Ifs, Ands, and Buts: An Incremental Truthmaker Semantics for Indicative Conditionals Stephen Yablo October 23, 2015 Abstract Indicative conditionals appear to lie on a continuum, with the subjective and information-based on one side, and the objective and fact-based on the other. Attempts to bring them all under the same theoretical umbrella usually start at the subjective end; conditionals get more objective as they come to be based in higher-quality, less parochial, information. I propose to go in the other direction, looking first for a class of "absolute" conditionals, then bringing in other conditionals by relaxing the constraints defining that class. (A plan of action is laid out at the end of section 4. The final footnote of each section sketches the contents of the next.) 1 1 Ifs Indicative conditionals A→C are epistemic and in some sense subjective, according to current orthodoxy Their assertability (or assent-ability) in a context is thought to reflect the properties of credal or information states, far as these might be from the apparent subject matters of A and C. Indicative conditionals do not even say anything, if the triviality arguments are to believed, of a type to be regarded as true or false. The most they can aspire to is to seem right to someone with the right kind of epistemic perspective. This is the view of, among many others, Ernest Adams, Allan Gibbard, Jonathan Bennett, and Dorothy Edgington. But of course, information states are not pulled out of a hat. They derive in many cases from empirical assumptions that are evaluable as true or false.2 If those assumptions are mistaken, indicative conditionals based on them are liable to be mistaken too, however acceptable at the time of utterance. Even if conditionals in the first instance express states of mind, they nevertheless in the second instance track the external facts to which those states of mind are answerable. Subjectivists wrestle with this in various places. Adams discusses the case of Alice collecting some possibly poisonous mushrooms. You will die if you eat those, Boris tells her, in the belief that they are poisonous, and she refrains. Later, when the mushrooms turn out to be harmless, You would have died if you'd eaten those strikes us as wrong. No surprise there, youu may say; counterfactuals track actual underlying mechanisms, whether anyone knows about them or not. What is slightly surprising is that the original indicative now seems wrong too. As does Boris's subsequent claim (he hasn't heard they were safe) that Alice did die, if she did eat the mushrooms.3 Boris's conditional probabilities, based as they were on bad information, are relevant to acceptability, but not, it seems, to truth. Adams is aware of these judgments but it is not clear how he wants to account for them. An opposite sort of case, due to Morgenbesser, features an initially unacceptable conditional that is vindicated by events. Alice is about to toss a coin; Boris was going to put his money on heads, but elects in the end not to bet. He would have won, Cloris rightly remarks when the coin comes up heads. This conditional is of course subjunctive. But speculation before the toss that he will win if he bets on heads is vindicated too, as is a later past-tense indicative to the effect that Boris did win if he, despite appearances, did bet on heads. Or imagine, as Edgington does, that Alice is taking a cab to the airport. 1Versions of this material were presented at NYU (the New York Institute of Philosophy Lectures), Yale University, UT Austin, the University of Amsterdam, Cambridge University, University of Massachusetts at Amherst, Ohio State University, and the Center for Study of Logic, Language, and Information in Amsterdam. Audiences at these places provided extremely useful feedback. Discussions with Justin Khoo about his (super interesting) manuscript "Remainders and Conditionals" got me started on the project. I owe particular thanks to Robert van Rooij, Frank Veltman, Kit Fine, Floris Roelofsen, Ivano Ciardelli, Jeroen Groenendijk, Stephen Pastan, Eileen Wagner, Bill Noble, Katharina Felka, Ken Gemes, Josh Dever and Daniel Rothschild. 2"We can [oftentimes] identify evidential relevance [of the antececedent to the consequent] with beliefs about connections between the facts" ([Stalnaker(1984)]). 3"The man in the street usually says that ... the [indicative] statement is plainly false because the mushrooms were actually non-poisonous" ([Adams(2005)], 7). 1 The driver has a premonition, which he keeps to himself, that she will die if she gets on that plane. Alice arrives too late to be boarded and the plane goes down without her. The cabbie admittedly didn't know that things would play out as he imagined. He was surely right, though, when he (however irresponsibly) said to himself that Alice will die if she gets on the plane.4 He is also right if he judges later, not having heard of the crash, that Alice did die if she did get on the plane. Not all conditionals are as factual-seeming as these ones. But wholly epistemic conditionals are not so common, either. Stalnaker suggests the following: a reliable oracle tells you that all the sentences on a certain whiteboard agree in truth-value. As it happens the first sentence is Goats eat cans, and the fifth is Fish never sleep. You conclude that if goats eat cans, then fish never sleep. The plausibility of this conditional is entirely information-based. Between the states of affairs themselves, there are no useful relations whatever. There are cases and cases, then. Epistemic and factual are not sui generis categories, one assumes, but variations on a theme. What is the theme?5 2 Subjective/Objective Conditionals are evaluated in light of bodies of information, according to the subjectivist-that the coin came up heads, that the mushroom was poisonous, that the sentences agree in truth-value. They look to the properties of this information for the explanation of relative objectivity Is the information true, or only believed to be? Does it stand out in itself or does it merely reflect our epistemic position? Does it explain why A&¬C must fail, as the coin coming up heads explains why Boris cannot both bet on heads and lose? Or is it only evidence that the combination doesn't obtain, as Goats eat cans and Fish never sleep appearing together on that whiteboard suggests, without shedding any light on the matter, that Goats eat cans and Fish sometimes sleep are not true together. These are some of the issues Stalnaker seems to be raising when he says, If we could filter out those aspects of our epistemic situation which derive more from our parochial perspective and less from the way we take the world to be, we might be able to explain the acceptance of conditional propositions in terms of [the indicatives] that would be acceptable in ideal contexts which abstract away from these aspects ([Stalnaker(1984)]). Stalnaker speaks of "filtering out aspects of our epistemic situation deriving from our parochial perspective." This idea has a history in philosophy. It calls to mind Bernard Williams's discussion (in [Williams(2005)]) of the absolute conception of reality. To conceive the world absolutely, Williams says, is to conceive it in a way that does not cater to any particular sensibility or point of view. There is no room for sweetness in the absolute conception, because (i) sweetness reveals itself only to creatures with a certain sensory apparatus, and (ii) it owes as much to the state of that apparatus as to the external world.6 Geometrical properties do better in these respects. Cylindrical pillars may seem to bulge slightly, and railway tracks may seem to converge in the distance, but we know how to control for these factors to arrive at a shared assessment of the objects' actual spatial features. Are conditional judgments like judgments of taste, or of shape? One can imagine arguments on both sides. Favoring the first hypothesis, perhaps, is that (i) if-appearances present themselves only to creatures with conditional credences, and (ii) they reflect those credences as much as goings-on in the outside world. Favoring the second is that conditional credences, like shape-appearances, are educable and apt to become less idiosyncratic as they become better informed. An up or down answer to this sort of question is not to be expected. Some conditionals are relatively objective; some are more parochial. The most we can hope for is to get them under the same theoretical umbrella. There are two obvious things to try if the goal is to make X and Y look like variations on the same theme: start with X and try to make it more Y -like, or vice versa. The options in this case are to start with subjective/epistemic conditionals and make them less parochial, or to start with absolute conditionals and try to work some perspective into them. The first strategy has the advantage that we have some idea how to do it. Epistemic conditionals are everywhere, and one sees more or less how "beliefs about connections between the facts" can come to play a larger role in the information states supporting them. One has by contrast little idea of what an absolute conditional is supposed to look like. I want to try nevertheless the strategy of reaching first for the absolute end of the stick. The next section tries to explain why.7 4"Someone who did make the corresponding forward-looking indicative conditional judgements in advance, however unreasonably, could claim to have been right, when the coin did land heads, and the plane did crash" ([Edgington(2013)]). 5Precis: Information-based conditionals become more factual according to the qualities of the information. This is the usual way to bring the two types of conditional together. It is not the only way, however, nor is it clearly the best. 6[Williams(2005)], 49-50. 7Precis of the next section: If conditionals are information-based, what kind of information I is an absolute conditional based on? I should be neutral, other things equal, on whether A, and also on whether C? That still leaves lots of options, obviously. Would it help for I to be maximal among information-states neutral on A and C? Not much; there are are lots of option here too. There is no such thing as the maximal subtheory T of A&X&C that is neutral on A and C. Shall it contain contain X≡C or X∨C? One or the other, by maximality, but not both, for the two together imply C. So we are left again with a choice. The only real option, it seems, is for absolute conditionals to be based on NO information. 2 3 Absoluteness To rise above our parochial perspective, we might try embellishing that perspective so that proprietary information gets lost in the crowd. A→C will be true, on this approach, just if it is acceptable "to a perfectly rational thinker with unlimited mental powers and knowledge of all the facts."8 A→C will be acceptable to this thinker if and only if he would come to accept the proposition expressed by C if he were to add the (hypothetical) belief that A to a non-question-begging but otherwise unrestricted stock of true beliefs and make whatever minimal adjustments are necessary for the sake of consistency (without sacrificing the belief that A in the process) (186) What counts as a "non-question-begging but otherwise unrestricted stock of true beliefs"? Our thinker will be guided by "general beliefs...central to our conception of the world" and "beliefs concerning 'relevant conditions'," but not "beliefs concerning the truth-values of A and C, or other beliefs which depend on them or on which they depend," except when one believes A to be true and C false (203). This is more of a wish list, though, than a plan. How are these judgments to be made (e.g. of "centrality" and "relevant conditions") if not on the basis of restricted world knowledge? The strategy that embellishes our epistemic situation - that gives us knowledge of "all the facts," except facts too caught up with whether A and whether C " - cannot really be carried out, because "too caught up with" has no clear non-epistemic meaning. The only other strategy I can think of is to impoverish our epistemic perspective by allowing us no factual information whatever. This is one of the attractions of the material conditional reading, I take it. One can evaluate A⊃C just on the basis of A and C ; nothing need be known about how A connects up with C. Which is of course also the problem with the material conditional reading. As already mentioned, we would like if possible to bracket the truth-values of A and C, concerning ourselves rather with the relation between them-how A connects up with C. This makes little sense if A→C is truth-functional. Why would we want to bracket the truth-values of A and C if they are the sole determinants of its truth-value? Now we seem to have painted ourselves into a corner. A's relation to C is an empirical matter, and empirical matters were to be avoided by making A→C truth-functional. This line of thought rests on an confusion, however. How we determine A→C 's truth-value is one thing; how the world determines it is another. Absoluteness