The Divergence Thesis

4.1 The thesis stated

With the apparatus of Chapter 3 in hand, T1 and T2 can be stated as a set of claims about any persistent conceptual aporia in the sense of §2.1.

(4.1) The aporia concerns a polyfunctional concept C, with functions F₁, …, Fₙ (n ≥ 2) that are distinct by the case-partition criterion of §3.3.

(4.2) Each thesis of the aporetic cluster is either an articulation of the application condition Aᵢ associated with some function Fᵢ, or a claim about the logical form of C (its arity, its transitivity, its determinacy) that some Fᵢ requires.

(4.3) The theses of the cluster are jointly satisfiable over the coincidence range of C and jointly unsatisfiable only over its divergence region.

(4.4) The cases that expose the inconsistency are divergence cases: possible situations constructed so that properties correlated in the environment of acquisition are decorrelated.

(4.5) The verdicts elicited by divergence cases are prioritisations. A verdict expresses which function the subject gives precedence to under the case's presentation, and it varies with presentation where presentation varies the salience of functions.

(4.6) The doctrinal positions on the aporia correspond to standing prioritisations of functions.

I now argue that (4.1) to (4.6) hold for each of four problems. For each I proceed in the same order: the functions of the concept, identified by the methods of §3.3; the coincidence of their application conditions in the ordinary run of cases; the divergence cases that have shaped the debate; the positions as prioritisations; and the fit with the explananda of §1.4. I have tried to keep the exposition of the debates themselves to what is needed for the argument, and I assume a reader who knows them.

4.2 Knowledge

Functions. Genealogy delivers a first function. Craig (1990) argues that a community with the generic need to acquire true beliefs from others would need to mark those individuals whose say-so on a question can be relied upon; the concept of knowledge, in its "prototypical" form, flags good informants. Call this F₁, the informant function. Its application condition is roughly: S is a good source on whether p, which requires that p be true and that S's belief be formed in a way that makes S reliable on such matters, in a way the inquirer can detect.

Subtraction reveals others. Were we to lose the concept, we would lose our standard way of marking the point at which inquiry into a question may properly stop. Kelp (2011) and Kappel (2010) argue that this inquiry-closing function, F₂, is central: to know whether p is to be in a position where further inquiry into p is pointless. We would also lose the norm that governs assertion. Williamson (2000, ch. 11) argues that one may assert p only if one knows p; whether or not that is the correct norm, the concept of knowledge is the concept we use to state candidate norms of this kind, and F₃, the assertion-licensing function, is a distinct job. Hawthorne and Stanley (2008) and Fantl and McGrath (2009) argue that knowledge is the norm for treating a proposition as a reason for action; F₄, the action-licensing function, requires that S's epistemic position be good enough, relative to what is at stake, for S to rely on p. Virtue epistemologists (Sosa 2007a; Greco 2010; Zagzebski 1996) treat knowledge as a cognitive achievement, a success creditable to the agent's competence; F₅. Anti-luck epistemologists (Pritchard 2005; Sosa 1999) treat knowledge as belief that is safe, that could not easily have been false; F₆. And there is a residual function inherited from the Cartesian tradition, that of marking what is certain or indubitable, F₇, which Unger (1975) takes to be the operative one and from which he derives scepticism.

Case-partition confirms that these are distinct: for each pair there is a possible case on which the two deliver different verdicts, and the cases are the ones that constitute the post-Gettier literature.

Coincidence. In the ordinary run of cases the seven conditions coincide. A belief formed by a competent perceiver in good conditions, or acquired from a reliable and sincere informant, is safe, is creditable to the believer's competence, marks the believer as a good source, closes inquiry, and licenses both assertion and action; and if it is not certain in the Cartesian sense, nothing in ordinary life turns on that. The correlation is not accidental. All seven are consequences of competent belief-formation in a cooperative environment where sources are honest, faculties work, and the world is not arranged to deceive. That is the environment in which the concept was learned, and it is why the concept could be learned as one.

Divergence cases. The history of post-Gettier epistemology is the history of the construction of cases in which the conditions come apart.

Gettier's (1963) cases decorrelate internal justification from safety and from credit. Smith has excellent grounds for believing that the man who will get the job has ten coins in his pocket; his belief is true; but it is true by luck, since his grounds concerned Jones and it is Smith himself who gets the job. Zagzebski (1994) showed that the decorrelation is structural: for any account on which the justification condition does not entail truth, one can construct a case in which bad luck would have made the justified belief false and good luck makes it true after all. In my terms, whenever a concept of knowledge is fallibilist, its justification-based conditions and its safety-based conditions have a non-empty divergence region.

Goldman's (1976) fake-barn case decorrelates achievement and informant-reliability from safety. Henry, driving through a county in which the locals have erected barn façades, looks at the one real barn and believes it is a barn. His faculties are working; he is a fine informant on barns in general; but he could very easily have been wrong.

Lottery cases (Hawthorne 2004) decorrelate high probability from safety. You have every reason to believe your ticket will lose, and it does, but you could easily have been wrong. Hawthorne's own version sharpens the divergence by adding an action-licensing element: you seem to know you will not be able to afford an African safari this year, which entails that your ticket loses.

DeRose's (1992) bank cases decorrelate action-licensing from the informant and inquiry-closing functions. The same evidence that the bank is open on Saturday suffices for action when little is at stake and seems not to when a great deal is; yet nothing about the speaker's reliability or the state of inquiry has changed.

Sceptical scenarios decorrelate certainty from everything else. A brain in a vat has beliefs that are, in every respect that the other six functions care about in the ordinary sense, as good as ours; only the Cartesian function is not served.

