. Quine’s two main examples of this are the puzzle of Frederic in The Pirates of Penzance (who has reached the age of 21 after passing only five birthdays) and the Barber Paradox, which Quine considers simply a sound proof that there can be no such barber as is described.1 A ‘falsidical’ paradox is one whose ‘proposition’ or conclusion is indeed obviously false or self-contradictory, but which contains a fallacy that is detectably responsible for delivering the absurd conclusion. We decide that a paradox is falsidical when we look carefully at the argument and spot the fallacy. Quine’s leading example here is De Morgan’s trick argument for the proposition that 2 = 1.2 Oddly, Quine does not mention a third related category, the obverse of a veridical paradox: the argument in question could have an obviously false or self-contradictory conclusion, yet rest on no error of reasoning however subtle – so long as it has a premiss that looks for all the world true until we let the argument itself show us that, and how, the premiss is false after all. For example, the Barber is classified as veridical because its conclusion is the truth that there is no barber who shaves all and only those who do not shave themselves; but turn it on its head, so that its conclusion is rather the absurdity that there is a barber who both does and does not shave her-/himself. There is still no fallacy, but only the innocent-seeming premiss that there is a barber who shaves all and only those who do not shave themselves.3 We might call this sort of paradox, for want of better, ‘premiss-flawed.’ Finally (returning to Quine), an ‘antinomy’ is an intractable paradox, one that we cannot see how to resolve in either of the foregoing two ways: the argument does not succeed in convincing us that its conclusion is true-despite-appearances (often because the conclusion is overtly contradictory or otherwise incoherent, yet we can find no fallacious move in the argument; nor is there a premiss that is shown false-despite-appearances. Antinomies, Quine says, ‘bring on the crises in thought’ (5); they show the need of drastic revision in our customary ways of looking at things. My main problem with Quine’s taxonomy is its heavy dependence on the current state of one’s knowledge and on one’s ability to figure things out. Let me explain. All a valid deductive argument shows, just in virtue of its validity, is that a certain set of propositions is internally inconsistent. If all we know about an argument is that it is valid, we do not thereby have even the slightest reason to believe that the conclusion C is true, even though doubtless the person who constructed the argument was reasoning, from P1, P2, … taken as premisses, to C. For unless we are independently moved to accept those premisses, their jointly implying C is of little interest; and even if we do already accept them, seeing that they imply C may make us reconsider one or more of them, rather than inclining us any the more strongly to endorse C as well.4 In this sense, an argument has no intrinsic direction; its direction has been imparted to it rhetorically by a speaker who has recruited it for a particular dialectical purpose. Intrinsically, the argument is just the inconsistent set {P1, P2, …, ∼C}. The relevance of that (I hope uncontroversial) point for Quine is that, faced with an apparently inconsistent set of plausible statements, anyone may choose which of the statements’ denials to single out as ‘the proposition’ or ‘conclusion’ of the corresponding paradoxical argument, either arbitrarily or on a ground of some sort. (Remember that a paradox in Quine’s sense has an already appalling conclusion, so there is not initially any epistemic reason, as opposed to expository reasons, why the conclusion appears as such rather than being negated and taken as a premiss.) And given a paradoxical argument, two theorists might well disagree on whether the paradox is veridical, because they may disagree as to which component propositions are more plausible than which; in particular, one theorist may find the argument veridical while the other finds the ‘conclusion’’s denial more plausible than one of the ‘premisses’. In Quine’s examples of veridical and falsidical arguments, the comparison of plausibility is sufficiently obvious and uncontroversial that in fact no one would dispute his judgements about them. But we should bear in mind that that is because we all think roughly alike on issues of Leap Year and birthdays and barbers and arithmetic; a person who for whatever reason had different background beliefs and very different interests might resist Quine’s judgements. In short, to classify a paradox as veridical is to assume that one’s own preferred way of resolving the paradox is the correct way. More generally, then, a Paradox (I mark my own proposed usage with the capital letter) is an inconsistent set of propositions, each of which is very plausible.5 And to resolve a Paradox is to decide on some principled grounds which of the propositions to abandon. One might then think of saying that when that decision is comparatively easy, we could call the Paradox either veridical or falsidical, depending on which of the component