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To appear in: Bob Hale, Alex Miller, and Crispin Wright, eds. Blackwell Companion to the Philosophy of Language, Blackwell. Pre-final draft. Conditionals Anthony S. Gillies Rutgers University 1 Introduction:
To appear in: Bob Hale, Alex Miller, and Crispin Wright, eds. Blackwell Companion to the Philosophy of Language, Blackwell. Pre-final draft. Conditionals Anthony S. Gillies Rutgers University 1 Introduction: Conditional Information I want very much that you have the information that the beer is gone. In fact, I want you to take action that requires it. (Let us also stipulate that that action is unavailable (insert your favorite constraint here: policy, prudence, politeness) if there is plenty of beer.) So I say something that gives voice to the state of affairs, beer-wise: (1) The beer is gone. I hope that you understand me and take that information on board and then do the right thing. But things won t work out if I m mistaken about the facts, for then what I am trying to pass off as information isn t that and my hopes will go unfulfilled. Nutshell: successful information exchange depends on the way things are. That s clear (enough) if the information I aim to get to you is plain (enough) and the linguistic vehicle simple (enough). Conditional information is a useful kind of information and it is no surprise that natural language has canonical ways of expressing it. That is what if s conditionals are for. So take a case of conditional information exchange where the information is less plain and the linguistic vehicle less simple: (2) a. If Jimbo is here, then he bought this round. b. If Jimbo is here, then he might buy this round. c. If Jimbo were here, then he would buy this round. The information at stake here is information about what is or might or might not be the case if Jimbo is here, and what would or wouldn t be the case if he had been. Nutshell: successful (conditional) information exchange depends 1 on the way things are but also on the ways things are in various alternative scenarios (the way things are if such-and-such). Not all conditional information is the same, and this is reflected in differences in conditionals. Take a so-called Adams pair (Adams 1975): (3) a. If Oswald didn t kill Kennedy, then someone else did. b. If Oswald hadn t killed Kennedy, someone else would have. The first conditional is an indicative conditional, saying what is the case if it turns out that Oswald wasn t involved. The second conditional is different. It has distinctive tense/aspect saying what would have been if, contrary to the facts, Oswald hadn t been involved. Such conditionals are counterfactual conditionals, but the name isn t a perfect fit. 1 The types of conditionals are genuinely different since (3a) and (3b) have the same antecedent and the same consequent, but one is true and the other false. Two not-unrelated bundles of questions will occupy us here, regardless of the type of conditional we are considering. First bundle: What is that conditional information that is conventionally carried by various conditional constructions in natural language? Answering this will involve saying something about a conditional s dependence both on how things are and on how 1 Counterfactuals are sometimes called subjunctive conditionals, but this is an even worse fit. A natural thought: the subjunctive marking goes exactly with counterfactuality. Alas, no, as an example from Anderson (1951) shows: (i) If Jones had taken arsenic, he would have shown just exactly those symptoms which he does in fact show. A doctor may well use a conditional like this to argue that what afflicts poor Jones is arsenic poisoning. If counterfactuality means anything like (the speaker is taking it that) the antecedent is false, then the distinctive marking isn t sufficient counterfactuality. Nor is it necessary. In sportscasterese, seemingly run-of-the-mill indicative conditionals are used to convey counterfactual meanings. It s the top of the ninth, the visiting team is down a run but they are down to their last out with a runner Speedy, as it happens on second. Slugger hits a double to the gap surely Speedy should score! but as Speedy rounds third he trips, falls, and is thrown out. The visitors lose in a shocker. The announcer almost can t believe it. In the aftermath he says: (ii) If Speedy stays on his feet, they probably win the game. The announcer isn t confused about how the game ended: he s sure Speedy would have scored and Slugger would likely have been hit in, too. See von Fintel 1998 (and the references therein) for the status of the connection between subjunctive marking and counterfactuality. 