Verdicts. The account predicts, by (3.6), that verdicts will be robust where a case falls near the coincidence range or where most functions side together, and unstable where the functions are balanced. The evidence bears this out.

Gettier verdicts are robust. Machery, Stich, Rose, and colleagues (2017) found that participants in Brazil, India, Japan, and the United States reliably denied knowledge in Gettier cases; Nagel, San Juan, and Mar (2013) and Turri (2013) found the same among lay participants in North America. On the present account this is what one should expect. In Smith's case five of the seven functions deliver "no": Smith is not safe, not creditable, not a good informant on this proposition, and no one would license assertion or action on grounds that are about the wrong man; only the internal-justification reading of F₁ and the inquiry-closing function, on a lenient reading, side with him. The case is near the edge of the divergence region, and the verdict is correspondingly firm. Early reports of cross-cultural variation in Gettier and related verdicts (Weinberg, Nichols, and Stich 2001) have not replicated (Kim and Yuan 2015; Seyedsayamdost 2015), and the robust finding is now the one to be explained.

Fake-barn verdicts are not robust. Colaço, Buckwalter, Stich, and Machery (2014) found that a substantial proportion of participants attributed knowledge to Henry, and the philosophical literature is divided: Pritchard denies knowledge on grounds of safety, Sosa's later work grants Henry an "animal" knowledge while denying him the reflective knowledge that would require his competence to be reliably exercised in that environment (Sosa 2007a, 2011), and others grant knowledge outright. Here the functions are balanced: achievement, informant-reliability, and action-licensing say "yes"; safety says "no". The split is predicted.

Bank-case verdicts shift with stakes, as the action-licensing function requires and the informant function does not, and the three main positions on the cases, contextualism (Cohen 1988; DeRose 1992), subject-sensitive invariantism (Stanley 2005; Fantl and McGrath 2009), and classical invariantism, are three ways of handling the divergence: the first locates the stakes-sensitivity in the semantics of "knows", the second in the metaphysics of knowledge, the third denies it and explains away the appearance.

Positions as prioritisations. Knowledge-first epistemology (Williamson 2000) treats the licensing functions, F₃ and F₄, as primary and refuses to analyse knowledge in terms of anything else; the refusal is intelligible if what one wants from the concept is a norm, since norms need not be decomposable. It is worth saying plainly that knowledge-first is the position least embarrassed by the present account and, read through it, the best motivated. If a concept's several application conditions coincide over the range in which it is learned, then no analysis framed in terms of any one of them will hold outside that range, and the unanalysability Williamson infers from the failure of the analyses is what polyfunctionality predicts. The difference is in what to conclude from it. Williamson takes the failure to show that knowledge is a mental state which the analyses were wrong to try to decompose; I take it to show that there was never one thing there to decompose, and that "knows" is unanalysable in the way that "cause" is unanalysable, not because it names something simple but because it names several things at once. Virtue epistemology prioritises F₅. Anti-luck epistemology prioritises F₆; Pritchard's (2012) "anti-luck virtue epistemology" is an explicit attempt to keep F₅ and F₆ together in one concept by making both necessary, which is to say an attempt to legislate for the divergence region by conjunction. The function-first tradition (Craig 1990; Hannon 2019) prioritises F₁ and reads the others as derived. Contextualism is the position that the prioritisation itself shifts with conversational context, which is close to a statement of (3.6) in semantic dress. And scepticism is the prioritisation of F₇ over all the rest: Unger's (1975) argument is precisely that "knows" is an absolute term whose application condition is certainty, and that nothing meets it. Fallibilism, which nearly every contemporary epistemologist accepts (Cohen 1988), is the standing decision to demote F₇.

Fit. The Gettier episode displays E3 (persistence without stagnation) in miniature: justified true belief was abandoned by nearly everyone, and what replaced it was not a successor analysis but a proliferation of positions, each prioritising a different function among those the case had separated. It displays E4: epistemologists across the doctrinal divide accept the distinctions, between justification and safety, between animal and reflective knowledge, between the epistemic position needed for assertion and that needed for action, while disagreeing about the headline question. It displays E6, as just shown. And Kaplan's (1985) argument that "it's not what you know that counts", that the concept of knowledge does no work in inquiry that justification and truth do not do, is, in my terms, the observation that once F₂ is separated from F₁ and F₃, the inquiry-closing function can be served without the word.

One feature of this case study bears more weight than its fit with any single explanandum, and I want to set it out separately. Analytic epistemology has spent six decades on a repair problem with an explicitly stated success condition, a set of conditions individually necessary and jointly sufficient for knowledge that no Gettier case defeats, and the problem is not merely unsolved but has not narrowed. Each proposed further condition has met a case that separates it from something else the concept does: the barn façades for conditions cast in terms of reliable discrimination, the lottery for conditions cast in terms of probability, the bank cases for conditions indifferent to what is at stake. On the standard reading this is a record of failure that calls for an excuse, and the excuses offered have been that the concept is unusually difficult, that the analysts were insufficiently ingenious, or that analysis is the wrong method. On the present account it is not a repair problem at all. A further condition is a proposal that one application condition should govern the region in which the others diverge; a counterexample to it is a case drawn from that region and presented so as to make a different function salient; and the supply of such counterexamples is therefore inexhaustible in principle, which is what the history shows. Zagzebski's (1994) result is the general form of the point: the divergence region of a fallibilist concept of knowledge is never empty, so there is always another case. What the account predicts is not that the repair will be hard but that it cannot succeed as posed, and that the absence of convergence on a question worked this hard for this long is not a scandal in need of an excuse but the datum from which to begin.