is threatened only if the second is not objective. The problem with A⊃C is not that, but that the world decides its truth-value the wrong way, focussing on A and C rather than the relations between them. Do the relations obtain? That is of course an empirical matter. Our concern is with what the relations are on which the truth of A→C depends. It's the identity of those relations that has to be kept non-empirical. Truth-functionality is certainly one way to arrange that A→C 's demands on a world can be identified non-empirically. But it is not the only way. It would be enough for absoluteness if A→C were truth-condition-functional, that is, if what A→C asked of a world were a function of what A and C asked of it. We know in fact of conditionals like this. If p and q are thoroughly independent, then what p→(p&q) asks of a world is that q should hold in it. I myself am inclined to read If you're tired, that makes two of us so that its truth-conditions are precisely I am tired. There may be other ways of understanding it as well; your tiredness could function as evidence that I am tired. But that is OK with the absolutist, indeed, it is to be expected. Absolute conditionals are meant to be a specially perspicuous subclass of indicative conditionals. Other conditionals are obtainable, we hope, by relaxing the conditions defining that subclass.9 4 Methodology How will these conditions be found? Absolute indicative are defined so far by a goal, not a set of truth-conditions. Is that a problem? I like to think of it as an opportunity. We are trying to work out how absolute indicatives, if there are going to be any, will have to behave. A→C is not being read absolutely, if it asks different things of a world depending on factors that ought not to matter. Suppose a parameter X can be identified that reflects something parochial: speaker information, say, or speaker preferences, or how the speaker "groks" the conditional. And suppose that A→C appears to ask different things of a world, depending on the value X takes. Then, since that kind of dependence is not allowed when A→C is read absolutely, the sought after reading will have to either ignore X or freeze it at some canonical value. An example might be this. Speakers have a certain amount of discretion about how much weight to lay on (i) C 's stand-alone plausibility, as against (ii) the comparative plausibility of C given A and on ¬A. Sometimes A is expected to bear more favorably on C than does ¬A. Other times it's allowed that C holds despite A or regardless of whether A. (This may or may not be marked linguistically, as in C even if A or C whether or not A.) Conditionals read the second way-socalled non-interference conditionals-owe their plausibility almost entirely to C. All we learn about A is that it doesn't pull 8[Pendlebury(1989)], 185. 9Precis of the next section: a method is sketched for identifying the conditions that A→C will have to satisfy, if it wants to be absolute. 3 the rug out from under C, and sometimes not even that. There are clearly a range of possibilities here, according to how A's bearing on C is traded off against C 's stand-alone plausibility. Suppose, for instance, that Donald Trump is reliably wrong on foreign policy matters. He may say, for instance, that any idiot could have negotiated a better deal with Iran than Obama did, when in truth (let us suppose) Obama got the best possible deal. Consider this conditional: Obama got the best deal, if Trump says so. I feel myself pulled two ways when I try to evaluate it. Sometimes I want to read it as a non-interference conditional; it is true simply because Obama got the best deal, whatever Trump may say. A reading that takes the antecedent more seriously will have it turning in part on whether Trump speaks the truth on matters like the Iran deal. Then it is apt to strike me as false; Trump is so anti-reliable on these matters that Obama probably did not get the best deal, if Trump thinks he did. This is the kind of interpretive discretion that will have to be blocked on an absolute reading. One approach would be to lean exceedingly far in the direction of non-interference conditionals. By this I mean that we could ignore A entirely, and treat A→C as just a long-winded way of saying that C. If our one and only goal were to avoid awkward tradeoffs between A's relevance to C and C 's stand-alone plausibility, confining our attention to C would accomplish that. Imagine though that we wanted the option of taking A seriously, as surely we do. Then the issue becomes how seriously to take it. Again we turn to absoluteness for guidance. The only non-discretionary response to How important is A's bearing on C, as against C itself? is this: we should ignore C 's antecedent plausibility, and let the issue turn entirely on the link (or not) between A and C -in this case on whether Trump is right or wrong about Iran. Assuming again that he is wrong, the conditional strikes us as false. Consider next If Trump says that Obama got the best deal, he is right. This time the missing link is that Obama got the best deal, and the conditional seems much more plausible.10 Plan of the paper: The problem of absolute "if" is reviewed in the next section. Absolute conditionals will have to be incremental, I claim. Their truth-value turns on that of C 's surplus content with respect to A, written ∆AC . Section 6 pits incremental conditionals against Lewis's Triviality Proof. Section 7 looks, through the lens of the Ramsey Test, at the state of mind underlying acceptance of indicative conditionals generally (it can be skipped without loss of continuity) Section 8 reframes the Triviality question in terms of Kauffman's distinction between global and local styles of interpretation. Section 9 considers the idea of surplus content in its own right. A connective ∼ is introduced-something like the inverse of conjunction- such that C∼A is the remainder when A is subtracted from C. Affinities between remainders and conditionals are noted in section 10. Formal details are supplied in section 11 and an Appendix. Section 12 introduces a notion of "conservativeness" borrowed from generalized quantifier theory. Absolute conditionals are not conservative, and, it is argued, marketplace conditionals are not conservative either, as existing theories say they should be. The remaining sections explore the prospects for understanding marketplace conditionals on the model of incremental conditionals.11 5 Surplus Content The notion of surplus content originates in 20th-century philosophy of science. Goodman wanted to distinguish inductive confirmation--the way This raven is black confirms All ravens are black-from "mere content-cutting"-the way The coin came up heads confirms It always comes up heads. The difference is supposed to be that This raven is black confirms the surplus content of All ravens are black, viz. All other ravens are black, while The coin came up heads does not confirm It comes up heads on every other toss = the surplus content of It always comes up heads. Popper and Miller use the surplus content idea against Goodman in a 1983 letter to Nature. If inductive evidence has got to support surplus content, then there is no such thing, they maintain. Their argument has two premises. First, H's surplus content relative to E is E⊃H.12 Second, evidential support is to be understood as positive probabilistic relevance. Far from supporting E⊃H. E bears negatively on it, since E⊃H is implied by ¬E. If the absolute conditional is to be understood in terms of surplus content, and the surplus content of C over A is A⊃C, then A→C read abolutely is just A⊃C. Is this a plausible result? What do we think of ⊃ as a candidate for the role of absolute "if"? A point in its favor is that A⊃C takes the shortest path from A to C, in this sense: it is the weakest statement that can be plugged in for R in the enthymeme A, R ∴ C to obtain a valid argument. But, as already noted, A→C should 10It might be objected that C 's truth-value is sometimes dispositive, notably in A-worlds. How on the proposed policy is it to be explained why A&C entails A→C, and A&¬C entails ¬(A→C)? This assumes, however, that the only thing distinguishing A&C -worlds from A&¬C is whether C. Another thing distinguishing them, though, is whether the link obtains, in this case whether Trump is right or wrong about Iran. If the link L between A and C is such that L holds in A-worlds where C holds, and L fails in A-worlds where C fails, then C 's antecedent plausibility is safely ignored. Such an L will be introduced below. 11Precis of next section: Two notions of surplus content are distinguished, associated respectively with Popper and Goodman. Popper's notion, A⊃C, takes the shortest path from A to C, if longer means stronger. Goodman's pick takes the shortest path if longer means more circuitous. Goodman's approach is more plausible but the notion of shortest-qua-directest (straightest?) logical path lacks a general definition. An initial suggestion is made as to how straightest logical paths may be understood. 12Hempel seems to be have this conception of surplus content in mind when he says that E⊃H "has no content in common with E," since its disjunction with E is a logical truth (Hempel 1960, 465). 4 concern itself with the relations between A and C, rather than the two taken separately. It should not be true just because A is false, or because C is true.13 Now, it is not as though A⊃C can only hold because A is false or C is true;. Goats eat cans ⊃ They eat cans and tinfoil might surely be true because goats eat tinfoil. Goats eating tinfoil is a third sort of truthmaker for the material conditional, over and above their not eating cans and their eating cans and tinfoil. The problem with A⊃C is not that it can't hold for the right reasons, but that it can hold for the wrong reasons. A⊃C will in fact be playing a crucial role. Absolute A→C 's reasons for being true will have to imply truthmakers for A⊃C, or we lose modus ponens. Its reasons for being true may indeed be truthmakers for A⊃C. It is just that they'll have to be a certain kind of truthmaker, the kind that targets the space between A and C, rather than either taken individually. An analogy may be helpful. Suppose we were looking, not for C 's added value relative to A, but 33's added value relative to 8. This added value is 33 8 = 25. Now, 25 has no more to do with 33, or with 8, than with any other numbers. That the numerals 33 and 8 figure in a characterization of 25 does not mean that the numbers 33 and 8 figure in the number 25. What goes for 33 and 8 goes for C and A as well. C and A do of course figure in our characterization C 's surplus content over A. But the contents of those sentences do not figure in the surplus content of one over the other, any more than 33 and 8 figure in 25. How can the shortest path from A to C not be A⊃C, if every other enthymeme-completer is stronger? Stronger does not mean longer. A longer path from A to C is one that is more indirect or circuitous. The path from p to p&q than which every other is stronger is ¬p∨(p&q), The path from p to p&q than which every other is longer is q. Likewise the shortest path from a generalization's observed instances to the generalization as a whole is by way of its other instances. This is presumably what Goodman had in mind, since he says that a black raven Rudy reflects favorably, not on All ravens are black if Rudy is black, but All ravens other than Rudy are black. What would a shortest-path conditional look like in general? This is what we are coming to, but we can at least say the following. It should be true for reasons that block A&¬C "as such." Anticipating a bit, let a targeted truth maker for A⊃C be a fact that's consistent with A and that "makes the most of A," getting all the help from it that it can. A→C is true (read absolutely) where a targeted truthmaker obtains for A⊃C, and because of that targeted truthmaker. (It is false where and because a targeted truthmaker obtains for A⊃¬C.) Absolute A→C expresses the (possibly partial) proposition whose truth goes with the disjunction of targeted truthmakers for A⊃C, and whose falsity goes with the disjunction of targeted truthmakers for A⊃¬C.14 To explain this properly, especially the bit about making the most of A, is not trivial (see section 11). But it is easier than explaining what perfect thinkers are and what constitutes a "minimal A-preserving adjustment to a non-question-begging but otherwise unrestricted stock of true beliefs." There is even a literature on the topic, growing out of Wittgenstein's question about what is left over when arm-rising is subtracted from arm-raising.15 The idea, though, is that R takes the shortest/straightest path from A to C just if R is the enthymeme-completer that takes maximal advantage of A.16 It is this shortest-path R that an un-parochial observer looking for A→C 's factual basis will be drawn to.17 6 Triviality Remainders are propositional and fact-stating. Indicative conditionals are supposedly neither, due to the triviality results. Let's review the first and simplest of these, due to Lewis. A→C 's propositional probability would have to be governed by standard probabilistic laws. π(A→C ) must by the Law of Total Probability be (1) π(A→C |B)×π(B) + π(A→C |¬B)×π(¬B).18 π(A→C |X ), we assume following Adams, ought to be π(C |A&X ).19 So π(A→C ) can also be written as follows: (2) π(C |A&B)×π(B) + π(C |A&¬B)×π(¬B) 13"Just because A is false" is a nod in the direction of truthmakers. Truthmakers for A→C should ideally be neutral on whether A. 14If a world contains targeted truthmakers both for A⊃C and A⊃¬C, then depending on the application we may want to regard A→C either as true and false, or neither true nor false. 