2 things might have been. It will also (but not invariably as we will see) involve saying when conditionals are true and when they are false. Second bundle: How do the various conditional constructions in natural language manage to carry that information? Answering this will involve saying something both about how a conditional s meaning arises from the parts of it and something about how a conditional s meaning interacts with and contributes to the meaning of embedded and embedding environments. The bundles are in principle separable, but in practice often enough go hand in hand. 2 The first thing is all about saying what conditionals mean (what semantic values they have) and how things that mean those things are used in well-run conversations (what their pragmatic profiles are). The second thing is all about how if interacts with the rest of our language how what specific conditionals mean is determined by the bits that make them up and about conditional constructions contribute whatever-it-is they mean to embedding environments in which they occur as proper parts. The aspiration is to at least see where some answers can be found. 3 2 There are complications and wrinkles galore. Here are just three, followed by executive decisions about the issues they raise. One: some conditional constructions do not seem to carry conditional information at all. (i) There are biscuits on the sideboard if you want them. (Austin 1956) These are so-called biscuit (or relevance) conditionals and are fine things to say but they don t normally (or obviously) express a what conditionals normally (or obviously) do. So let s agree to set them aside. Two: there are ways to conventionally express conditional information in natural language without resorting to if. This is especially clear with conditional imperatives: (ii) a. Be on time or text me! b. Keep it up and I ll turn this car around! These do manage to express some conditional kind of meaning (iffiness but if lessness!). Still, set them aside. (While we re at it: we will also not have anything to say about conditional imperatives or conditional questions.) And three: Some languages (apparently) lack lexicalized if -constructions altogether. We will largely ignore cross-linguistic pressures, too. The hope is that the executive decisions won t distort things (too much). 3 Though this is a survey, it can t (and won t) pretend to be either exhaustive or unopinionated. See Section 9 for references that may provide balance on either horn. 3 sentence p context c semantic value p c index w truth-value p c,w = 0/1/# Figure 1 Variable But Simple Semantic Values 2 Preliminaries Distinguish between (i) conditional sentences (indicative or counterfactual) of natural language and (ii) conditional connectives of some formal language that serves to represent the relevant conditional sentences. The aim is to associate an if -in-a-natural-language with an if -in-the-formal-language that then gets associated with its semantic value. We will assume that the role of the formal language can be adequately played by a simple propositional language with the usual sentential connectives (,,, ) plus a binary sentential connective (if )( ). Context will disambiguate which sort of conditional (if )( ) represents. 4 (When the time comes, we will also have use for modal operators of the usual sort like and and perhaps a few other things.) This two-step route to assigning meanings to conditionals isn t obligatory and is in-principle dispensable (assuming that the mapping from natural language to the formal language (we won t fuss with this one) and then the mapping from the formal language to the universe of meanings (we will fuss with this one) are well-behaved). But it does make for a clearer view of the landscape. A (formal) language is only useful if it is interpreted. Whether or not truth-in-english has been achieved by an utterance of (4) Jimbo has to be washing the dishes. 4 Whether conditionals in natural language can be represented by a binary conditional connective in a regimented intermediate language is also, as we ll see, up for grabs. 4 depends on the facts how things actually are at the world where it is uttered. It also depends on the state of the conversation when it is uttered: for instance, whether what is being claimed is a claim about Jimbo s obligations or about what we know about his current activities. And it also depends on how things are in various other situations: for instance, whether at those relevant situations the prejacent Jimbo is washing the dishes is true. So truth-in-english is sensitive to both a context of utterance and an index of evaluation. One of the things we re after is a way of systematically saying when a sentence p is true in a context at a world. 5 So let s suppose that semantic values (whatever they are) determine truth values at points of evaluation with respect to contexts. We will not need to say just what contexts are. It will be enough to carve out what role they play in the set-up. Similarly, our indices will be worlds, but the most we need to know about them is that they are the kinds of things at which sentences are true or false. 