4.3 Free will

Functions. The concept at issue is that of the freedom of will or action that is required for moral responsibility; the aporia is about that, not about freedom in the political or the phenomenological sense. Genealogy here is less developed than for knowledge, but Strawson's (1962) reconstruction of the practice of holding responsible from the reactive attitudes, resentment, gratitude, indignation, and their kin, plays the role that Craig's state of nature plays for knowledge: it identifies the practice the concept serves and asks what the practice requires.

Four functions have been articulated by the debate itself, and case-partition confirms their distinctness.

F₁, leeway: the concept marks agents who could have done otherwise, in a sense that connects with the principle that "ought" implies "can", with the fairness of blame, and with the openness of alternatives in deliberation. Its application condition is the existence of accessible alternatives.

F₂, reasons-responsiveness: the concept marks agents whose actions issue from a mechanism that recognises and responds to reasons, so that praise and blame can function as reasons to them. Its application condition is what Fischer and Ravizza (1998) call guidance control; Wolf (1990) and Frankfurt (1971) articulate cognate conditions.

F₃, sourcehood: the concept marks agents who are the origin of their actions in a way that is not itself the product of factors beyond their control. Kane (1996) calls the condition "ultimate responsibility"; Pereboom (2001) makes it the basis of his source incompatibilism; Strawson's (1994) Basic Argument is that nothing could meet it.

F₄, reactive aptness: the concept marks agents towards whom the reactive attitudes are appropriate. Its application condition is given by the internal standards of the practice of holding responsible, and Strawson (1962) argued that these standards neither require nor could be undermined by a general metaphysical thesis.

Coincidence. In the normal exercise of adult human agency the four coincide, and they coincide for a reason. An agent who deliberates and acts on her own reasons, in circumstances in which no one is manipulating her, is an agent for whom alternatives were open in every sense that ordinary life recognises, who is as much the source of her action as anyone ever is, and towards whom resentment or gratitude is apt. The correlation is so tight that pre-philosophical thought has no separate words for the four.

Divergence cases. The debate has constructed, in sequence, cases that decorrelate each function from the others.

The thesis of determinism, made precise in van Inwagen's (1983) Consequence Argument, decorrelates F₁ from F₂. If our actions are consequences of the remote past and the laws, and if no one has power over either, then no one has power over her actions in the sense that would make alternatives accessible; yet a determined agent may be perfectly reasons-responsive. Classical compatibilism (Moore 1912, ch. 6; Ayer 1954) tried to prevent the divergence by reinterpreting "could have done otherwise" conditionally, so that F₁ would coincide with F₂ after all; the failure of the conditional analysis is generally acknowledged, and the compatibilist response has since taken a different form.

Frankfurt's (1969) case decorrelates F₁ from F₂ directly, without appeal to determinism. Jones decides on his own to do something; Black, who wants him to do it, stands ready to intervene if Jones shows signs of deciding otherwise, but never has to. Jones acts on his own reasons and could not have done otherwise. The dominant verdict is that Jones is responsible, and Frankfurt drew the conclusion that F₁ is not required. The minority response, the "flicker of freedom" and "dilemma" defences (Widerker 1995; Kane 1996), argues that the case cannot be constructed without either leaving Jones a residual alternative or presupposing determinism; in my terms, it denies that the divergence region is reachable. The dispute over whether it is reachable has itself been stable for thirty years.

Manipulation cases decorrelate F₂ from F₃. Pereboom's (2001, ch. 4) four-case argument presents Professor Plum, who kills for reasons and through a reasons-responsive mechanism, but whose reasons-responsive character was installed by neuroscientists, by a programme of childhood conditioning, or, in the fourth case, by ordinary deterministic causation. Mele's (2006, ch. 7) zygote argument makes the same point with a designed zygote. Plum meets every compatibilist condition on F₂ and fails F₃. The compatibilist may take the "hard line" (McKenna 2008), holding that Plum is responsible in every case, or the "soft line", adding a historical condition to distinguish the cases; the incompatibilist holds that the cases are alike and Plum is responsible in none. Each response is a prioritisation of F₂ or F₃, and the dispute has not moved.

The luck objection to libertarianism (van Inwagen 2000; Mele 2006) constructs a divergence case within the incompatibilist camp, decorrelating F₁ from F₃: if an undetermined choice is "rolled back" and replayed, it sometimes goes one way and sometimes the other, so that the agent has alternatives but seems not to control which is realised. Leeway and sourcehood, which coincided for the libertarian in ordinary cases, come apart, and libertarians divide over which to save.

Verdicts. The experimental literature on free will provides the clearest evidence for (3.6) in any domain. Nahmias, Morris, Nadelhoffer, and Turner (2005) found that when participants were presented with a concrete agent in a deterministic universe, most judged the agent free and responsible. Nichols and Knobe (2007) found that when the question was posed abstractly (whether anyone in a deterministic universe could be responsible), most participants gave the incompatibilist answer, while a concrete, affectively charged case (a man who kills his family) elicited the compatibilist one. Sarkissian and colleagues (2010) found the abstract incompatibilist judgement across participants in the United States, Hong Kong, India, and Colombia. Murray and Nahmias (2014) found that a portion of the incompatibilist judgements were driven by participants' reading determinism as "bypassing" the agent's deliberation, which suggests that the abstract framing had engaged F₃ specifically.

