Disjunctive Antecedent Conditionals ∼ Justin Khoo ∼ Forthcoming, Synthese Abstract Disjunctive antecedent conditionals (DACs)-conditionals of the form if A or B, C-sometimes seem to entail both of their simplications (if A, C; if B, C) and sometimes seem not to. I argue that this behavior reveals a genuine ambiguity in DACs. Along the way, I discuss a new observation about the role of focal stress in distinguishing the two interpretations of DACs. I propose a new theory, according to which the surface form of a DAC underdetermines its logical form: on one possible logical form, if A or B, C does entail both of its simplications, while on the other, it does not. An outstanding issue in the logic of conditionals concerns the status of the principle of simplification of disjunctive antecedents: simplification of disjunctive antecedents (SDA): if A or B, C entails if A, C and if B, C. e controversy over SDA stems from the fact that there is evidence both for its validity and for its non-validity. On the one hand, assertions of simple disjunctive antecedent conditionals (DACs) suggest that SDA is valid: (1) If Amy or Beth comes to the party, it will be fun. a. If Amy comes to the party, it will be fun. b. If Beth comes to the party, it will be fun. ere is a strong intuition that (1) entails both (1-a) and (1-b). Someone endorsing (1) seems to commit themselves to endorsing both (1-a) and (1-b). For instance, compare A's two responses to B in the following dialogue: A: If Amy or Beth comes to the party, it will be fun. B: I disagree. If Beth comes, it will be horrible – Beth always ruins parties. A: Oh I did not know that. I guess I was wrong! / #Oh I did not know that. I stand by what I said. Disjunctive Antecedent Conditionals It is natural for someone who has assertively uttered (1) to retract her claim aer accepting that one of its simplications is false. By contrast, would be very odd for that person to stand by her claim in such circumstances. is is evidence that (1) entails both (1-a) and (1-b), and thus evidence that SDA is valid. On the other hand, assertions of specicational DACs like (2) (which state which of their antecedent disjuncts will obtain) seem to suggest that SDA is not valid: (2) If the US spends more than half its budget on defense or education, it will spend more than half its budget on defense. a. If the US spends more than half its budget on defense, it will spendmore than half its budget on defense. b. If the US spends more than half its budget on education, it will spend more than half its budget on defense. In contrast with (1), (2) seems not to entail both of its simplications-in particular, it seems not to entail the obviously false (2-b), since the falsity of (2-b) does not seem sucient for the falsity of (2). Put dierently, someone may endorse (2) without thereby being committed to (2-b). How should we reconcile these conicting considerations? One strategy is to hold that SDA is valid, and then account for the apparent counterexamples by appealing to pragmatic shis in context (this strategy is endorsed by Fine 2012a,b, Willer 2015). Alternatively, we might hold that SDA is invalid, and try to account for the evidence supporting it by appealing to implicatures (see Bennett 2003, Klinedinst 2007, van Rooij 2010, Franke 2011). Or, we might hold that DACs are genuinely ambiguous between an SDA-valid interpretation and an SDA-invalid interpretation (see Alonso-Ovalle 2009, Santorio forthcoming). Given the possibility of a univocal theory, it may seem that positing an ambiguity in DACs should be our last resort. However, in this paper, I argue that there are actually reasons to favor an ambiguity theory of DACs. In particular, I propose a theory of DACs on which their surface forms underdetermine their logical forms: if A or B, C can have as its logical form either of the following:1 1A brief note about my choice of notation for what follows. I use italics to denote expressions of English (if, or, etc) and uppercase italic letters for variables over sentences (A, B, C, . . . ). I use sans serif to denote the logical forms of English expressions (if, or, etc) and uppercase sans serif letters for variables over the logical forms of English sentences (A, B, C, . . . ). Finally, I use uppercase boldface letters to denote the propositional contents of the corresponding sentences (A, B, C, . . . ) at the relevant contexts. I also assume that propositions are just sets of possible worlds. Combinations of particular expressions and variables are to be read as having invisible corner quotes: these are sentence