15[Hudson(1975)], [Jaeger(1976)], [Humberstone(1981)],[Fuhrmann(1996)], [Fuhrmann(1999)], [Humberstone(2000)]. 16[Yablo(2014)] 17Precis of the next section: Incremental conditionals have truth-conditions. But indicative conditionals supposedly do not, due to certain well-known triviality results. The results assume adams' thesis-π(A→C) = π(C|A)-which implies independence: A→C is probabilistically independent of A. independence is prima facie puzzling; why can't an oracle tell us that A and A→C stand or fall together? When it comes to incremental conditionals, however, it's absurd. A→A&X on an incremental interpretation is generally just X ; for A to be independent of A→A&X thus means that A is independent of X. But X could be almost anything! independence has the result that everything is probabilistically independent of virtually everything else. 18Assuming π(B) is not 0 or 1. 19[Adams(1975)] 5 The problem is that this yields different results, some of them clearly unwanted, for different values of B. Lewis, setting B equal to C, gets that π(A→C ) is (3) π(C |A&C )×π(C ) + π(C |A&¬C )×π(¬C ) which, since π(C |A&C ) = 1 and π(C |A&¬C ) = 0, simplifies to 1 × π(C ) + 0 × π(¬C ) = π(C ). A→C is equal in probability to C ? How could that be? What is the antecedent doing there, if it is just going to be tossed out?20 Other choices of B lead to equally strange results. If B is A&C, we get that π(A→C ) = π(A&C ). If B is C⊃A, we get that π(A→C ) = π(A⊃C ). If B is A∨B, π(A→C ) becomes π(C |A)×π(A∨C ) + undefined. If B is A≡B, we get π(C≡A) + undefined.21 This is not as big a problem as it once seemed. Opinion has been converging anyway on a certain response to Lewis's argument, one that seems particularly compelling from an incrementalist perspective. The response starts with a number of seeming counterexamples to π(A→C ) = π(C |A) (the thesis, a special case of (1)). Here is a stripped-down version of Vann McGee's game-show case.22 You are a contestant on "To Tell the Truth." Your job is to identify the panelists, all in disguise, on the basis of how they respond to cleverly posed questions. Convinced as you are that #1 is Holmes, how likely is it that if #1 says the killer was Mrs Hudson, the killer was Mrs Hudson? Highly likely, you might think, since Holmes is hardly ever mistaken. For that very reason, #1 is not expected to accuse Mrs. Hudson in the first place. (Mrs Hudson is the kindly landlady.) Indeed if he were to accuse her, rather than accept that she was guilty, we would reconsider our view of #1 as Holmes. The probability of If #1 says the killer was Mrs Hudson, the killer was Mrs Hudson is thus lower on the assumption of its antecedent than considered alone. The thesis can't allow this, for it entails the principle of 23 independence π(A→C |A) = π(A→C ) independence seems clearly wrong in the game show case; 'Mrs Hudson' → Mrs Hudson is less probable, we have said, conditional on its antecedent, than absolutely.24 Now, whatever one thinks of this particular example (there will be others below), the idea of A→C having to be probabilistically independent of A ought to strike us as surprising. Is there precedent for this sort of thing elsewhere in logic/epistemology, where sentences X and Y are prevented just by their form from bearing evidentially on each another? I know only of precedent for the opposite. A is favorably relevant to A&C just because it recurs as a conjunct. A is relevant to A∨C because it recurs as a disjunct. One might then expect A to be relevant to A→C, since A recurs as an antecedent. This doesn't follow, let's agree. But look what we are being asked to accept: that the opposite follows. What could possibly prevent an oracle from informing us that that A→C is (im)probable given A? (Maybe they are both on the whiteboard that we've been told contains statements all of the same truth-value.) If independence holds only sometimes, that points in the direction of incrementalism, for the nature of the mistake comes out particularly clearly on an incrementalist perspective. A→C read incrementally is just a completely new proposition, whose probabilistic relations with A are as may be. If C is A&Z, for instance, then A→C is A→(A&Z ), which is equivalent for present purposes to Z. independence says that A and Z are probabilistically independent. But A and Z could be anything; they were chosen more or less at random. If nothing is ever favorably relevant to anything, A→C 's failure to express a proposition is the least of our problems.2526 20We'll be charging current theories with a related, though lesser, offense, viz. tossing out the information encoded in C 's behavior away from the antecedent. 21One natural thought at this point is that truth-functional combinations of A and C are bad things to partition on when assigning a probability to A→C, for they in different ways screen off or otherwise distort A's influence on C ; more on this below. 22Adams, Stalnaker, and Pollock had cases like these, but may not have attached the same importance to them. An example of Kauffman's is given later. 23Proof of independence from the equation: π(A→C) = π(C|A) = π(C|A&A) = π(A→C |A). 24This is an indicative analogue of the so-called conditional fallacy, discussed in the next section. 25Another proof of triviality, due to Richard Bradley, proceeds from a weaker assumption, Preservation: If π(A) > 0 and π(C) = 0, then π(A→C) = 0. This again is not plausible if π(A→C) is the probability of C∼A. Suppose π(Kennedy was killed) = 1 while π(Oswald killed Kennedy) = .9. Preservation says π(If Oswald didn't kill Kennedy, he wasn't killed) = 0. But If Oswald didn't kill Kennedy, he wasn't killed read incrementally is No one other than Oswald killed him. This is highly probable on anyone's account; it is 81% likely if we are 90% confident there weren't two killers. 26Precis of the next section: If conditional probabilities determine update counterfactuals (as they should for the Bayesian) while the probability of the conditional goes with update dispositions (as maintained by Stalnaker among others) then thesis violations are inevitable, due to what used to be called the Conditional Fallacy. A finkish disposition-one whose basis is undermined, or implanted, by its trigger-by definition does not align with the corresponding counterfactual. The disposition constituting acceptance of an incremental conditional is finkish if learning A undermines belief in the surplus content. The finkishness of that disposition explains how acceptance of A→C can happily coexist with low conditional probability of C on A. 6 7 Independence and Acceptance Consider what is involved in accepting a conditional, absolute or otherwise. The usual story, or story-outline, is due to Ramsey. "If two people are arguing 'If A will C ?' and both are in doubt as to A, they are adding A hypothetically to their stock of knowledge and arguing on that basis about Q ... We can say that they are fixing their degrees of belief in C given A." (Ramsey 1931, p. 249). Ramsey attempts to indicate here the state or condition of mind that goes with accepting a conditional more or less strongly. What is that state of mind, in his view? The quote doesn't pin it down exactly, but the main candidates appear to be (DIS) the strength of one's disposition to believe C, on becoming certain of A. (CFL) the confidence one would have in Q, were one to become certain of A. (CCR) one's conditional credence in C given A (that is, π(C |A)). The last of these, (CCR), is known as Adams' Thesis, or just27 the thesis π(A→C ) = π(C |A). The first is dispositionalism, as sometimes advocated by Stalnaker: To be disposed to accept C on learning A is to accept C conditionally on A, or to accept that if A, then B ([Stalnaker(1984)], 103) That leaves (CFL), or counterfactualism. Counterfactualism is perhaps closest to the text. "Adding A hypothetically to one's stock of knowledge" sounds a lot like supposing in a counterfactual spirit that A is known. "Arguing on that basis about C " sounds like arguing about how much confidence, on that counterfactual supposition, to repose in C. Now, for reasons indicated in the last section, the thesis has been called into question lately. An example of Kauffman's is this. Suppose that I am to draw a ball from one of two urns, each with a hundred balls in it. The question is, will the ball be shiny, if it is red? Urn 1 contains 10 red balls, all of which are shiny, and 90 green balls; Urn 2 contains 90 red balls, all of which are dull, and 10 green balls. The urn before me, the one I am reaching into, is 90% likely to be Urn 1. How confident am I that If the ball is red, it is shiny? Very confident, because the ball is very likely from Urn 1, and all the red balls in Urn 1 are shiny. The probability of the ball being shiny, conditional on its being red, is π(S|R) = π(S|R&U1)×π(U1|R) + π(S|R&U2)×π(U2|R) Given that 100% of the red balls in Urn 1 are shiny, π(S|R&U1) = 1. π(S|R&U2) by similar reasoning is 0. We know by Bayes' Theorem that π(U1|R) = π(R|U1)×π(U1)/π(R) = (1/10 × 9/10) ÷ 18/100 = 50%. π(S|R) = π(S|R&U1)×π(U1|R) + π(S|R&U2)×π(U2|R) is (1 × 50%) + (0 × 50%) = 50% If the probability of drawing a shiny ball, conditional on its being red, is 50%, then π(S|R) is a far sight short of π(R→S), which was found to be 90%. The thesis thus appears to be off by a factor of almost half in equating π(R→S) with π(S|R). What about (CFL)? Our agent is a Bayesian, let's assume, who updates by conditional credence.28 Her confidence in S, when she learns that R, is her prior confidence in S given R. But then we run into the same sort of problem as before. The confidence she would have in the ball's being shiny, on learning that it is red, is low, for she found it antecedently unlikely that a ball would be shiny, given that it was red. But although the counterfactual Had she learned the ball was red, she would have concluded it was shiny is mistaken, she is by hypothesis 90% sure that the (still undrawn) ball is shiny if red. If the counterfactual account falls to Kauffman examples, then the dispositional acount might seem vulnerable too- for the disposition on x 's part to do such and such when suitably triggered lines up in many cases with the truth of a corresponding counterfactual: x would do such and such were it triggered. Stalnaker however explicitly distances himself from this idea. (DIS) is not meant to imply "that the rational agent who accepts If A then C will always come to accept C on learning A" (104)-for the learning experience may erode her confidence in the conditional. There are at least two ways this could come about. One may wind up learning more than A, where the totality of what is learnt argues against C. I don't know about it, if my friend is lying to me; he is pretty convincing. Would I accept the consequent-that I don't know about his lying- on learning (just) that he is lying to me? The question is hard to make sense of since I would almost certainly learn, in addition, that I was learning it, which ruins the experiment. 27Hall and Hajek call it the Conditional Construal of Conditional Probability (CCCP) ([Hájek and Hall(1994)]). Edgington speaks of the Equation, because it equates two probabilities, the monadic probability of a conditional and the conditional probability of the consequent on the antecedent. 28πE(A) = π(A|E). 