On the present account the pattern is not a puzzle to be explained away but a prediction confirmed. The abstract framing makes the metaphysical functions, F₁ and F₃, salient, and they deliver the incompatibilist verdict. The concrete framing makes the practice function, F₄, salient, and it delivers the compatibilist one. The same participants are not confused; they are giving each function its due when it is foregrounded.

One finding in that literature is harder for the account than the paragraph above allows, and it is one I have just used. Murray and Nahmias (2014) do not merely report that abstract framings engage sourcehood. They argue that a substantial part of the incompatibilist response is error: participants read a deterministic description as one in which the agent's deliberation makes no difference to what she does, and when judgements about such bypassing are controlled for, the incompatibilist response weakens. If that reading is right, the abstract framing is not making a function salient; it is inducing a misunderstanding of the case, and a misunderstanding is a measurement of nothing.

The account has a reply, and I do not think it is decisive. The reply is that bypassing is not a foreign intrusion into the sourcehood function but very nearly a statement of it: what F₃ demands is that the agent, rather than something upstream of her, be where the action originates, and a participant who reads determinism as leaving her deliberation idle has grasped that demand and judged that determinism defeats it. What would separate the two readings is a manipulation that makes sourcehood salient while closing off the bypassing misreading, in which the agent's deliberation is stipulated to be causally efficacious and also stipulated to be the product of factors she did not choose. Parts of the manipulation literature approach this, and Pereboom's Plum is built to it. But the published record does not distinguish the two readings, and I record the consequence rather than argue past it: (3.6) is better supported by Williams's first-person and third-person framings of a single identity case, where no misreading is available, than by the compatibilism findings, where one is.

Positions as prioritisations. Classical compatibilism tried to hold F₁ and F₂ together by analysis. Frankfurt-style and Fischer–Ravizza compatibilism demotes F₁ and prioritises F₂. Strawsonian compatibilism prioritises F₄ and treats the metaphysical functions as confusions. Source incompatibilism (Pereboom) prioritises F₃; leeway incompatibilism (van Inwagen) prioritises F₁; Kane wants both and constructs an account of "self-forming actions" to secure them together. Hard incompatibilism and Strawson's (1994) impossibilism prioritise F₃ and conclude that nothing satisfies it. Dennett's (1984) Elbow Room, whose subtitle is "the varieties of free will worth wanting", is in effect an inventory of functions and an argument that the ones worth wanting are compatibilist; Vargas's (2013) "revisionism" is an explicit proposal to redesign the concept of responsibility around F₂ and F₄ and to abandon the folk commitment to F₁ and F₃. Pereboom's (2014) distinction between responsibility in the "basic desert" sense and in forward-looking senses, and Watson's (1996) and Shoemaker's (2011) distinctions among attributability, answerability, and accountability, are disentanglements of the concept of responsibility that the debate has produced and that all parties now use.

I have given this debate more room than the other three, and the reason is evidential rather than doctrinal. It is the aporia in which all four kinds of evidence the account needs are in the best supply: a cluster whose theses the parties state explicitly (§2.3); a documented sequence of divergence cases, each separating a pair of functions that no earlier case had separated; two rounds of a survey of professional opinion taken a decade apart (§1.2); and an experimental literature large enough to have produced replications, and disputes about the replications. If the account is to be embarrassed by evidence, it should be embarrassed here first, which is why the decomposed survey proposed in §7.9 is specified for this debate and not another.

Fit. The free will debate displays E5 more sharply than any other: the survey figures of §1.2 show the distribution of professional opinion across the prioritisations to be almost perfectly stable across a decade in which every one of the cases above was intensively discussed. It displays E4: compatibilists and incompatibilists agree that the Frankfurt case separates two things that had been conflated, that the manipulation cases separate two more, and that the luck objection separates two within libertarianism; what they do not agree on is which to keep. It displays E6, as just argued. And it displays E1 with particular clarity: each of the four functions is something our practice of holding responsible really does track, and each thesis of the aporetic cluster of §2.3 is compelling for that reason.

4.4 Personal identity

Functions. Locke (1690, II.xxvii.26) supplied the genealogy in a sentence: "person" is "a forensic term, appropriating actions and their merit". The concept's core function, F₁, is forensic reidentification: to mark the individual who is to be held responsible, rewarded, or repaid later as the same individual who acted, was promised, or lent. Subtraction reveals the rest. F₂, prudential anticipation: the concept marks the future individual whose experiences I am to anticipate in the special first-personal way, whose pains I dread and pleasures I look forward to as mine; Parfit (1984) made this function central by asking what "matters" in survival. F₃, organismic tracking: the concept marks the continuing human animal, the thing that is born, grows, and dies (Olson 1997). F₄, characterisation: the concept marks what is truly mine among my traits, commitments, and history, the self of self-understanding; Schechtman (1996) distinguished this "characterisation question" from the "reidentification question" and argued that most of the literature had conflated them. F₅, agential unity: the concept marks the unity that deliberation and commitment over time require, the standpoint from which one can make a promise now that binds one later (Korsgaard 1989). And there is a social function, F₆, of recognition by others: the concept marks the individual whom others continue to treat as one and the same, in the network of relations that Schechtman (2014) calls a "person life".

Coincidence. In every actual human life, with the exceptions to be noted, the six coincide. The animal that was born is the psychological continuant that anticipates, the forensic subject that is held to account, the narrative self that understands itself, the agent that binds itself, and the individual others recognise. They coincide because human psychology is realised in a single human organism, organisms do not divide or fuse, memory is continuous, and the social world tracks bodies. The correlation is so complete that the question "which of these am I?" does not arise.