6 The set-up is summarized in Figure 1: a sentence p in a context c gets associated with a semantic value p c, which combines with an index w to deliver a truth-value (if the sentence has one) of the sentence at that world in that context, p c,w = 1 or p c,w = 0 or p c,w = # (as the case might be). These aren t entirely innocent assumptions, but we can start here and suspend them when it suits us. Theories of conditionals are constrained by the patterns of intuitive entailment they participate in. So what-intuitively-entails-what is important data and we want to explain its patterns as best we can. For conditionals this might tie what (if )( ) might mean very tightly to what we can and can t say about entailment. For instance, take the basic deduction theorem. Deduction Theorem X, p q iff X ( if p )( q ) Taking this as a constraint, what we say about (if )( ) impacts what we can say about and vice versa. (This holds for various weakenings of the deduction theorem too, wherein the weakness lies in how the set of premises A and the premise p are combined.) The point is just that entailment is as much part of the theoretical machinery as is anything. So it s up for grabs whether what we say about if s bends to the will of entailment or whether 5 Lewis (1980) argued that contexts and indices are both needed and that neither can do the work of the other. 6 For now. Later, when we flirt with various dynamic theories, all we will need to know about them is that they are the kind of thing that atomic sentences (of our intermediate language) are true or false at. If an atomic a is true (false) at w, we ll say that w(a) = 1 (w(a) = 0). 5 what we say about entailment bends to the will of the if s. There is very little on stone tablets. That said, let s try to skirt issues about entailment when we can. The simplest (and in a precise sense the weakest) conditional is the material conditional: p q is true iff either p is false or q is true. Accordingly, the simplest (and in a precise sense the weakest) theory of if takes it to simply be the material conditional. 7 Definition 1 (Horseshoe Theory). Conditionals are material conditionals: ( if p )( q ) c,w = 1 iff either p c,w = 0 or q c,w = 1. This is, so far, neutral about whether the target conditionals are indicative or counterfactual. That matters and we will return to it. In any case, this treats conditionals as truth-functional (and, by the way, context-invariant): the truth-value of a conditional at a world is entirely determined by the truth-value of its antecedent and consequent at that world. In fact, this is the only truth-functional option available for the conditional. Fact 1. If (if )( ) is truth-functional then (if )( ) =. Here s why. Suppose p = The die came up six and q = the die came up even. Our theory has to then render ( if p )( q ) true no matter how the roll came out. So, in particular it s true if it came up six (antecedent true, consequent true), if it came up four (antecedent false, consequent true), and if it came up three (antecedent false, consequent false). The rub is that since by hypothesis ( if p )( q ) only depends on the (actual) truth values of p and q, any p and q with the same truth-values can be substituted in for them and the resulting conditional has to still be true. So: if a truth-functional conditional has a false antecedent or true consequent it is true. Now assume some conditional is false. It can only be because it has a true antecedent and false consequent. Given truth-functionality, this then holds for all conditionals. So: if a truth-functional conditional is false it has a true antecedent and false consequent. As a theory of counterfactuals, this is an obvious non-starter. Conceptually, counterfactuals the real deal ones with false antecedents ask us to consider other, non-actual ways things might have been. But a truthfunctional theory says that such considerings can t be relevant. The empirical 7 So-called because the material conditional is sometimes symbolized by the horseshoe. 6 coverage is also terrible. At least some counterfactuals with false antecedents are (contingently) true and thus worth arguing about. The material conditional rules that out. (5) a. If Alex had come to the party, she would have arrived before 8. b. If Alex had come to the party, she wouldn t have arrived before 8. These conditionals cannot both be true, and speakers using them seemingly disagree. That is not what you d expect if counterfactuals were horseshoes. When it comes to indicative conditionals things are different: the horseshoe isn t widely adopted, but it is not without defenders. The main difficulty is that the material conditional is, from a logical point of view, weak. Taking indicatives to be horseshoes thus predicts that there are more entailments to conditionals than there seems to be. Among them: the paradoxes of material implication. (6) a. Carl came alone.??so: if Carl came with Lenny, neither came. b. Billy got here first.??so: if Alex got here before Billy, Billy got here first. These don t strike as entailments even though the truth of either p or q at w secures the truth of p q at w. The thing that has to be said is that while these are genuine entailments, there are pragmatic reasons derived from how conditionals and surrounding sentences are reasonably and appropriately deployed in conversation why they strike us as weird. For instance: the conditional conclusions in (6) are weird because in each case a speaker in the position to assert the unconditional premise has no use for (and hence would mislead by using) the logically weaker conditional conclusion. So there is a clash between an implicature of the conclusion and the initial premise. 