7 Worse still, the new knowledge may erode one's confidence in the conditional even if one learns exactly A. I am thinking here of what used to be known as the conditional fallacy.29 An undropped vase is still fragile, even if its guardian angel has resolve to surround it in bubble wrap, or tinker with its molecular structure, should it be dropped. The disposition would not be destroyed by this decision but only masked ; one should not expect a disposition to manifest itself counterfactually if the trigger would undermine its basis. Going in the other direction, an undropped styrofoam cup does not become fragile when an avenging angel decides that she would fragilize its underlying structure were it dropped. The disposition in this case is mimicked but not really there. Dispositions to believe are like any other dispositions in this respect. I would indeed not conclude the ball was shiny, on learning that it was red. But that does not show I lack the disposition to pronounce it shiny if it is red. The disposition would not be manifested, because it would not survive that discovery.30 This is a mimicking-style counterexample to the thesis, and to counterfactualism, but not as far as I can see to dispositionalism. How far are these subtleties reflected in the probabilities? That I would expect the ball to be dull is predicted by my conditional probabilities, not the probability I assign to If the ball is red, it will be dull. I assign a low probability to that conditional and lack the update disposition as well, even if I would acquire it when I learned the antecedent. That I would not expect the ball to be shiny is again predicted by my conditional probabilities, not the probability I assign to If the ball red, then it is shiny. I assign a high probability to that conditional and possess the update disposition as well, even if I would lose it when I learned the antecedent. The outcome in either case is this, where α>γ means that the agent would believe C if she were to learn that A, and αγ means that she is disposed to believe C on learning that A: we shouldn't expect the thesis to hold in full generality if π(C |A) lines up with α > γ (an update counterfactual), while π(A→C ) lines up with α γ (an update disposition), for counterfactuals don't always line up with dispositions. This is shown in the diagram by the "¬iff" between α>γ and αγ on the left, and the corresponding "¬iff" on the right between π(C |A)≈1 and π(A→C )≈1. ↵>  ⇡(C |A) ⇡ 1 ↵   ⇡(A!C ) ⇡ 1 i↵ ¬i↵ i↵ ¬i↵ 1 Figure 1: Conditionals' probabilities stand to conditional probabilities as dispositions stand to counterfactuals When they do line up-when the antecedent is probabilistically independent of the facts thought to underly the disposition- the thesis holds. It fails when, as in the case of If red then shiny, the antecedent disconfirms the underlying assumptions, or, as in the case of If red then dull, it makes them for the first time believable. Now, the truth of A→C, read absolutely, is grounded in the fact that ∆AC .31 Suppose with Stalnaker that acceptance of A→C is something like a disposition to believe that C on acquiring the belief that A. It stands to reason that this should be a grounded disposition underwritten by belief in ∆AC . This makes for a neat overall package in which the fact making A→C true = the fact belief in which makes A→C acceptable. fact that  AC truth of A ! C belief that  AC accepting A ! C grounds tracks grounds tracks 1 Figure 2: A fact ∆ grounds the truth of A→C ; belief in ∆ grounds (the disposition constituting) belief in A→C 29[Shope(1978)] 30Huw Price makes a related suggestion about conditional probability itself, based on a remark of Ramsey's: "The degree of belief in p given q is not the same as the degree to which a subject would believe p, if he believed q for certain; for knowledge of q might for psychological reasons profoundly alter his whole system of beliefs" ([Ramsey(1931)], 180). Ramsey is talking about irrational update behavior, however. Conditional probability does predict the update behavior of rational Bayesians with respect to p. Update behavior is not predicted, I'm saying, even for the rational Bayesian, by the prior probability of If q then p, because there is nothing irrational in losing confidence in If q then p on learning that p. 31∆AC is a different and possibly more suggestive notation for C∼A, the "difference" between A and C. 8 The parallel depicted here, between the grounding of conditional truths in ∆AC and of the corresponding acceptance-states in awareness of ∆AC , has consequences both for the logic of absolute conditionals and their epistemology. The one we'll eventually be focusing on (section 12) is that A→C cannot be settled just by asking how many A-worlds verify C ; for ∆AC cannot be settled in that fashion.32 8 Local/Global The probability of A→C is not in general the probability of C conditional on A. Examples have been given in the last two sections. The probability of #1 says Mrs Hudson did it → She did it is a great deal higher than the probability of She did it conditional on #1 says she did it is low. Where do we get our confidence in #1 says she did it → She did it if not from the conditional probability? We seem to reason like this (following [Kaufmann(2004)]): There are two possible states of the world: #1 is Holmes or #1 is Watson. Since the probability that Mrs Hudson did it conditional on #1 saying so is quite different, depending on who #1 is, we adopt the following strategy. Step One: Divide logical space between the #1 is Holmes scenario and the #1 is Watson scenario. Step Two: Estimate the probability of Mrs Hudson did it conditional on #1 accused her once for each scenario. Step Three: Obtain the overall conditional probability as a function of these two local conditional probabilities. How likely is it that Mrs Hudson did it, conditional on Holmes accusing her? Highly likely, say 99%. How likely is it that Mrs Hudson did it conditional on Watson accusing her? Not likely at all, maybe 2%. π(Mrs Hudson did it | #1 says it was Mrs Hudson & #1 is Holmes) = .99 π(Mrs Hudson did it | #1 says it was Mrs Hudson & #1 is Watson) = .02 So should we just take the average of .99 and .02? Certainly not. That would be to treat both hypotheses about the identity of #1 as equally probable, when the Holmes hypothesis is much better supported. We want rather the weighted average, where .99 and .02 are multiplied respectively by the prior probability of # 1 is Holmes (= 90%) and that #1 is Watson (= 10%). Our credence that If #1 says it was Mrs Hudson, then it was Mrs Hudson is π(M|'M'&H)×π(H) + π(M|'M'&W )×π(¬H) = (.9×.99) + (.02×.01) = about 90%. This weighted sum, divided between two hypotheses about the (relatively) objective set-up that controls conditional chance, is called the local conditional, after recent work by Kaufmann. The idea behind it-the idea of treating conditionals as rational guesstimates of "real" conditional probability-goes back to earliest days.33 The fact is that we are inclined to speculate on the underlying mechanisms that make C more or less likely given A.34 Let the mechanisms between which we're undecided be the X is, and the set of them (the partition) be X. Then the localized conditional probability is 32Precis of next section: thesis failures can be traced quite often to a nonstandard way of calculating conditional probabilities, recently emphasized by Kauffman. There are two possible states of the world, H and H; C 's probability conditional on A is one thing in the H-region, another in the H-region; we take the average of those two conditional probabilities, weighted by the likelihood respectively of H and H. 33Adams considers a counterfactual analogue in Chapter 4 of The Logic of Conditionals (1970). Here is Edgington's version of Adams' example ([Edgington(2014)]): John has a rare disease. There are two drugs, D and E, which would help him. If he takes just one, he'll get better. If he takes both or neither, he'll get worse (though the harmful effect of taking both is not well known...). Both are in extremely short supply, and it's very unlikely that he'll get either, but it's about 100 times less likely that he'll get E than that he'll get D, and immensely unlikely that he'll get both. I think If John takes E, he'll get better. Now John does get better. Much the most likely explanation is that he got D. So now I think 'If he had taken E he would have got worse.' In thinking that John would have got worse if he'd taken E, I am disagreeing, and in a foreseeable way, with my earlier judgment that If John takes E, he'll get better. Reflection suggests that the earlier judgment was mistaken, or that it has a reading on which it was it was mistaken. But that original judgment is what you'd expect given my prior conditional probabilities. The prior probability of John getting better, given that he takes E, is high, say 99%. And yet If John takes E, he'll get better was, given that John does get better, probably in fact false-and we have as yet no explanation of that. Adams suggests one: A rather simple minded generalization of our prior conditional probability representation [π(B|E)] which would accommodate [this] is as follows.... we may suppose that there are mutually exclusive and exhaustive states D and ¬D which are causally independent of whether John takes E,.... In this case the probability of the [conditional] is plausibly given by π(E → B) = π(B|D&E)×π(D) + π(B|¬D&E)×π(¬D) ([Adams(1975)]). The chances are (since John got better) that he has taken D. So the first term in the expression for π(E→B) dominates. π(B|D&E) is low since the two drugs together will make John worse. This is why our confidence in E→B declines after learning that John gets better. The thought more generally is that, rather than using a simple conditional probability π(C |A), we take the weighted sum of conditional probabilities as we vary the underlying probability-determining facts. (Adams only toys with the local rule as he attempts to deal with a certain puzzle. The rule he prefers is the one that figures in the thesis: π(E→B) = π(B|E). 34Skyrms calls these mechanism-driven probabilities "propensities." 9 LOCAL: πX(C |A) = Σi π(C|A&Xi) × π(X i). This agrees in some cases, but not all, with the global conditional probability π(C |A). Standard rules allow us to rewrite that global probability as GLOBAL: π(C |A) = Σi π(C|A&Xi) × π(X i|A). The two formulas come to the same if the antececent A is probabilistically independent of the issue of which mechanism in fact obtains; π(X i|A) is the same as π(X i). This is not so in the Holmes case, because whether #1 accuses Mrs Hudson is not probabilistically independent of whether #1 is Holmes. It is not so with the Urn example, because whether the ball is red is not probabilistically independent of which urn it came from. Or consider an example of Roger White's: If Brad's book (advocating 4-dimensionalism) is correct, then nothing changes. Whether a book is correct is not independent of whether it says that nothing changes, and so not independent of whether If Brad's book is correct, then nothing changes.35 9 Buts Where does this leave us? The plan was to try in an experimental spirit to build our theory around special conditionals we called absolute. What would decide an absolute conditional's truth-value, however? This led to surplus content, otherwise known as what C adds to A. Surplus content, we decided, could be characterized in terms of enthymemes; it's the R that slots perfectly into the enthymeme A, ??? ∴ C. Part of slotting perfectly in is taking full advantage of A. "Takes full advantage" is only a metaphor, however. I propose to turn now to the rest of the title- "ands" and "buts"- in the hope that this may clarify things. The usual view among philosophers is that "but" implies "and," adding to it only some kind of pragmatic hint that the second conjunct is somehow unexpected given the first. This is crude ([Kripke(2011)]), [Toosarvandani(2014)]) but not in a way that matters here. For we are interested in a kind of "but" that does not meet even that minimal requirement-that works in fact against conjunction. You see it in sentences like Every Justice asked a question but Thomas. Every Justice asked a question but Thomas does not seem to be saying that every Justice asked a question and Thomas asked one. One reason to doubt this interpretation is that the conjunct would be redundant; to say that all of the Justices asked questions already implies that Thomas did. Does the speaker perhaps want to drive that implication home? If so then the "but Thomas" achieves the exact opposite of what she wants. It functions intuitively to cancel any implication that Thomas asked a question.36 When "but" is used to carve out exceptions, it expresses something like the inverse of conjunction; see below. Everyone asked a question but Thomas falls short of Everyone asked a question precisely by Thomas answering a question; it was obtained by subtractingThomas asked a question from Everyone did. The claim so far is that buts are, or can be, exception-makers along the lines of "present company excepted" or "leaving aside the typos" or "bracketing any requirements still under review."37 The usual examples involve quantification. But I will understand the class more broadly than this, allowing buts in their exceptive sense to function as main connective. This subsumes the quantificational use for a reason already mentioned. Everyone but Thomas asked a question says that Everyone asked a question, apart for the bit about Thomas asking one. So, we are interested in a logical connective P∼Q whereby, if P implies Q, that implication is cancelled or waived or stripped away.38 We can think of ∼ as undoing the effect of conjunction, or better, as the operator such that conjunction undoes its effect: (P∼Q)&Q is true in the same worlds as P.39 I'm not sure there is a perfectly colloquial way to formulate these kinds of statements, but the following are in the ballpark: P, except maybe not Q, P, apart from the bit about Q, and P, but for Q. 1. Every Justice asked a question, with the possible exception of Thomas. 