Divergence cases. The exceptions are where the debate lives, and unusually for a philosophical problem several of them are actual.

Locke (1690, II.xxvii.15) constructed the first: the soul of a prince, carrying the prince's consciousness, enters the body of a cobbler. F₁ and F₂ follow the consciousness; F₃ stays with the body. Locke's own treatment of the drunkard who does not remember what he did (§22) is a divergence case between the forensic ideal and legal practice: human law punishes the sober man for what the drunk did, Locke observes, because it "cannot distinguish certainly what is real, what counterfeit", so that F₁ as the law must apply it comes apart from F₁ as it ideally would. Reid's (1785, Essay III, ch. 6) brave officer, who remembers his schoolboy flogging as a young soldier and his soldierly exploit as an old general but not the flogging, exhibits a failure of transitivity that the forensic function cannot tolerate and the psychological criterion, in its simple memory form, entails.

Williams (1970) constructed the case that most directly confirms (3.6). Two people, A and B, are to have their psychological contents exchanged, and one of the resulting persons will be tortured. Described in the third person, as a "body-swap", the case elicits the verdict that A goes where A's psychology goes, and should choose that the B-body person be tortured. Described in the first person, as a sequence of things that will be done to you, your memories erased, new ones installed, and then torture, the same case elicits the verdict that you remain with your body throughout, and the torture is yours to fear. Williams took the divergence of verdicts to be the problem. On the present account it is the datum: the third-person description makes F₁ and F₄ salient, and they follow the psychology; the first-person description makes F₂ salient, and prudential anticipation, faced with a continuous body about to be hurt, follows the body.

Parfit's (1971, 1984) fission case decorrelates F₂ from the logical form that F₁ requires. My brain is divided and each half is transplanted into a new body; each resulting person is psychologically continuous with me. Identity is one–one, so I cannot be both. But everything that matters to me in survival, on the prudential understanding, is preserved twice over. Parfit's conclusion, that identity is not what matters, is the explicit disentanglement of F₂ from F₁: what the prudential function tracks is Relation R, psychological connectedness and continuity with any cause, and R is not one–one and need not be determinate. The alternatives, the no-branching clause, Nozick's (1981) closest-continuer schema, and Lewis's (1976) proposal that there were two persons sharing a body all along, are attempts to keep F₁ and F₂ together by adjusting the logical form of identity or the counting of persons.

The actual divergence cases are medical, and I take them briefly because §6.5 returns to them. The ventilator decorrelated F₃ from F₂ and F₄, by making it possible for a human organism to continue while all brain function had irreversibly ceased, and the dispute it left between whole-brain, higher-brain, and cardiopulmonary criteria of death is a dispute over which function "death" should track. Dementia decorrelates F₂ and F₄ from F₃ and F₆ over time, and the dispute between Dworkin (1993), who argues that the earlier competent person's "precedent autonomy" should govern the treatment of the later demented one, and Dresser (1995), who argues that the present patient's interests should, is a dispute about whether the person of the advance directive is the person in the bed. Nagel (1971) argued that split-brain patients decorrelate the unity of consciousness from the unity of the person.

Positions as prioritisations. Psychological continuity theories prioritise F₁, F₂, and F₄, which is why they dominated for three centuries: those are the functions that ordinary reflection on identity most often engages. Animalism (Olson 1997) prioritises F₃, and its "too many thinkers" argument is the complaint that the psychological theorist has severed the concept from the organism that, in every actual case, is the thinker. Narrative theories (Schechtman 1996) prioritise F₄; Korsgaard (1989) prioritises F₅, arguing against Parfit that the unity of the person is a practical presupposition of agency and not a metaphysical discovery. The "simple" or "further-fact" view (Chisholm 1976; Swinburne in Shoemaker and Swinburne 1984) insists that there is a single fact of identity which all the functions track, and I return to it in §7.3 as the realist rejoinder in its natural habitat. Parfit's own reductionism is the denial that any such fact is needed: the functions are all there is, and they come apart.

Fit. The debate displays E2 with unusual purity: nearly every case that has shaped it is physically impossible or medically extreme, because human organisms do not divide and consciousness does not migrate, and only by imagining that they do can the correlated functions be prised apart. It displays E4: animalists and psychological theorists alike accept Schechtman's two questions and Parfit's distinction between identity and what matters as clarifications, whatever they think of the answers. And it displays E8: the actual divergence cases in medicine were settled, to the extent they have been, by legislation rather than by discovery, and the legislation has the form of a decision about which function to privilege.

The weakest of the four. I should say plainly that this is the thinnest of the case studies, and that where it is thin is where my own criterion applies. In §4.2 and §4.3 I said that case-partition confirms the distinctness of the functions listed. I cannot say it here for all six. Fission separates F₁ from F₂. The prince and the cobbler separates F₃ from F₁ and F₂. Williams's case separates F₂ from F₁ and F₄ under two descriptions of one situation. Dementia separates F₂ and F₄ from F₃ and F₆ over time. But I know of no case that separates F₅, agential unity, from F₂ and F₄ taken together, and none that separates F₆, social recognition, from F₁ except by separating both of them from the organism at once. By the criterion of §3.3, functions that no possible case separates are one function under several descriptions, and I have therefore probably overcounted. Korsgaard's argument for F₅ and Schechtman's for F₆ are arguments that the practices are distinct, that deliberation presupposes something reidentification does not, that being treated as one and the same is not the same achievement as being one and the same, and I have accepted them at the level of practices rather than at the level of cases. For my purposes that is the wrong level, for a reason §7.5 takes up. Four functions would be the defensible count here on my own criterion; six is what the literature supplies, and I have followed the literature.