propositions have been designated as ‘premisses’ and which has been denied by way of ‘conclusion’, though when the choice of culprit is difficult or controversial, we call the Paradox an antinomy, as Quine does. But that translation of Quine’s terminology into mine would not be accurate, for in regard to my notion of Paradox, Quine’s ‘veridical’ and ‘falsidical’ are not natural opposites. A Paradox would count as veridical in the proposed new sense just in case (a) one of its members turns out to be clearly less plausible than the others, and (b) the Paradox is already (for whatever reason) cast in the form of a deductive argument having the culprit’s denial as its conclusion. But there would be no such thing as a falsidical Paradox, for a falsidical argument in Quine’s sense is fallacious, and the apparent inconsistency corresponding to it is not real. In my terms, then, a falsidical paradox in Quine’s sense is only an apparent Paradox.6 But on my view the ‘veridical’/‘falsidical’ distinction really loses its point. Rather, there are inconsistent sets containing identifiable culprits – call those ‘tractable’ Paradoxes – and there are the antinomies. It is better just to drop Quine’s terms. There is a further complication. I have defined ‘Paradox’ in terms of actual inconsistency, namely, as a set of propositions that is in fact inconsistent. But (a) a set can be inconsistent without anyone’s knowing that it is or even being able to know that it is; and more importantly, (b) we may have reason to think that a set is inconsistent when actually it is not. Case (b), Quine’s falsidical again, is not uncommon; we have some plausible premisses and we use what seems to be a sound principle of reasoning to deduce our absurdity, but in fact the absurdity does not follow and the fault is in the principle of reasoning rather than in any of the premisses. (De Morgan divides through by a number that is covertly equal to 0; elsewhere (Lycan 1993, 2001) I have argued that when one is reasoning in English rather than in a truth-functional calculus, reliance on the rule Modus Ponens can lead from perfectly acceptable premisses to contradictory conclusions and so must be rejected.) Now, to circumvent this complication and keep the terminology neat, let us dispense with every even faintly dubious principle of reasoning. That is, let us require that for a Paradox to be worthy of its capital ‘P’, it must be formulated truth-functionally, with every initially non-truth-functional principle of reasoning replaced by that principle’s corresponding material conditional inserted as a member of the inconsistent set that constitutes the Paradox in question; thus we shall make every possibly controversial inference principle explicit. And, owing to the semantical completeness of the propositional calculus, any Paradox is provably as well as model-theoretically inconsistent.7 It must be conceded that, in at least three ways, even the propositional calculus is ‘controversial’. First, there is the question of relevant implication. Relevance logicians8 brand the truth-functional calculus as libertine, charging that some of the inferences it sanctions (notably Disjunctive Syllogism) introduce informational irrelevancies and are therefore not ones that English speakers would or should make. But however we feel about this as a complaint against the analysis of English or any other natural language in truth-functional terms, it does not apply to my idea of reconstructing paradoxes into Paradoxes; I am here using the propositional calculus and its libertine notion of validity only as a tool for strict truth-preservation in inferential moves between propositions that are already formulated in the truth-functional idiom. Disjunctive Syllogism may not be a valid principle in the logic of English (any more than are Antecedent-Strengthening and Modus Ponens), but it is valid trivially and by definition for the tilde and the vel. Second, paraconsistent systems have been offered as rivals to standard contradiction-shunning logic,9 and in particular dialetheists led by Graham Priest (1987, 1995) have maintained that there are actually true contradictions. Indeed, Priest cites some familiar paradoxes as examples; there is nothing to ‘resolve,’ because the contradictions simply are true. But this view is no opponent of my characterization of a Paradox. It merely takes a refreshing attitude toward Paradoxes once they are identified. Third, Quine, Putnam and others have suggested that even elementary logic may be brought into question by exotic scientific developments such as in quantum mechanics. Once again we must distinguish between logical laws intended as representing the logic of English and the theorems of a formal system whose truth-theoretic semantics has been officially and stipulatively assigned. My notion of Paradox is tied to the latter, and has no implications regarding the former. But if(!) I understand quantum logicians correctly, they mean to impugn even standard propositional logic understood as formulated in terms of the traditional truth-defined connectives. Depending on one’s view of analyticity,10 and once we have distinguished the matter of epistemic revisability from the