2. Kennedy was killed by someone other than Oswald, or for that matter by Oswald. 3. The king is in the counting house, though don't hold me to that Nigel guy really being the king. 4. These two triangles are congruent, waiving the requirement of being the same size. 5. Pete won, ignoring the possibility that he folded. 35Precis of next section: Surplus contents have not yet been fully explained. Inspiration is sought in "exceptive constructions" like I agree, but for the bit about China. A binary connective ∼ is introduced on a model of "but for." The new connective is an inverse to conjunction insofar as (X∼Y )&Y is equivalent to X. 36Some may think it adds the opposite implication. I will assume not. Any suggestion that Thomas in fact didn't ask a question will be treated as an implicature. 37[Gajewski(2008)], [Von Fintel(1993)] 38The operator still applies if P does not I'mply Q, only its action cannot be described in this case as cancelling an implication. 39(P&Q)∼Q is going to be trivial if P=Q; it would be P if ∼ just undid the effect of conjunction. 10 What do these statements say? How is the proposition to be identified? An arithmetical analogy may be helpful. Consider the problem of identifying m minus n on the basis of m and n. This is solved, we know, by looking for the unique number r such that adding r to n gets you back to m. An analogous proposal about C∼A would be this: it's the R that has to be added to A to get C. The problem is that there might be any number of Rs such that A and R imply C, ranging from A⊃C to A≡C to C itself. Some of these R's are, once again, better than others. The R we want is the hypothesis that completes the enthymeme A, ??? ∴ C in the A-beholden way. How on this way of doing things are we to find Every Justice asked a question∼Justice Thomas asked a question? First let's set out the relevant enthymeme. Thomas asked a question. ⇒ ??? Every Justice asked a question. To complete this in a maximally A-beholden way, we look for the feature of certain Thomas asked a question-worlds whereby they are Every Justice asked a question-worlds. That feature is, I take it, that every other Justice asked a question in them. Again, what is the shortest path from Nigel is the king to The king is in the counting house? What is going on in certain Nigel is the king -worlds to make them The king is in the counting house-worlds. Nigel is in the counting house in those worlds. The king is in the counting house∼Nigel is the king is therefore Nigel is in the counting house. Let's try finally to strip Pete won of the implication that Pete called. What is going on in certain Pete called-worlds to make them Pete won-worlds? Well, Pete had the better hand in those Pete called-worlds. Pete won∼Pete called is therefore Pete had the better hand. A more precise statement, in terms of truthmakers, is given in section 11.40 10 Affinities Remainders and conditionals have a good deal in common. Remainders are apt indeed to be formulated in conditional terms without anyone noticing or remarking on the fact. Consider the view called if-thenism in the philosophy of mathematics. One explanation I have heard of this view says that a statement like The number of stars is finite is used to express, not its full number-involving content, but what that content adds to the assumption that there are numbers in the first place-presumably that there are finitely many stars. Another says that The number of stars is finite is used to express, not its full literal content, but that content conditioned on the assumption of numbers, viz. that the number of stars is finite, if (or on the assumption that) there are numbers. The label if/thenism puts us in mind of the second, conditional, formulation-which is odd, because the conditional formulation is risky; it leaves room for the (surely unwanted) thought that the existence of numbers might be evidentially relevant to how many stars there are. (An oracle might have told you that more stars means fewer abstract objects, and vice versa.) The worry about evidential relevance strikes us as ridiculous and misconceived.41 That it is ridiculous if the conditional is remainder-like suggests that if/thenists are implicitly understanding If the number of stars exists, then it is finite, as what the consequent adds to the antecedent,, viz.that There are finitely many stars. A second affinity is to do with the phenomenon of non-catastrophic presupposition failure. It has long been recognized that while the falsity of a presupposition P may sometimes make a sentence intuitively unevaluable-as The king of France is bald seems unevaluable-there are plenty of such sentences that strike us as false: The king of France is bald and pigs fly, for instance, and The king of France is sitting in this chair. S will strike us as false, despite the failure of its presupposition P, if S adds something to P-Pigs fly, The chair is not empty-which is independently evaluable and turns out to be false, that is, if S∼P is false. S∼P is a remainder, obviously, but one is strongly tempted to reformulate it as a conditional ([Lasersohn(1993)]): Even if France has a king, still, it is not the case that: He is bald & pigs fly. Even if France has a king, still, he is not sitting in this chair. 40Precis of next section: noindent Remainders C∼A and conditionals have a number of affinities, ranging from the language used to express them to the graphical representations that suggest themselves when we try to calculate their truth-conditions. 41Horgan's counterfactual style of if-thenism runs into an analogous problem ([Horgan(1984)]). Hellman suggests a "non-interference proviso": "we must stipulate from the outset that the only possibilities we entertain in employing the [modal operator] are such as to leave the actual world entirely intact." ([Hellman(1989)], 99). 11 These too are liable to be misunderstood. Imagine someone convinces us, never mind how, that the king of France is a master illusionist who is sitting undetectably in that chair, if he exists. If The king of France is sitting in this chair seemed false before, it continues to now. The appearance of falsity is based on the fact that the chair is empty. Whether it is still empty on the hypothesis that France has a king is irrelevant; all that matters is that the sentence adds something false to its presupposition. This is just to say that the reformulated test is tempting only insofar as we are reading the conditionals incrementally. Clearly it is tempting; so that must be how we are reading them. This kind of rewriteability extends as well to the remainders mentioned above-Every Justice with the possible exception of Thomas asked a question, The king is in the counting house, albeit Nigel might not really be king. The corresponding conditionals are 1. If Thomas asked a question, then all the Justices asked questions. 2. If Oswald didn't shoot Kennedy, then someone else did. 3. If Nigel is the king, then King Nigel is in the counting house. 4. If these triangles are the same size, they're congruent. 5. If Pete called, he won. These conditionals seem in each case to admit a non-evidential interpretation whereby they stand or fall with C∼A. The Oswald conditional is decided by whether Kennedy was killed. The Thomas conditional is decided by whether everyone other than Thomas asked a question. The Nigel conditional is decided by whether Nigel is in the counting house. The Pete conditional is decided by whether Pete had the winning hand. That all the conditionals can be read incrementally is a second point of analogy between "if" and "but." A further point of analogy between A→C and C∼A is that they lend themselves to a similar sort of graphical representation (see below). Both require us to extrapolate in some way from the distribution of C in "home"-worlds (A-worlds) to its likely distribution in "away"-worlds (¬A-worlds). They both agree with C in the home-region. The challenge with conditionals and remainders alike is to find a way to model their away behavior on their behavior at home. "home" "away" C A A→C C ~ A Doubts are often expressed about the possibility of carrying out this extrapolation in a principled way. What are we to think of If Cora is in France, she's in Lyon in a world where Cora is in Antarctica?42 How is one to extend the principle behind Cora's car is dark red to worlds where the car is green? What does I raised my arm ask of worlds where my arm does not go up? To say nothing of extending It's five o'clock here in Cambridge to worlds where "here" is not Cambridge!43 To gather the structural affinities between A→C and C∼A into one place, Both are implied by A&C. Both are inconsistent with A&¬C.44 Both attempt to extend C from "home" (the A-region) to "away" (the ¬A-region). Both extensions are apt to be problematic. Neither extension is inherently problematic.45 42Skepticism about the away-behavior of conditionals is expressed in [Quine(1982)], [Belnap(1970)], [McDermott(1996)], and [Cantwell(2008)]. 43See, for doubts about the away behavior of remainders, [Jaeger(1973)] and [Jackson(1977)]. 44Both are intermediate in strength, then, between A&C and A⊃C. 45A&B∼A is in many cases just B. 12 These analogies are what license(d) us in postulating a conditional connective →α understood so that A→αC holds (i) in the same worlds as C∼A, (ii) on account of their being C∼A-worlds. Given the connection with ∼, a better notation for this conditional might be ;; ; is what I will use henceforth. Pete called;He won holds where, and because, Pete had the better hand. Nigel is the king;The king is in the counting house holds where Nigel is in the counting house, because of his being there. The claim is that we have in A;C a prime contender for the role of absolute conditional. A further point in support of this is that if one looks for an extrapolation strategy suited to C∼A, one winds up with something like the suggestion above that A→C holds absolutely in w if a fact X obtains in w that explains why w would be C there, given that it is A. This is explained in the next section.46 11 Extrapolation How in graphical terms are we to solve "C∼A = R" for R? R may be understood informally as the result of extending C 's behavior at home, where A holds, to the away region, where A is false. From this it seems that R's truth-value in the away region ought to be controlled by the same sorts of factors as distinguish A&C -worlds from A&¬C worlds. The proposal is again going to involve truthmakers for A⊃C. But we do it more carefully now, with an eye especially to making the most of A. Looking back at the home vs away diagram, we can see that extrapolation is going to require two types of rule. "Home" rules speak to R's behavior at home, that is, within the A-region. An R that continues C beyond that region should at a minimum follow C 's lead within it. "Away" rules speak to R's behavior outside the A-region. If R is to divide up away-worlds on the same principle as at home, its away behavior should agree in some still undetermined way with its behavior at home. A second, cross-cutting, distinction is between "classifying" conditions, which are to do with whether R is true in a given world, and "rationalizing" conditions are to do with how and why R is true in a world. This gives us four types of condition overall-home-classifying, home-rationalizing, away-classifying, and away-rationalizing. The following order seems logical: 1. HC, 2. HR, 3. AR, 4. AC. Here in broad outline is how the construction is going to go: 1. Where is C∼A true at home? Where C is true in addition to A 2. Why is C∼A true at home? For the reason(s) that C is true-given-A 3. Why is C∼A true away? For the same reason(s) as it is true at home 4. Where is C∼A true away? Where it has reason to be true So, let's do it, beginning at home and extending patterns established there into the away-region. R = "what C adds to A" should have the same truth-value as C in the A-region. It should be true if C &A, and false if ¬C &A. The requirement on R is to agree truth-value-wise with C at home, so we call this condition Agreement (1) R is { true false } at home, in the A-region, in exactly the same worlds as C is { true false } . 