Two further weaknesses have the same source. The mapping from positions to prioritisations is looser here than in the other three studies: animalism and narrative theory prioritise F₃ and F₄ cleanly enough, but the psychological continuity theory prioritises three functions at once and is not, as a position, a single weighting at all. And the individuation is carried almost entirely by fission, which leaves this study exposed in a way the causation study is not, where pre-emption, double prevention, omission, and transitivity failure are four independent classes of case pointing the same way.

The overcount has a consequence elsewhere that I had better state than leave for a reader to find. §3.5 offers "person" as the specimen of a high-load concept, bearing at least five functions where "uncle" bears one, and the account predicts that the depth and persistence of a problem vary with functional load. On the defensible count "person" bears four, which is what "cause" bears and fewer than the seven I attribute to "knowledge". What §3.5 claims about the number, then, is not established by this chapter; only what it claims about the stakes is, and the stakes are not in doubt, since they are legal rights, medical decisions, and the object of a lifetime's concern. The cost falls on a correlation that §7.2 offers as a check on the account, not on T1 or T2, but it is a cost and the check is weaker for it.

What the overcount does not touch is worth being equally plain about, because the two are easy to run together. F₅ and F₆ do no work in the evidence this section supplies for (3.6). Williams's case turns on F₁, F₂, and F₄, all of which survive the recount, which is why §4.3 was able to retreat to it when the compatibilism findings turned out to admit a misreading. The weakness here is in how many functions "person" has, not in whether presentation selects among them.

So the plain statement, which §4.7 will repeat in a sentence and which belongs here in full. Personal identity is the case study to attack first, and it is thin in the one place I cannot dismiss, namely where my own criterion of §3.3 is applied to my own list. Strengthening it needs cases the literature has not constructed: cases in which agential unity or social recognition comes apart from forensic reidentification, which is to say cases about commitment and about who one is taken to be, rather than further cases about the persistence of a thing. Until such a case exists I am entitled to four functions here and have written six, and the reader should discount the section by the difference.

4.5 Causation

Functions. Hume gave two definitions of cause and treated them as equivalent: an object followed by another, "where all the objects similar to the first are followed by objects similar to the second"; "or in other words where, if the first object had not been, the second never had existed" (1748, §VII.ii). They are not equivalent, as Lewis (1973, 556) observed at the opening of the paper that founded the counterfactual tradition, and the gap between them is the oldest documented case of latent polyfunctionality in modern philosophy.

Four functions have been articulated. F₁, production: the concept marks the process by which one event brings about another, the mechanism or transfer of energy or "derivativeness" (Anscombe 1971) that connects them; process theories (Salmon 1984; Dowe 2000) make this the whole of causation. F₂, dependence: the concept marks what the effect depends on, what would have to have been different for the effect not to occur, and thereby what one would intervene on to prevent or produce it; the counterfactual (Lewis 1973, 1986, 2000) and interventionist (Woodward 2003) traditions make this central, and Menzies and Price (1993) trace it to the agent's need to identify levers. F₃, inference: the concept marks what licenses inference from one event to another; Hume's first definition and the regularity tradition (Mackie 1974) articulate it. F₄, selection: the concept marks the cause among the many conditions without which the effect would not have occurred, the one to which explanation, praise, blame, or liability is to be attached. Mill (1843, III.v.3) noted that ordinary usage selects among conditions on grounds that have nothing to do with their causal contribution; Hart and Honoré (1959) showed in detail how the law's selection is governed by the requirements of assigning responsibility.

Coincidence. In an ordinary causal sequence, a single chain, uncontested, with no back-ups and no absences doing work, the four coincide. What produced the effect is what it depended on, is what licenses the inference to it, and is what we single out. That is the environment in which the concept was learned, and the reason the concept has one word.

Divergence cases. The literature of the last fifty years is a catalogue of decorrelations, and Hall (2004) drew the general conclusion from them.

Pre-emption decorrelates F₁ from F₂. Suzy and Billy each throw a rock at a bottle; Suzy's arrives first and shatters it; Billy's sails through the space where the bottle was. Suzy's throw produced the shattering, but the shattering did not depend on it, since Billy's throw would have done the job. Late pre-emption of this kind, and the "trumping" pre-emption of Schaffer (2000), in which Merlin's spell and Morgana's spell both enchant the prince but the earlier spell takes precedence by law, showed that the simple counterfactual analysis could not track production, and Lewis's (2000) revision to "influence" was an attempt to recover it.

Double prevention decorrelates F₂ from F₁ in the opposite direction. Suzy is flying a bombing mission; Enemy would have shot her down; Billy, escorting her, shoots Enemy down first. The bombing depends on Billy's action, and the interventionist would identify Billy's action as something to intervene on to prevent it; but no process connects what Billy did to what Suzy did, and the locality and intrinsicness that production requires are absent.

Omissions decorrelate F₄ from F₁ and expose the norm-sensitivity of F₄. My plants died because the gardener failed to water them; they did not die because the Queen failed to water them, though the counterfactual dependence is the same (Beebee 2004; McGrath 2005). Knobe and Fraser (2008) found that when a professor who is forbidden to take pens from the department office and an administrative assistant who is permitted to both take pens, and the last pen is thereby unavailable, participants judge the professor to have caused the problem; Hitchcock and Knobe (2009) argue that normative considerations enter causal selection systematically. The selection function is doing what it is for, tracking responsibility, and it is doing it in a way that the production and dependence functions, which are norm-indifferent, cannot follow.