schemas whose instances are particular sentences of English, replacing the sentence variables with English sentences. So, if A, C is a sentence schema whose instances include: 2 Disjunctive Antecedent Conditionals ▸ [ [ if A or B ] C ] ▸ [ [ if A or if B ] C ] us, SDA can be divided into two distinct logical principles, one for each of these logical forms: (SDA1): [ [ if A or B ] C ] entails [ [ if A ] C ] and [ [ if B ] C ]. (SDA2): [ [ if A or if B ] C ] entails [ [ if A ] C ] and [ [ if B ] C ]. My claim is then that (SDA1) is invalid and (SDA2) is valid.2 e paper is structured as follows. In §1, I provide new independent evidence that DACs are ambiguous between a strong interpretation (that entails both simpli- cations), and a weak interpretation (that only entails the disjunction of their simplications). en, in §2, I discuss a new observation that focal stress can be used to disambiguate DACs. I relate this observation to an observation of Will Starr's (Starr 2014) about DACs coordinating two if-clauses-sentences like: (3) If John draws a gold coin or if John draws a silver coin, he will win. (i) a. If John comes to the party, Sue will come to the party. b. If Amy or Beth comes to the party, it will be fun. I will generally be loose regarding the instances of the scope of disjunction when talking about DACs. So among the instances of if A or B, C will include: (ii) If Amy comes to the party or Beth comes to the party, it will be fun. as well as (iii) If Amy or Beth comes to the party, it will be fun. Finally, I will be restricting attention to indicative conditionals throughout. Historically, the discussion of disjunctive antecedents has focused on subjunctive conditionals. However, to simplify the discussion and to sidestep some complications arising from the probabilities of subjunctive conditionals, I will focus on indicative conditionals. Since indicative and subjunctive DACs behave analogously, the reader is welcome to import my conclusions to the subjunctive domain. 2Strictly speaking, my theory predicts that SDA is invalid, since it quanties over every instance of the sentence schema if A or B, C (in every context) and my theory predicts that some instances of that schema will, in some contexts, be an instance of (SDA1). But I think it is helpful to think about the upshot of my theory in terms of the bifurcated SDA-principles. 3 Disjunctive Antecedent Conditionals I connect these two data points by proposing a new hypothesis about focal stress (The FocusHypothesis). In §3, I statemy theory ofDACs and showhow it predicts the data discussed in §1-2. Finally, I conclude in §4 with remarks on future research. 1 New data for DACs Existing theories of DACs can be classied broadly into three kinds: Univocal Strong if A or B, C entails both if A, C and if B, C. . (Fine 2012a,b, Willer 2015, Forthcoming) UnivocalWeak if A or B, C does not entail both if A, C and if B, C. . (Loewer 1976, Bennett 2003, Klinedinst 2007, van Rooij 2010, Franke 2011) Ambiguity if A or B, C is ambiguous between two readings: (i) Strong: if A or B, C entails both if A, C and if B, C.. (ii) Weak: if A or B, C does not entail both if A, C and if B, C.. . (van Rooij 2005, Alonso-Ovalle 2009, Santorio forthcoming) Versions of all three kinds of theories are positioned to account for the observations about DACs discussed in the introduction, despite reaching dierent conclusions about the status of SDA. Start with Univocal Strong theories, which entail that SDA is valid. Such theories easily account for the observation that DACs like (1) seem to entail both of their simplications. (1) If Amy or Beth comes to the party, it will be fun. Wenow turn to see howoneparticularUnivocal Strong theory, due to Fine 2012a,b, Willer 2015, accounts for the intuitions surrounding specicational DACs like (2). Roughly, according to this kind of theory, although (2) entails (2-b), this does not prevent (2) from being true as uttered in certain contexts. (2) If the US spends more than half its budget on defense or education, it will spend more than half its budget on defense. a. If the US spends more than half its budget on defense, it will spendmore than half its budget on defense. 