46Precis of next section: C∼A may be understood informally as the result of extending C 's behavior at home, where A holds, to the away region, where A is false. Rules are suggested for extrapolating from home to away. 13 Next, given that a home-world w is R, how does it acquire that status? To answer a question with a question, why is w on the A&C side of the line rather than the A&C side? (Why is it a The # of dragons = 0 world rather than one where The # of dragons 6= 0)? That is why R holds true in w. R's reason for being true (false) in a home-world w is whatever makes C true (false) in w, given that A is true there.47 (Whatever that means-see below.) Reasons (2) R is { true false } at home for the same reason(s) as C is { true false } given A. Next we have to specify R's ways of being true/false in away worlds. A hypothesis "goes on in the same way" from the home-region if its truth is controlled by the same factors. R should not acquire new truthmakers and falsemakers when it leaves home. Integrity (3) R is { true false } for the same reasons away as it was { true false } at home. R's truth-value(s) in away-worlds is a function, finally, of the available reasons for R to be true/false in such worlds. Projection (4) R is { true false } in an away-world just if it has reason to be { true false } in that world. Reasons could stand to be clarified. What are C 's reasons for being true given A, in an A-world? A reason for C to be true-given-A is a truthmaker X for A⊃C that is consistent with A and makes the fullest possible use of A. X does that, the claim is, if it minimizes the extent to which B⊃C is also implied. for B weaker than A. The official definition is in three steps. A− in the first step ranges over hypotheses weaker than-asymmetrically implied by-A.48 1. X ′ uses more of A than X if {A−| A−, X′ 2 C} & {A− |A−, X 2 C} 2. X is wasteful in w if a X′ holding in w uses more of A than X does. 3. X is wasteful simpliciter if it is wasteful in every A-world where it obtains. A truthmaker for A⊃C is targeted, finally, if it's compatible with A-it doesn't falsify the antecedent-and doesn't waste A-with the result in most cases that it doesn't imply the consequent.49 Truthmakers of this sort will also be called difference-makers as between the antecedent and the consequent. C∼A is true (false) in w iff A⊃C (A⊃¬C ) has a targeted truthmaker in w and A⊃¬C doesn't, that is, a differencemaker obtains in w for A and C, but not A and ¬C. This agrees, more or less, with the truth-conditions given earlier (in section 10) for absolute conditionals; A→αC is true in w, we said, iff a fact obtains there that explains why a world stipulated to be A would be moreover C.50 12 Conservativeness Recall the notion of conservativeness from generalized quantifier theory. Predicates G and H are F -equivalent iff F &G is equivalent to F &H, that is, if all and only G s are H's, when we restrict attention to the F s. A quantifier Q is conservative iff If G and H are F -equivalent, then Q(Fs are Gs) iff Q(Fs are Hs) 47In this case: It doesn't have any dragons. 48Alternatively we could let it range over A's parts; this paper is noir about content-parts, however, so we use the definition in the main text. 49Targeted truthmakers should take advantage of A, unless there is no advantage there to be taken. Then they have no option but to imply C. This occurs, for instance, with If the balloon is colored, it's red and If it's not red, it's nevertheless colored. 50Precis of next section: Existing theories make indicative conditionals conservative in the sense that consequents C and D are freely substitutable provided only that they agree on A-worlds. This seems like the wrong prediction both for marketplace conditionals and incremental ones. The latter have the advantage of being non-conservative for clear, understandable reasons; we may want to look to them for a model that holds more generally. 14 So for instance, ALL is conservative because whether all dogs are friendly turns entirely on what goes on with the dogs; which non-dogs might be friendly is completely irrelevant. SOME is conservative because whether some dogs are friendly again turns entirely on what goes on with the dogs; the non-dogs you can feel free to ignore. MOST is conservative because whether most dogs are friendly is determined by the proportion of friendly dogs to non-friendly dogs; it doesn't matter how friendliness is distributed among cats. Conservativeness is sometimes thought to be a semantic universal for quantifiers; no natural language quantifiers care in the slightest about things that are not F. The corresponding property for conditionals A→C is that if the same A-worlds are C as D, then A→C is equivalent to A→D. Using A-equivalence for the property of equivalence over the A-worlds, If C and D are A-equivalent, then A→C is true (correct, probable,....) iff A→D is true (correct, probable,....). The two kinds of conservativeness are related insofar as the truthor correctness conditions of conditionals are given in quantificational terms, which they generally are all A-worlds are C -worlds most A-worlds are C -worlds the closest A-worlds are C -worlds Existing theories, their many other differences notwithstanding, somehow wind up agreeing on on this one point. They all make if/then out to be conservative: it is only C 's overlap with A that matters to the correctness of A→ C. The reasons vary, of course. A→C holds, according to Goodman, Adams, Kratzer, Stalnaker, etc., just if truths cotenable with A entail A⊃C (Goodman) ....which is the same as truths cotenable with A entailing A⊃(A&C ) the consequent C is highly probable, conditional on A (Adams) .....which is the same as A&C being highly probable conditional on A the consequent C holds in all (the best) A-worlds (Kratzer) .... which is the same as A&C holding in all (the best) A-worlds the consequent C holds in the closest A-world in the context set (Stalnaker) ... which is the same as A&C holding in the closest A-world in the context set the consequent is probable given A for every probability function π in the context set (Yalcin) ... which is the same as A&C being probable conditional on A for every π in the context set. Obliviousness to C 's away behavior is not a problem, obviously, if away-behavior doesn't matter. First an analogy to show how it could matter. The analogy is with vector subtraction. That −→c and −→d agree in how they project onto −→a does not mean that −→c –−→a = −→d –−→a (−→r ac = −→r ad). d a c rac rad Wednesday, March 12, 2014 Conservativeness is implicated in a number of well known puzzles. Sly Pete examples, for instance, in which pairs of conditionals A→C and A→¬C both seem right, are hard to handle if → is conservative, since A-worlds cannot be both C and ¬C. Agglomeration failures, in which A→C and A→D seem not to imply A→(C &D), should be impossible; but If Bizet and Verdi are the same height, Verdi is short and If Bizet and Verdi are the same height, Bizet is tall do not seem to imply If Bizet and Verdi are the same height, Verdi is short and Bizet is tall. And there are just straight-up intuitive counterexamples: If Oswald didn't kill Kennedy, someone else killed him ought to come out equivalent to If Oswald didn't kill Kennedy, Kennedy was killed. Conservatives have their responses, of course! But responses might not be needed if a non-conservative (progressive?) conditional could be made out.51 51Precis of next section: Examples are given of intuitively inequivalent conditionals A→C and A→Dthat should on conservative theories be equivalent, since their consequents C and D are true in the same A-worlds. 15 13 Incremental progress Incremental conditionals are away-dependent, or progressive, right out of the box. Just as the −→r that gets you from −→a to −→c may not fit snugly into the gap between −→a and −→d , the shortest path from A to C may not be the shortest path from A to D. 52 They are not conservative because their truthmakers are supposed to pick up where A leaves off, and yet take us to different destinations (C and D). A preference for picking up where A leaves off is not peculiar to them, however. Here is a string of examples that hopefully bear this out. A→C (in the first column) is predicted by standard theories to stand or fall with A→D (in the second). The third column indicates how their truth-values can come apart. A bit of commentary may be helpful, in case some of the examples are not clear. The first, Donnellan-style, example, ought to be familiar. The structure is this: If a=b, then Fa says, if we make most of the antecedent, that Fb, while If a=b, then Fb says that Fa. Fa can come apart truth-value-wise from Fb in worlds where a isn't b. The second needs some background. It is agreed, let us say, that brain states necessitate mental states. But there is controversy about which necessitate which. Our currently best psychophysical theory, theory T, says that X-fiber firings necessitate anxious feelings. I have no idea what my brain state was at noon, but I take the theory's word for it that If my X-fibers were firing at noon, I felt anxious then. I am aware of course that I have other ways of determining whether I was anxious at noon, like trying to remember. Suppose I determine that I was anxious. Then the theory was right about the mental state corresponding to X-fiber firings, if mine were firing at noon. Suppose on the other hand that I was not anxious. Then the theory was wrong if mine were firing at noon. Still convinced of the theory, I in fact recall I was not anxious, whence my X-fibers were probably not firing at noon. Still this conditional seems true: If they were firing at noon, then the theory was wrong about them. While this one still seems true as well: If they were firing at noon, then I was feeling anxious at noon. In worlds where they were firing, however, the theory is right iff I was anxious, because that's what the theory says. The third has an analogous structure to the first. If a agrees H-wise with b, then b is H-wise thus and so, turns on whether a is H-wise thus and so; mutatis mutandis for If a agrees H-wise with b, then a is H-wise thus and so. In worlds where a and b do not agree H-wise, there is no reason that a is H-wise thus and so cannot be true b is H-wise thus and so is false. According to a certain book, nothing ever changes. If the book is correct, nothing ever changes is true, because that by hypothesis is what the book says. If the book is correct, it says that nothing ever changes is false; to be correct, it should say that things change. The consequents are equivalent in worlds where the book is correct. Let me now skip ahead to the "loving" example. The choices before me are to love you or leave you; one is right and the other wrong. I don't want to be right is equivalent in loving-you-is-wrong worlds to I don't want to leave you, and I want to love you is equivalent in such worlds to I prefer wrong to right. If loving you is wrong, I want to be wrong is true, because I want to love you. If loving you is wrong, I want to love you is false, because as between wrong and right, I prefer right.53 Conditionals are away-dependent; the truth-value or correctness-value or probability of A→C is not always a function of C 's behavior in A-worlds.54 A couple of models have been suggested of how away-dependent conditionals might be accommodated. One, suited especially to absolute conditionals, involves truthmakers. The other, discussed in section 8, suited to local conditionals, involves partitions. These are not as different as they appear, I claim. The conditions on a suitable partition are akin to the conditions on truthmakers.55 In the next we consider the two models in turn. 14 Cells and truthmakers On the truthmaker model, A→C is true in an away-world w just in case an (A⊃C )-implying fact obtains there that (i) is consistent with A, and (ii) makes the most of A, that is, minimizes the extent to which A−⊃C is also implied. A fact implying A⊃C the (i)-(ii) way is one thing, a fact implying A⊃D in that way is another-even if C and D are true in the same A-worlds. Nothing changes) is, for instance, The book is correct-equivalent to The book says that nothing changes. Imagine a world where things change, while the book says they don't change. That the book says (we're imagining) that 52If this is true, it ought to show up somehow in our extrapolation rules. Look again at Reasons: R is true (false) in a A-world w for the same reasons as C is true (false) there given A. Suppose that D hold in the same A-worlds as C. Must it hold, given A, for the same reasons? Not at all. Your brother is an idiot holds, given that that guy is your brother, because that guy is an idiot. Is it also because that guy is an idiot that That guy is an idiot holds, given that that guy is your brother? No, that may be why That guy is an idiot holds simpliciter, but its reason for holding given that that guy is your brother is different. The reason D holds given A is a targeted truthmaker for A⊃D; the fact that D is not targeted, because it wastes the antecedent. 53Precis of next section: Away-dependence can be arranged either with partitions or truthmakers. Truthmakers for us are sets of worlds, and partitions have sets of worlds as their cells. Some partitions are of course better than others, and some truthmakers-the targeted ones-are better than others. The good-making features of cells turn out to be not unrelated to the desiderata for a good truth maker. Incremental conditionals can as a result be seen as local conditionals par excellence. 54[Gabbay(1972)]. Gabbay mentions unpublished work by Kamp 55Cells and truthmakers are both conceived here as sets of possible worlds. 