Transitivity failures (Hall 2000; Hitchcock 2001) decorrelate the logical form that F₁ seems to require from the verdicts F₂ delivers: a boulder dislodged towards a hiker causes him to duck, and the ducking causes his survival, but the boulder's fall does not cause his survival.

Hall's (2004) conclusion is that the theses of §2.3, transitivity, locality, intrinsicness, dependence, and the causal efficacy of omissions, are jointly unsatisfiable, and that they partition cleanly: production is transitive, local, and intrinsic, and does not admit omissions; dependence is none of these and does. There are, he concludes, two concepts. Cartwright (2004) reached a more radical version of the same conclusion in a paper whose title is a statement of T1 for this domain: "Causation: One Word, Many Things". Godfrey-Smith (2009) surveys the varieties of causal pluralism that have followed.

Positions as prioritisations. Process theories prioritise F₁, and their treatment of omissions and double prevention as not "really" causal is the demotion of F₂ and F₄. Counterfactual and interventionist theories prioritise F₂, and Woodward's insistence that his account is meant for the special sciences and for the identification of levers is an explicit statement of which practice the concept is being designed for. Regularity theories prioritise F₃. Legal theories of causation (Hart and Honoré 1959; Moore 2009) prioritise F₄ and build the norm-sensitivity in. Contrastivism (Schaffer 2005) relocates F₄ into the semantics, holding that causal claims are always implicitly contrastive, so that selection is not a distortion of the concept but part of its logical form; this is the causal analogue of contextualism about knowledge. Russell's (1913) argument that the concept of cause should be eliminated from advanced physics is the judgement that physics has no need of any of the four functions in the form the folk concept supplies them, which is, if correct, a reason for physics to drop the word rather than a discovery about causation.

Fit. Causation is the domain in which the field has come closest to accepting T1 explicitly, and I take that to be evidence that the diagnosis is natural rather than imposed. It displays E4 completely: process theorists and counterfactual theorists agree that pre-emption and double prevention separate two relations, and they agree on which theses go with which. It displays E8: legal causation is the selection function institutionalised, and its doctrines of proximate cause are legislation for the divergence region. And it displays E1: each of Hall's five theses is compelling because each articulates something that the concept, in its coincidence range, does.

4.6 Beyond the four

The four case studies were chosen because they are central and because the fit can be checked in detail. The account is not confined to them, and I sketch here, more briefly, how it applies elsewhere. I make no claim to have established (4.1) to (4.6) for these further cases; I claim that the shape is recognisably the same.

Truth. Tarski (1944) observed that the ordinary concept of truth, governed by the schema "'p' is true if and only if p" and applied in a language that contains its own truth predicate, is inconsistent, and confined his definition to formalised languages for that reason. The functions of "true" include disquotation and generalisation (Quine 1970; Horwich 1990), the marking of the aim of inquiry and belief, the marking of correspondence with how things are, and the licensing of assertion. These coincide except under self-reference. The Liar is a divergence case: a sentence for which the disquotational function delivers a verdict and the groundedness that the other functions presuppose is absent. Kripke's (1975) theory locates the divergence exactly, by identifying the grounded sentences as those for which the functions coincide. Scharp's (2013) replacement of "true" by two concepts is disentanglement; Lynch's (2009) functionalism, on which truth is one functional property variously realised, is an attempt to keep one concept by moving up a level of abstraction, a move I discuss in §6.4.

Liberty. Berlin's (1958) distinction between negative liberty, the absence of interference, and positive liberty, the presence of self-mastery or of the means to act, is a disentanglement of two functions of "free" that coincide for an unimpeded, resourced, self-governing agent and come apart for the impeded rich, the unimpeded destitute, and the addict. MacCallum's (1967) triadic analysis, on which every freedom claim has the form "x is free from y to do z", is a re-unification at a higher level (§6.4).

Moral rightness. Ross (1930) proposed that "right" answers to several prima facie duties, fidelity, beneficence, non-maleficence, justice, and the rest, no one of which is overriding, and that they coincide in ordinary cases and conflict in hard ones. The trolley cases (Foot 1967; Thomson 1976, 1985) are divergence constructors: they decorrelate the harm-minimising function of "wrong" from its constraint-marking function, and the doctrine of double effect and its successors are attempts to legislate for the region. The stability of the consequentialist–deontological dispute and the sensitivity of trolley verdicts to framing are what the account predicts.

Existence. Carnap's (1950b) distinction between internal and external questions, Thomasson's (2015) "easy" ontology, on which existence questions are settled by the application conditions of the relevant sortal, and Sider's (2011) insistence that the interesting ontological questions concern what is fundamental or "carves at the joints", can be read as identifying two functions of "exists": the quantificational and inferential function, which Thomasson takes to exhaust the concept, and the function of marking what is basic in reality, which Sider takes to be what ontology was always after. Hirsch's (2011) quantifier variance is the claim that the divergence region here is wholly verbal. Van Inwagen's (1990) Special Composition Question, pursued through cases of gluing, fusing, and living together, is a series of divergence constructions for "object", and Schaffer's (2009) proposal to shift ontology from existence to grounding is a disentanglement.

Set. Russell's paradox (Russell 1903, ch. 10) is usually presented as a pure derivation, but it is a divergence case in the sense of (2.4): the naive concept of set has the function of providing an object for every predicate's extension (comprehension) and the function of providing objects that can themselves be members (eligibility for membership), and these coincide for every set anyone had occasion to consider before Russell constructed the set of all sets that are not members of themselves. The iterative conception (Boolos 1971) disentangles them by legislating that membership-eligibility takes precedence, and the convergence of mathematical practice on ZFC, in contrast with the non-convergence of philosophy, is explained in §7.7.