8 Explaining away unwanted entailments by appeal to implicatures is tricky since this strategy has nothing to say about conditionals that occur unasserted in embedded environments. But conditionals occur in such 8 This is Grice s (1975) strategy and is taken up and extended by Lewis (1976). Another pragmatic defense, developed by Jackson (1991), says that when you use an indicative ( )( ) if p q, it conventionally implicates that your credence in its truth (that is, the truth of p q) conditional on p is high. (Lewis, in the Postscript to his 1976, drops the conversational story and instead goes for a slight variant of Jackson s.) There is little independent evidence in favor of this stipulated conventional implicature. 7 environments and when they do the horseshoe theory makes some unhappy predictions. Negated indicatives are a case in point. The issue is that since material conditionals are so weak, their negations are correspondingly strong. (7) It s not so that if the gardener didn t do it then the butler did. I can be signed-up for (7) without being signed-up for the truth of the gardener didn t do it. After all, it might have been the driver. Maybe we need a pragmatic defense of our pragmatic defense. Perhaps what we have in (7) is a denial of a conditional rather than an assertion of a negated conditional, and so the negation is not a negation. I doubt it since the issue can be pushed where the negated conditional itself occurs embedded and thus unasserted and thus not open to this defense. For instance: the argument in (8) is a disaster but predicted to be an entailment by the horseshoe. (8) If there is no god, then it is not the case that if I pray, my prayers will be answered. I don t pray.??so: there is a god. Is there another pragmatic defense to save these pragmatic defenses of the earlier pragmatic defense? It s possible, but this rescue is quickly becoming a wheels-within-wheels situation. 3 Strict Conditionals The material conditional is extreme in its myopia: only how things are at w matter to the truth of a material conditional at w. At the other end of the spectrum lie strict conditionals: these survey all possibilities. That is a quantificational claim and so, as with a lot of quantificational claims, this one may be restricted. 9 We will care about the if -relevant worlds in a context c. A strict conditional is a (restricted) universal quantifier, saying that all the if -relevant possibilities at which the antecedent is true are possibilities at which the consequent true. When the only job of contexts is to provide such worlds as it is now let s simply identify a context c with the selection function that delivers the if -relevant for each world w. 9 All the beer is gone! often does not mean that the universe is out of beer but something more modest like the relevant beer supply is out. 8 Definition 2 (Strict Conditionals). Let c(w) be the set of if -relevant worlds at w (in c). ( if p )( q ) c,w = 1 iff if v c(w) and p c,v = 1 then q c,v = 1. This is equivalent to saying that ( if p )( q ) is true at w with respect to c iff the material conditional p q is true at every world in c(w). What does it take to be an if -relevant world? We haven t said (that s by design). Thus depending on what we say about what it takes for a world to be an if -relevant world at w (that is, depending on what we say about the function c given an argument w), we get a different strict conditional. Here are some possibilities (not exhaustive): For any w, only w ever matters (for every w : c(w) = {w}). The resulting strict conditional is the material conditional. For any w, all worlds (unrestricted!) always matter (for every w : c(w) = W ). The resulting strict conditional is strict implication (true iff the antecedent entails the consequent). For any w, all worlds compatible with what is known in w matter (for every w : c(w) = {v : if X is known at w then v X}). The resulting strict conditional is a sort of epistemic strict conditional. For any w, all worlds similar to w to fixed degree d matter (for every w : c(w) = { v : v is similar to w to at least degree d } ). The resulting strict conditional is a sort of similarity-based strict conditional. Notice that for each such c(w) there is a corresponding (restricted) necessity operator c(w) and a strict conditional ( if p )( q ) with respect to c(w) amounts to claiming that the corresponding material conditional p q is c(w) -necessary. The strictness of two strict conditionals can be compared: if the set of if -relevant worlds (at a world) for one strict conditional is included in the set of
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