metaphysical issue of truth by virtue of stipulated meaning, this may not make sense. But even if it does and we are thereby forced to admit a notion of falsidical Paradox after all, at least my format will make the point of contention as explicit as anything could. Two objections have been made to me; oddly (but fortunately) they oppose each other. The first11 is that most or at least many people think of paradoxes as arguments, hence, contrary to my conception, as being intrinsically directional, from premisses to conclusion. It may be that many people do so; and that way of thinking is well represented by Quine’s model. But I have pointed out that arguments themselves are not intrinsically directional, save by prior commitments of their proponents. To repeat, the speaker who has deployed the argument has chosen to present one or more propositions as its premisses and infer another proposition as its conclusion, but that is rhetoric; we can equally argue from the ‘conclusion’s’ negation to that of one of the ‘premisses’, and nothing about the argument itself tells us which should be preferred. The second objection is that paradigmatic paradoxes are single sentences or statements such as the Liar, hence not inconsistent sets of propositions (and, n.b., not arguments either).12 Whether or not the Liar is paradigmatic, it is a single sentence rather than an inconsistent set, and it does suggest mild readjustment of our formula. I said that a Paradox is an inconsistent set of propositions ‘each of which is very plausible,’ but on its face the Liar cannot be so described. It can be expressed as an inconsistent set, {‘(L) is true,’ ‘(L) is false,’ ‘(L) is either true or false but not both’}, but none of those three propositions has great intuitive appeal. What is true is at best that each of the propositions has a strong argument in its defence.13 Those three arguments have premisses, such as ‘What (L) says is that (L) is false’ and ‘If S says that P then S is true iff P,’ so a Paradox as I originally defined it would feature those undefended premisses, not the lemmas as above. So, either we can say that the Paradox is that more complicated set, which is not very natural (also, we would get different versions depending on exactly how the ultimate premisses were formulated), or we can liberalize the definition by replacing ‘each of which is very plausible’ by ‘each of which either is very plausible in its own right or has a seemingly conclusive argument for it’. If we stick by my notion of Paradox, we will regard the distinction between ‘tractable’ Paradoxes and antinomies as theory-infected and somewhat presumptuous, but above all a distinction of degree. There are actually two matters of degree involved: the disparity in plausibility between a putative culprit proposition and the other, more plausible propositions in the set, and the average degree of plausibility all around. The higher both of these degrees go, the more readily we will see a Paradox as intractable; we get a real antinomy when the first is near zero and the second is still high. If the second is low, we have only a mild Paradox, and I am loath to call it antinomic, but only an array of competing theories.14
In 1936, 200 of the world's top medical scientists met in Brussels to address an emerging epidemic. Since the turn of the century, cancer had been steadily growing as a major cause of death. Naturally, researchers looked first to environmental factors, particularly agents of the Industrial Age, such as asbestos, road tar, ionizing radiation, and synthetic dyes. Since then, however, attention has been diverted away from such important environmental causes of cancer, especially where economic interests are at stake, according to Dr. Devra Davis in The Secret History of the War on Cancer (1). This expansive, ambitious book spans nearly a century and dozens of controversies, anecdotes, and personal stories. Davis devotes a chapter to describing the growth of the eugenics movement and Nazi medicine. Another chapter details the evolution of the nation's leading voluntary anticancer organization, from the grassroots anticancer advocacy of the American Society for the Control of Cancer's Women's Field Army to the corporate Lasker-era American Cancer Society. She describes how professional turf battles delayed the introduction of the Papanicolaou smear for over a decade. In subsequent chapters, she warns of the possible hazards to physician-researchers in working with novel compounds, the overselling of routine mammography, the role of paid experts in environmental tort litigation, the politics of establishing regulatory standards for environmental and occupational exposures, and the potential as-yet-unproven hazards of cell phones. A key theme running throughout the book is how financial interests, professional allegiances, and political ideology can manipulate the scientific process. Archives and court documents reveal how well-meaning scientists have at times become complicit in defending industrial interests at the expense of public health. There is no “secret history” here, however, as these histories are largely drawn from the work of other scholars. One exception is Davis's own study of