4 Disjunctive Antecedent Conditionals b. If the US spends more than half its budget on education, it will spend more than half its budget on defense. e strategy holds that (2) is true as uttered in a context in which it is not epistemically possible that theUSwill spendmore than half its budget on education-indeed, asserting (2) is a way of indicating this. In that same context, (2-b) is predicted to be trivially true. But (2-b) is not assertable in a context in which it is not epistemically possible that the US will spend more than half its budget on education-therefore, uttering (2-b) tends to change the context by accommodation, which expands the space of epistemic possibilities to include ones in which the US spends more than half its budget on education.en, in this new context, (2-b) is false.us, according to this strategy, (2) seems true and (2-b) seems false because they are most naturally evaluated in dierent contexts. Next, turn to Univocal Weak theories, which, by contrast, easily account for the behavior of specicational DACs like (2), since they predict that SDA is invalid. However, to account for the behavior of DACs like (1), which do seem to entail both of their simplications, UnivocalWeak theorists typically oer a pragmatic story. I will focus on a theory inspired by ideas in Bennett 2003, Klinedinst 2007. is theory begins with a variably strict theory of conditionals (?Lewis 1973). On such a theory, if A, C is true i all of the closestA-worlds are C-worlds. is kind of theory predicts that DACs do not entail both of their simplications. Suppose that all of the closest A ∪ B-worlds are A-worlds, and all of these are C-worlds, so that if A or B, C is predicted to be true. is may be the case even if none of the closest B-worlds are C-worlds, in which case if B, C is predicted to be false. However, there is also an interpretation of if A or B, C (given this semantics) on which it is true only if both of its simplications are true. For any world w at which there are both A-worlds and B-worlds among the closest A ∪ B-worlds to w, if A or B, Cwill be true atw only if both if A, C and if B, C are true atw.3 So, given the extra premise that the set of closest A ∪ B-worlds contains both Aand B-worlds, if A or B, Cwill entail both of its simplications (on this kind of semantics). is fact opens up space for the Univocal Weak theorist to account for the simplifying behavior of ordinary DACs. e idea is that for certain reasons we, by default, interpret the closeness relation underlying if A or B, C in such a way that it entails both of its 3Here is the derivation. Suppose w is such that (i) there are both A-worlds and B-worlds among the closest A ∪ B-worlds to w, and (ii) if A or B, C is true at w. From (i), it follows that the closest A/B-worlds to w are a subset of the closest A ∪ B-worlds to w. And from (ii) it follows that all of the closest A ∪ B-worlds to w are C-worlds. But then all of the closest A-worlds to w are C-worlds, in which case if A, C is true at w; and likewise all of the closest B-worlds to w are C-worlds, in which case if B, C is true at w. 5 Disjunctive Antecedent Conditionals simplications.4 Finally, Ambiguity theories can account for the behavior of both (1) and (2): these theories predict that DACs are ambiguous between an SDA-valid reading and an SDA-invalid reading. However, ambiguity theories face the threat of overgeneration: they predict that (1) has an SDA-invalid reading and that (2) has an SDAvalid reading, contrary to how things seem. To avoid the threat of overgeneration, Alonso-Ovalle 2009 holds that DACs have their strong interpretation by default, and their weak interpretation arises only as a "last resort strategy to avoid interpreting examples like [specicational DACs, e.g, (2)] as contradictions" (Alonso-Ovalle 2009: 239). is predicts that the weak interpretation of a DAC is available only if its strong interpretation is necessarily false (given the relevant background assumptions). Since (1) is not necessarily false, it is predicted to only have the strong reading. And since (assuming it is epistemically possible that theUSwill spendmore than half of its budget on education) (2-b) is necessarily false, this means that (2) would also be necessarily false on the strong interpretation, so it is predicted to have only the weak interpretation. us, existing versions of each kind of theory of DACs have strategies for handling the behavior of DACs like (1) and (2). To advance the dialectic, we will have to look at a broader range of data. In the rest of this section, I aim to do this by looking at data about the probabilities of DACs.5 1.1 Probabilities of DACs Start with the following case: 4For some representatives of this strategy, see Bennett 2003, Klinedinst 2007, Franke 2011. 