16 Antecedents are identical Consequents agree at home Conditionals disagree away from home C. Smith's murderer is insane. TRUE cuz: That guy is insane IF A. that guy is Smith's murderer, THEN D. that guy is insane. FALSE cuz: Smith's murderer is NOT insane. C. I was anxious. TRUE cuz: The theory says so. IF A. my X-fibers were firing, THEN D. the theory is right about X-fibers FALSE cuz: I was not feeling anxious. C. Verdi is short. TRUE cuz: Bizet is short. IF A. Bizet and Verdi are the same height, THEN D. Bizet is short. FALSE cuz: Verdi is tall. C. nothing ever changes TRUE cuz: That's what the book says ... IF A. This book is correct THEN D. It says nothing ever changes.. FALSE cuz: Things change. C. he won. TRUE cuz: Pete had the better hand. IF A. Pete called, THEN D. his hand was better. FALSE cuz: Pete didn't know Mr Stone's hand. C. Mr Stone lost. TRUE cuz: Pete was playing Mr Stone. IF A. Pete won, THEN D. he was playing Mr Stone. FALSE cuz: He can only beat Mr Scone C. I don't want to be right. TRUE cuz: I can't give you up. IF A. loving you is wrong, THEN D. I want to love you. FALSE cuz: I want to be right C. I want better than your best TRUE cuz: I want better than that IF A. that's your best, THEN D. I want better than that FALSE cuz: I don't want the impossible. C. they will hatch. TRUE cuz: They are viable. IF A. Hettie sits on those eggs, THEN D. they're viable FALSE cuz: Hettie is a poor judge of viability. Table 1: Away-dependent conditionals; C and D agree in A-worlds but A→C is true and A→D is false nothing changes is a targeted truthmaker for The book is correct⊃Nothing changes. But The book is correct⊃It says nothing changes has no such truthmaker. That nothing changes would do it, but that is not a fact that obtains in this world. Correct→Stasis is true in the world in question, while Correct⊃Book says "Stasis" is false. This is a clear case of away-dependence, since in worlds where the the book is correct, it says that nothing changes just if nothing indeed changes. Next the partition-based, or "local," route to away-dependence. To read A→C locally is to associate it, not with a single conditional probability π(C |A), but a bunch of them π(C|A&Hi), Hi ranging over hypotheses about the objective set-up that affect the prevalence of C -worlds among A-worlds. These are then brought together into a weighted sum, the weights given by the prior probabilities of each hypothesis: π(A→C ) = πH(C |A) = Σi π(C|A&Hi) × π(Hi). Now, why would π(A→C ) come apart from π(A→D) (when C and D are A-equivalent) on this way of doing it? Our confidence in A→D is, one might think, π(A→D) = πH(D|A) = Σi π(D|A&Hi) × π(Hi). But, given that C and D are true in the same A-worlds, π(D|A&Hi) = π(C|A&Hi), whence πH(C |A) is the same quantity as πH(D|A). It must be, then, that the partition itself changes when we switch to A→D. π(A→D) = πH′(D|A) = Σi π(D|A&H'i) × π(H'i). So far so good- but why does the partition change when the consequent changes from C to D? For the same sort of reason as the targeted truthmakers for A⊃Z change when Z changes from C to D. A targeted truthmaker for A⊃C (A⊃D) has got to make the most of A, which means, in the first case, not implying C all by itself, and in the second, not implying D all by itself. C is an unacceptable partition cell for A→C because it renders A irrelevant, the very reason it cannot serve as a targeted truthmaker for A→C 56 56Imagine for instance that A→C is If your mood ring is accurate, you are jealous. A natural partition is H1 = D = Your mood ring glows green and H2 = ¬D = Your mood ring does not glow green. The ring is accurate exerts one kind of control over You are jealous in Green-worlds and another in worlds where the ring does not glow green. Shall we stick with the same partition when it comes to evaluating If your ring is accurate it must be glowing green? Surely not, for Your ring is accurate exerts no control whatever over the ring's color in either cell of {Green, Green}. The ring glows green holds regardless of accuracy in the Green-region, and fails regardless of accuracy in the Green-region. The partition we use to assess Your ring is not mistaken→You are jealous is {Green, Green}. (Reliability makes different demands on your mood depending on which mood the ring is currently indicating.) To assess The ring is accurate→It glows green, we use {Jealous, Jealous}. (The probability that it says you're jealous, if it's accurate, is the probability that you are jealous.) To partition in the first case on whether you are jealous, or in the second on whether the ring glows green, would be silly. One might as well try to determine the relevance of smoking to cancer by asking whether smokers with (without) cancer are likelier to have cancer than non-smokers with (without) cancer. 17 So-one way of getting A→C to care about C 's away games is to read it incrementally: π(A→C ) = π(C∼A), for short, π(∆), Another is to partition according to the factors controlling conditional probability between which you are undecided. The first approach can seen as a special case of the second; for partitioning on ∆ vs ∆ is the best one could possibly do on the score of controlling for factors affecting conditional probability. What do I mean by this? In one cell the conditional probability is π(C |A&∆) = 1. In the other it's π(C |A&∆)) = 0.57 The partition ∆ vs ∆ thus completely settles A's probabilistic influence on C, driving it up to 1 in the ∆ (= the C∼A) -region, and down to zero in its complement. 15 Proper partitioning The absolute/local analogy is our first glimpse of how the manifold of conditionals might hang together. To take the analogy further, we would need to know how (on the local approach) the partition into different possible scenarios Hi is arrived at. This is a delicate matter for several reasons. The first and most obvious is that A→C 's local probability is going to vary with the partition. Second, the triviality results suggest that some partitions are just bad. The local probability of A→C is π(C ), for instance, if the partition is C, C. Third, A's probabilistic bearing on C may be uniformly positive across cells, or uniformly negative. This might seem to clarify matters, except that the polarities are oftentimes reversible by cutting cells up more finely. The logic is that of Simpson's paradox. Female applicants, the story goes, are less likely to be admitted to grad school at Berkeley than men. But they are more likely to be admitted to any particular department; it is just that they apply to more competitive departments. One can imagine that they are again less likely to be admitted, in each department, both to MA programs and to PhD programs; their advantage in departments as a whole is because they tend to apply for the less competitive degree. And so on.58 Fourth, local probabilities are going to be more or less objective depending on the range of acceptable partitions.59 Incremental conditionals fare best by this standard because the one and only allowable partition is between worlds where C≈A is true and worlds where it is false. Otherwise the principles guiding partition choice are unclear. How did we hit on #1 is Holmes and #1 is not Holmes as our partition for He accused Mrs Hudson → She did it? The problems with She did it, She didn't do it, have already been mentioned; we return to them below. What about #1 accused Mrs Hudson vs #1 did not accuse Mrs Hudson? Or #1 said she did it ⊃ She did it vs ¬(#1 said Mrs Hudson did it ⊃ She did it)? The reasons differ but they are reminiscent, in each case, of the constraints on targeted truthmakers: they should be consistent with A and should not waste A. Suppose we tried to break A's bearing on C into the difference it makes in C -worlds, and the difference it makes in ¬C -worlds. π(C |A&C )×π(C ) + π(C |A&¬C )×π(¬C ) = π(C ). A just drops out of the picture on this partition, because C and ¬C both screen A off from the consequent. (If we divide the Berkeley applicants into those that were admitted and the rest, we will find that gender makes no statistical difference in either group.) The point is that we want C 's truth-value to vary in each cell, lest A's contribution go unrecognized. This ought to remind us of how a targeted truthmaker for A;C (A;¬C ) is supposed to assign A the greatest possible responsibility for the fact that A with its help entails C (¬C ). Or suppose we tried to break A's bearing on C into the difference A makes in A-worlds, and the difference it makes in ¬A-worlds. π(C |A&A)×π(A) + π(C |A&¬A)×π(¬A) = π(A&C) + undefined. π(A&C) tells us nothing about the proportion of C -worlds in the A-region (for it is silent the prevalence of ¬C -worlds in that region). This ought to remind us, again, of how a truthmaker for A;C (A;¬C ) is supposed to be consistent with A and to lean on it to the maximum extent possible. A truthmaker that repeats things already there in A (as facts implying A do) is taking on for itself a job that A could have done. So much for what we don't want; what do we want? I don't exactly know, but it is something like the following. We want to partition the space of worlds into scenarios that fix the "channels" over which A sends its signal to C. A partition is better, other things equal, if A's distribution in a cell is more predictive of C 's in that same cell. Incremental conditionals are, again, the limiting case of this, where A's truth-value is completely predictive of C 's both in the ∆-region and its complement: C holds in all A-worlds of the first kind and none of the second kind. 57C∼A is true, when it is, on account of a truthmaker for A⊃C. A&(C∼A) therefore implies C, whence the probability of C conditional on A in the C∼A cell is 1. C∼A is false, when it is, on account of a truthmaker for A⊃¬C. A&¬(C∼A) therefore implies ¬C, whence the probability of C conditional on A in the ¬(C∼A) cell is 0. 58Which statistic is relevant for any particular applicant may depend on where she is in her decision process. Stanford may be a better choice for those not decided on a department; Berkeley is a better choice if they are applying to Computer Science; the advantage goes to Stanford, maybe, if they're seeking a PhD. This is on the principle that we choose on the basis of "stable" properties of the options, properties that they retain whichever option is chosen. 59[Douven(2008)] 18 16 Unification (?) But, you may say, ∆ versus ∆ is not the only partition with this property. Take again the Oswald conditional: Oswald didn't shoot Kennedy→Someone else did. The obvious partition here is between Kennedy was killed and Kennedy was not killed. Partitioning on was vs was not is just like partitioning on C∼A and C∼A as far as the math goes. Was killed-worlds where Oswald didn't do it are all Someone else did-worlds, while Wasn't killed-worlds where Oswald didn't do it are all No one else did-worlds. Yet the existential fact that Kennedy was killed can hardly be considered a truthmaker for Oswald didn't do it⊃Someone else did ; on the contrary Kennedy was killed by virtue of Oswald's killing him. The hypothesis of Kennedy being killed plays instead a belief-making role. Our confidence that Kennedy was killed is based on the film; it does not derived from any particular hypothesis about who killed him. Our confidence that he was killed is thus well-positioned to ground our confidence in Oswald didn't⊃Someone else did. The story is just like before except that truth-grounds have been replaced by epistemic grounds. Go back to the whiteboard case and pretend this time that the only two sentences on the whiteboard are fish never sleep (F ) and Goats eat tinfoil (G ). Cloris tells you F and G agree in truth-value, but this time it's not clear whether to believe her. Cloris might be a knight, who always tells the truth, or else a knave, who always lies. Partitioning on the knight and knave hypotheses is just like partitioning on C∼A and C∼A as far as the math goes. F -worlds in the knight cell are all G, while F -worlds in the knave cell are all ¬G. Just as probability of Brad is right→Nothing changes comes down to probability of Brad's claiming that nothing changes, the probability of F→G is just the probability that Cloris is a knight. Yet F→G is the opposite of an absolute conditional. The technical point is undeniable, and points us in the proper direction. The Cloris example it can serve as a template for the understanding of marketplace conditionals generally. We start by listing the conditions on cells X, X that make for an absolute reading of the conditional; these are no different than the conditions on targeted truthmakers. To obtain nonabsolute conditionals, we relax one or more of these conditions.60. So: X vs X is a partition fit for absolute conditionals iff X and X are targeted truthmakers for A⊃C and A⊃¬C respectively. Teasing the components of targeted-truthmaker-hood apart a bit to enable pointwise adjustments, X vs X is an absolute partition iff 1. X entails A⊃C and X entails A⊃¬C (since they are truthmakers) 2. X explains A⊃C and X explains A⊃¬C (since they are truthmakers)61 3. X is consistent with A and X is consistent with A (A-consistency is part of being targeted) 4. X is consistent with ¬A and X is consistent with ¬A (in order to make the most of A)62 5. X is consistent with ¬C and X is consistent with ¬C (or A will have been wasted)63 6. X (X) maximizes A's responsibility for X and A entailing C (¬C ) (the preciser truth behind 4. and 5.)64 Weaker than 1 would be a condition requiring X and A to probabilify C instead of entailing it, and similarly for X, A, and ¬C. This is what happens in Kauffman's Urn example. If X and A hold (= you draw a red ball from Urn 1) then it is 90% likely that S (the ball will be shiny). If T and A hold (if you draw a red ball from Urn 2) then it is 90% likely that ¬C (that the ball will be dull).65 Weaker than 2 would be a condition allowing X to be, say, epistemically prior to A⊃C rather than explanatorily prior. This is what we get with the Oswald example, as already discussed. Our belief that Kennedy was killed is not based on a belief regarding any particular person that s/he killed him. That Kennedy was killed by someone or other does not explanatorily underwrite his being killed by another person if not by this one; rather the belief that he was killed epistemically underwrites the belief that another person killed him if this one didn't.66 60In some cases tweaking others to maintain coherence 61Explanation here is something like the converse of holding in virtue of. 62If X (or X) entailed A itself, it would be doing itself what A might have, whence it would not likely be making the most of A. 63If X entailed C itself, or X entailed ¬C, it would not be drawing on A at all, whence it would not likely be making the most of A. 64I say that 6. is the preciser truth behind 4. and 5. because 4. and 5. hold only when the antecedent is usable. The only Tom is coloredconsistent truthmaker I can think of for Tom is colored⊃Tom is red is the fact that Tom is red. This is not consistent either with Tom being uncolored (contra 4.) or Tom not being red (contra 5.) It is typical of determinables D and their determinates D* that a truthmaker X for D⊃D* cannot assign any responsibility to D for the fact that X and D imply D*. 