the papers of Robert Kehoe, a key figure in the development of the occupational health field who conducted a wide range of toxicology research under contract for various industries. These documents provide a unique case study of how one academic laboratory functioned and interacted with its industry sponsors, and one wishes they were more fully explored here. Unsurprisingly, tobacco figures big in Davis's story. She touts German physician Franz H. Müller's 1939 case-control study of lung cancer and smoking (previously described in Robert Proctor's The Nazi War on Cancer (2)) as “the first irrefutable modern proof that smoking causes lung cancer in humans” (1, p. 61). This work was largely ignored by American and British researchers conducting their own case-control studies 10 years later. Davis faults these latecomers with failing to immediately denounce cigarettes, whereas Müller had definitively declared tobacco to be “the single most important cause of the rising incidence of lung cancer” (1, p. 53). Davis's argument here and repeated throughout the book is that, when the bar for medical proof is set too high, life-saving public health action is delayed. By the time of the 1964 Surgeon General's report (3), an unprecedented wealth of evidence had been amassed, including seven cohort studies and over 30 case-control studies implicating cigarettes as a cause of cancer. However, public health interventions often have to be taken on lesser evidence. Davis criticizes reliance on epidemiology as the “gold standard,” demanding an alternative to “waiting for enough bodies to drop or sicken before we decide we've got a problem” (1, pp. 399–400). Yet how much and what kind of evidence should be required to act? Unfortunately, Davis stops short of proposing any concrete answers to this question. There are a number of factual and historical errors in the book that detract from its force. For example, Davis devotes a page to describing Clarence C. Little's tenure as “the first Director of the fledgling National Cancer Institute” (1, pp. 120–121). In fact, he never held this post, although he did serve as one of six original members of the National Advisory Cancer Council. She claims that the Council for Tobacco Research gave money “directly” to Wilhelm Hueper and Tom Mancuso to study the environmental and occupational causes of cancer. However, the document she cites for this makes it clear that funds were given to Mancuso but not to Hueper (1, p. 153). She has both R. A. Fisher and Nathan Mantel working “directly” for the tobacco industry in 1967; this is unlikely as Fisher died in 1962 and Mantel was still at the National Cancer Institute (1, p. 189). Davis is at her strongest when telling her own story. She provides engaging, first-hand accounts of her early career in the 1980s as a junior epidemiologist tackling big issues, such as assessing the disease burden for victims of exposure to hazardous wastes and studying whether cancer rates were increasing. Davis began working on the latter question under the guidance of Abe Lilienfeld at The John Hopkins University. She found that the incidence of multiple myeloma and brain cancer in men over 45 years of age had grown by more than a third in less than two decades, and a resulting paper in the Lancet (4) drew major headlines. Davis describes an encounter with Richard Doll, in which he told her that she had made a “colossal error” and that her findings would be explained by improved diagnosis and record keeping for these particular cancers (1, p. 257). She held to her story and gathered the evidence to prove Doll's hypothesis wrong. She also incorporates personal stories and encounters with cancer, including her own brush with a suspicious mammography reading, her family's exposure to the legacy of environmental pollution in Pennsylvania, and the experiences of friends and colleagues facing difficult decisions and questions about treatment options. Throughout, she also raises questions about what might have been done differently to prevent these cancers. In the end, however, the book is frustrating for its lack of explicit conclusions or recommendations about how things should be done differently. After 480 pages describing many twists and turns in the politics of cancer research, there are no substantive conclusions to tie it all together. Davis offers broad statements—“we need to open a new front” in the war against cancer (1, p. xviii)—but she fails to offer concrete proposals. In the book's closing pages, she briefly suggests the need for an independent commission for the assessment of toxic hazards and medical monitoring programs for exposed populations, but these suggestions are not developed. “I am not smart enough,” she claims, “to know what kind of system will best identify and address the preventable causes of cancer in our environment” (1, p. 430). On this point, she is clearly wrong. As her own autobiographic accounts illustrate, there are few people privileged with the range of scientific and real-world policy experience Davis has to be in a better position to offer some potential solutions. Indeed, this is what makes her failure to do so, so disappointing. Conflict of interest: none declared.