Scepticism. The Cartesian sceptic exploits the divergence between the certainty function of "knows" and its other six; fallibilism is the community's standing decision in the divergence region; and the sceptic's residual claim, that the word should be reserved for certainty, is a claim in conceptual ethics, which is how Unger (1975) in effect presents it.

Punishment. Rawls (1955) distinguished the justification of the practice of punishment, which he took to be utilitarian, from the justification of a particular punishment within the practice, which he took to be retributive, and argued that the classical dispute between utilitarians and retributivists conflated the two; Hart (1968) made the same distinction between the "general justifying aim" of punishment and its "distribution". These are disentanglements of the functions of "punishment", and the stability of the retributivist–consequentialist dispute in the divergence region, where the two justifications diverge (punishing the innocent for deterrent effect; refraining from useful punishment on grounds of desert), is what the account predicts.

4.7 The abductive case

I return to the explananda of §1.4 and assess the fit.

E1 (plausibility of each horn). Each thesis of an aporetic cluster is compelling because it articulates a function the concept in fact performs, and reflective users of the concept can feel it perform that function in the coincidence range. The principle of alternate possibilities is compelling because our concept of responsibility does, across ordinary cases, apply only where alternatives were open. The one–oneness of identity is compelling because the forensic function requires it. The sufficiency of dependence is compelling because the practice of finding levers requires it. No rival account explains this. The pessimist takes plausibility as given; the deflationist treats it as an illusion, which does not explain why the illusion has the specific content it does; the optimist has no account of it at all.

E2 (indispensability of exotic cases). The cases must be exotic because they must decorrelate properties that are correlated in the ordinary world for good reasons, and only unusual, extreme, or physically impossible situations do that. The account explains not only that the cases are exotic but how: each is constructed by holding one correlated property fixed and removing another. No rival account explains why philosophy should need such cases at all.

E3 (persistence without stagnation). The headline question persists because it is a question about the concept's application in the divergence region, where the concept's functions conflict and nothing in the concept settles the conflict. The literature does not stagnate because each new case individuates a new function, each new function is a new distinction, and distinctions accumulate. The pessimist explains persistence but not the accumulation; the optimist explains the accumulation but must redefine progress to account for the persistence; the present account explains both from a single mechanism.

E4 (convergence on decompositions, divergence on headlines). The parties agree on the distinctions because the distinctions are individuations of functions by case-partition, and case-partition is a matter of fact: either the case separates the verdicts or it does not. They disagree on the headline because the headline asks which function should govern, and that is not a matter the cases settle. This is the pattern Chalmers (2015) noticed when he observed that philosophy's agreed results are negative and conditional, and it is here given a reason.

E5 (stability of the distribution). The distribution of opinion is stable because it is a distribution of prioritisations, prioritisations reflect the weights individuals give to the practices the functions serve, and the accumulation of arguments about cases does not change those weights; it only makes clearer what the choice is. That the free will figures should be as stable as they are is, on any rival account, an embarrassment. On this one it is what should happen.

E6 (pattern of experimental findings). Paradigm and near-paradigm verdicts are robust because in the coincidence range and at its edge the functions agree or nearly all agree. Verdicts in balanced divergence cases are unstable and framing-sensitive because the functions conflict and presentation selects among them. The free will findings of Nichols and Knobe (2007) and the identity findings of Williams (1970) are direct confirmation of (3.6). Accounts that treat framing effects as noise to be filtered out (Nagel 2012; Sosa 2007b) and accounts that treat them as discrediting the case method (Weinberg, Nichols, and Stich 2001; Machery 2017) both miss what the effects are: measurements of function salience.

E7 (substantive to the parties, verbal to the observer). The parties are right that they are disagreeing about something, because which function a load-bearing concept should serve is a substantive question with practical consequences. The observer is right that the parties are, at one level, talking past each other, because at the level of the first-order question ("does Jones have free will?") each is applying a different application condition and there is no shared fact for them to disagree about. Chapter 5 develops this.

E8 (recurrence outside philosophy). Pluto, brain death, the tomato, the corporation, the gene, and the species are divergence cases for concepts with several functions, and they are handled by legislation because in those domains a single practice has authority to legislate: the International Astronomical Union, a medical commission, a court, a scientific community with a dominant aim. Philosophy's concepts are load-bearing for several practices at once, none of which has authority over the others, and that is why philosophy's divergence cases do not get legislated. Chapter 7 returns to this.

I conclude that T1 and T2 are well supported, though not equally by all four studies, and the differences are worth recording. Free will is the strongest case, for the reasons given in §4.3: the cluster is stated by the parties themselves, the sequence of divergence cases is documented, and there is an experimental literature against which (3.6) can be tested. Knowledge is next, and its strength is of a different kind, six decades of a clearly specified repair problem that has not narrowed. Causation is the case in which the diagnosis was reached independently of me, which makes it the most persuasive to a reader who suspects the account of having been fitted to its examples, and the least informative about whether the account adds anything to it. Personal identity is weak, and §4.4 says how and where. Three of four, with one of the three arrived at by someone else's route, is what the abductive case rests on. The concepts at the centre of philosophy's persistent problems are polyfunctional; the problems arise where the functions diverge; and the characteristic features of those problems, from the plausibility of each horn to the pattern of experimental findings, follow from that fact. The next two chapters draw out what this implies about disagreement and about progress.