65Or take the game show conditional: If #1 accuses Mrs Hudson, she did it. The deciders are #1 is Holmes, #1 is Watson. These are consistent with the antecedent (#1 accuses Mrs Watson) and make good use of it But they do not combine with (#1 accuses Mrs Watson to definitively resolve the consequent (She did it); Holmes is not always right nor Watson always wrong. The deciders here do somewhat resemble truthmakers in being explanatory, though. For #1 to be Holmes does some distance toward explaining why if he accuses her then she did it. 66Another example goes back to the whiteboard case. Pretend this time that the only two sentences on the whiteboard are (G) = Goats eat cans and (H) = fish never sleep. Cloris tells you that G and H agree in truth-value, but this time it's not clear whether to believe her. Cloris might be a knight, who always tells the truth, or else a knave, who always lies. Partitioning on the knight and knave hypotheses is just like partitioning on C∼A and ¬(C∼A) as far as the math goes. G -worlds in the knight cell are all H, while G -worlds in the knave cell are all ¬H. G→H is not absolute because Cloris being a knight provides only an epistemic block to goats eating cans while fish sometimes sleep; it doesn't 19 Conditions 3 and 4 between them say that X and X are logically independent of A. If condition 1 is weakened to allow probabilification instead of entailment, then 3 and 4 should perhaps require probabilistic independence instead of logical. This take us in the direction of Adams conditionals, since his thesis has probabilistic independence as a consequence. Weaker than 5 would be a condition that allowed X to entail C itself, with no help from A. This opens up room for non-interference conditionals along the lines of Whether she ate the mushroom or not, she's still very much alive. (The same example works for condition 6.) Generalizing wildly from these examples, let us try a focal meaning analysis of conditionals, in which certain sorts of examples are primo and others are understood as departing in various ways from the standard they set.67 Suppose we are trying to evaluate C conditional on A relative to such and such a partition. The partition appropriate to absolute conditionals has various special properties that make it unique of its kind. Other sorts of partition fall away from these standards in one way or another. Their cells do still support A⊃C or A⊃¬C. There are differences though in the amount of support, and the kind, and their attitude toward the antecedent. Every step away from targeted-truthmaker deciders is a step away from objectivity and toward parochiality. I spoke earlier of a continuum of conditionals but that turns out to be not quite right. The deciders for the Oswald conditional fall short in explanatoriness; the deciders for the Urn conditional fall short in decisiveness; the deciders for the Holmes conditional are not decisive but do seem somewhat explanatory. The deciders for a non-interference conditional are decisive and explanatory, but not targeted, since they require no help from the antecedent. We should really speak of a spider web of conditionals, then, with surplus content conditionals at the center. Which I hope in the fullness of time becomes a taxonomy in which conditionals are classified by how far short their truthmakers fall of the ideal, and which components of the ideal are compromised. Appendix: Formal Details Here is one way of implementing the proposal formally. The first two steps recapitulate van Fraassen's truthmaker semantics in "Facts and Tautological Entailment." Models A model M of propositional language L is a triple < W,F,V > where 1. W is a set of worlds; 2. F is a collection of privileged subsets of W (the "facts"), closed under certain operations 3. V is a function assigning to each atomic sentence a a fact a Truthmakers τ is a truthmaker (falsemaker) for φ iff 1. S is an atomic sentence a, and τ = {a} (τ = {a})68 2. S is ¬P, and τ is a falsemaker for P (τ is a truthmaker for P) 3. S is P∨Q, and τ makes P true or Q true (τ is the union of falsemakers for P and Q) 4. S is P&Q, and τ is the union of truthmakers for P and Q (τ makes P false or Q false) S tautologically entails P, van Fraassen shows, if every truthmaker for S contains a truthmaker for P. Requiring further that every truthmaker for P is contained in one S gets us (what in Aboutness I called) inclusive entailment.69 Now we expand L in the usual way to a language L+ any two of whose sentences, even those with conditionals in them, can be combined into a further conditional P;Q. A set of truthmakers and falsemakers must be assigned to each of the new sentences. A truthmaker τ for P⊃Q is wasteful in w iff (it obtains there and) a σ obtains in w that uses strictly more of P in this sense: for some P− implied by P σ does not nontrivially imply P−⊃Q and τ does; but not vice versa. A truthmaker is efficient iff it is not necessarily wasteful. Assume that P and Q have already been endowed with truthmakers and falsemakers; they are true where they have reason to be and no reason to be false; they are false where they have reason to be and no reason to be true. tell us what objectively stands in the way of the combination, and indeed presumably nothing does. This is crucial if absolute conditionals are to be aperspectival, because surplus content is unique and reasons for belief are not unique. Disagreements about A→C lose their objectivity if you are working with one sort of reason (Cloris is a knight) and me another (Doris, who wrote the sentences down, has a firm policy of not mixing truths with falsehoods. (This is one lesson of the Sly Pete example.) 67The original such analysis being Aristotle's of health. 68a is the set of worlds not in a. 69Compare [Fine(2013)] 20 Difference-Makers A difference-maker for P and Q (¬Q) is a nontrivial, efficient, truthmaker for P⊃Q (P⊃¬Q). Conditionals P;Q is true (false) in w if a difference-maker for P and Q (P and ¬Q) obtains there.70 Examples 1. p ; p&q................ is true in the same worlds as q Goats eat cans;They eat cans and bottles ⇔ Goats eat bottles 2. p⊃q ; p≡q ...........is true in the same worlds as q⊃p Goats eat cans if bottles;They eat cans iff they eat bottles ⇔ Goats eat bottles if they eat cans. 3. p∨q ; p ............... is true in the same worlds as ¬q∨ p Goats eat cans or bottles;Goats eat cans ⇔ Goats either do not eat bottles or do eat cans 4. p∨q ; p∨q71 ........ is true in the same worlds as ¬(p&q) Goats eat cans or bottles;They eat ONE of cans and bottles ⇔ They don't eat both cans and bottles 5. p∨r ; pq∨r ...........is true in the same worlds as q∨r Goats eat cans or bottles;They eat cans and tinfoil, or else bottles ⇔ They eat tinfoil or bottles Goats eat cans or foil;Goats & sheep eat cans, or else foil ⇔ Both eat cans or both foil or sheep eat cans and foil 6. p≡q ; p≡r ............agrees in all worlds with q≡r Goats eat cans iff they eat bottles;They eat cans iff tinfoil ⇔ They eat bottles iff they eat tinfoil References [Adams(1975)] Ernest W. Adams. The logic of conditionals: An application of probability to deductive logic, volume 86. Springer, 1975. [Adams(2005)] Ernest W Adams. What is at stake in the controversy over conditionals? In Wilhelm Rödder Gabriele Kern-Isberner and Friedhelm Kulmann, editors, Conditionals, information, and inference, pages 1–11. Springer, 2005. [Belnap(1970)] Nuel D. Belnap. Conditional assertion and restricted quantification. Noûs, 4(1):pp. 1–12, 1970. [Cantwell(2008)] John Cantwell. Indicative conditionals: Factual or epistemic? Studia Logica, 88(1):157–194, 2008. [Douven(2008)] Igor Douven. Kaufmann on the probabilities of conditionals. Journal of Philosophical Logic, 37(3):259–266, 2008. [Edgington(2013)] Dorothy Edgington. Estimating conditional chances and evaluating counterfactuals. Studia Logica, pages 1–17, 2013. [Edgington(2014)] Dorothy Edgington. Estimating conditional chances and evaluating counterfactuals. Studia Logica, 102(4):691–707, 2014. [Fine(2013)] Kit Fine. Angellic content. to appear, 2013. [Fuhrmann(1996)] André Fuhrmann. An essay on contraction. University of Chicago Press, 1996. [Fuhrmann(1999)] André Fuhrmann. When hyperpropositions meet. Journal of Philosophical Logic, 28(6):559–574, 1999. [Gabbay(1972)] Dov M Gabbay. A general theory of the conditional in terms of a ternary operator. Theoria, 38(3):97–104, 1972. [Gajewski(2008)] John Gajewski. NPI any and connected exceptive phrases. Natural Language Semantics, 16(1):69–110, 2008. [Hájek and Hall(1994)] Alan Hájek and N Hall. The hypothesis of the conditional construitl of conditional probability. Probability and conditionals: Belief revision and rational decision, page 75, 1994. [Hellman(1989)] Geoffrey Hellman. Mathematics without numbers: Towards a modal-structural interpretation. Clarendon Press; Oxford, New York, 1989. [Horgan(1984)] Terence Horgan. Science nominalized. Philosophy of Science, pages 529–549, 1984. [Hudson(1975)] James L. Hudson. Logical subtraction. Analysis, 35(4):130–135, 1975. [Humberstone(1981)] Lloyd Humberstone. Logical subtraction: Problems and prospects. Melbourne (typescript), 1981. [Humberstone(2000)] Lloyd Humberstone. Parts and partitions. Theoria, 66(1):41–82, 2000 2000. English. [Jackson(1977)] Frank Jackson. Perception: A representative theory. Cambridge University Press, 1977. [Jaeger(1973)] Robert A Jaeger. Action and subtraction. The Philosophical Review, 82(3):320–329, 1973. [Jaeger(1976)] Robert A. Jaeger. Logical subtraction and the analysis of action. Analysis, 1976. Mr 76; 36: 141 146 0003 2638 English. 70This allows P;Q and P;¬Q to hold together. We could alternatively count these cases gappy. 71(∨ is exclusive disjunction 21 [Kaufmann(2004)] Stefan Kaufmann. Conditioning against the grain. Journal of Philosophical Logic, 33(6):583–606, 2004. [Kripke(2011)] Saul A. Kripke. And and But: A note. ms., 2011. [Lasersohn(1993)] Peter Lasersohn. Existence presuppositions and background knowledge. Journal of semantics, 10(2):113, 1993. [McDermott(1996)] Michael McDermott. On the truth conditions of certain'if'-sentences. The Philosophical Review, pages 1–37, 1996. [Pendlebury(1989)] Michael Pendlebury. The Projection Strategy and the Truth: Conditions of Conditional Statements. Mind, XCVIII(390): 179–205, 1989. [Quine(1982)] Willard Van Orman Quine. Methods of logic. Harvard University Press, 1982. [Ramsey(1931)] Frank P. Ramsey. Truth and probability (1926). The foundations of mathematics and other logical essays, pages 156–198, 1931. [Shope(1978)] Robert K. Shope. The conditional fallacy in contemporary philosophy. The Journal of Philosophy, pages 397–413, 1978. [Stalnaker(1984)] Robert C. Stalnaker. Inquiry. The MIT Press, Cambridge, MA, 1984. [Toosarvandani(2014)] Maziar Toosarvandani. Contrast and the structure of discourse. Semantics and Pragmatics, 7(4):1–57, April 2014. [Von Fintel(1993)] Kai Von Fintel. Exceptive constructions. Natural Language Semantics, 1(2):123–148, 1993. [Williams(2005)] Bernard Arthur Owen Williams. Descartes: The project of pure enquiry. Psychology Press, 2005. [Yablo(2014)] Stephen Yablo. Aboutness. Princeton University Press, 2014.