5An alternative avenue to pursue would be to look at dierent kinds of simplifying conditionals. e rst are free-choice antecedent conditionals, which clearly only carry the strong, entailing, reading: (i) If John draws any metal coin, he will win. e second are negated conjunctive antecedent conditionals. By DeMorgan's Law, A∨B is equivalent to ¬(¬A ∧ ¬B). So, the antecedent of the following conditional is epistemically equivalent to the antecedent of (4): (ii) If it is not the case that John both does not draw gold and does not draw silver, he will win. Unlike (4), (ii) is very dicult to evaluate, so it is hard to say whether it has a strong (simplifying) reading. Using more natural examples, Ciardelli et al. forthcoming nd that ordinary speaker intuitions suggest that conditionals like (ii) behave dierently from DACs like (4). I set aside further discussion of these issues for future work. 6 Disjunctive Antecedent Conditionals Coins.ere are 100 coins in an urn: 90 gold, 9 silver, and 1 plastic.e gold coins are all winners, while the silver and plastic coins are losers. Given this information, how probable is the following DAC? (4) If John draws a gold coin or a silver coin, he will win. It seems to me that there are two equally good answers to this question: ▸ Answer 1: (4) is highly probable. ▸ Answer 2: (4) is certainly false. ere are precedents for both of these answers in the conditionals literature. We have already seen the precedent for Answer 2: non-specicational DACs seem to entail both of their simplications, but since one of (4)'s simplications, (5-b), is certainly false, we may thus conclude that (4) must be certainly false (since a sentence cannot be more probable than something it entails). (5) a. If John draws a gold coin, he will win. b. If John draws a silver coin, he will win. Indeed, additional evidence that Answer 2 is a reasonable conclusion to draw about the probability of (4) is that it is natural to reason from the falsity of one of its simplications to the falsity of (4): (6) is much is certain: if John draws a silver coin, he will lose. So, we may conclude that it is certainly false that if John draws a gold coin or a silver coin, he will win. A precedent for Answer 1 comes from The Thesis:6 The Thesis e probability of if A, C is equal to the probability of C given A. Suppose I roll a fair die and ask you how probable it is that if the die landed on a prime, it landed on an odd; it seems that the correct answer is two-thirds. But this 6An early incarnation of The Thesis appeared in Ramsey 1931, but the modern version of the claim is due to Stalnaker 1970. ere is a great deal of empirical support for The Thesis (see for instance Douven & Verbrugge 2013), and most theorists think that some version of The Thesis is correct (perhaps a restricted version, or one about assertability or degrees of belief): see Adams 1975, Edgington 1995, Rothschild 2013b,a, Bacon 2015. 7 Disjunctive Antecedent Conditionals just is the conditional probability that the die landed on an odd, given that it landed on a prime. Examples like this provide evidence supporting The Thesis. Given The Thesis, we predict that the probability of (4) should equal the probability that John will win, given that he draws a gold or silver coin. And this probability is high; in fact, it is approximately equal to .91.7 Is it plausible that The Thesis should apply to DACs like (4)? I think it is. e following reasoning about (4), which is an informal version of the calculation of (4)'s probability given The Thesis, seems quite natural: (7) If John draws gold or silver, he will very probably draw gold; then, since all the gold coins are winners, he will win. So, we may conclude that it is likely that if John draws a gold coin or a silver coin, he will win. So, we have some reason to think that there are two, equally reasonable but incompatible, answers to the question how probable (4) is. If this is right, this is evidence that (4) has two distinct interpretations: a weak interpretation on which it is highly probable and a strong interpretation on which is certainly false. To explore whether these judgments are shared by native speakers of English, I conducted an experiment. 1.1.1 Experiment 1: Probabilities Participants for this experiment were recruited through AmazonMechanical Turk.8 Each participant was presented with a second person version of the Coins scenario, and then was told to keep in mind one of the following primes: Weak: Since there are mostly gold coins, if you draw one of the gold or silver coins, you will very likely draw a gold coin, and hence win. Strong: If you draw a gold, you will win; if you draw a silver coin, you will lose. 7Here is the calculation: P(W∣G ∪ S) = P(W∣G) ⋅ P(G∣G ∪ S) + P(W∣S) ⋅ P(S∣G ∪ S) = 1 ⋅ 90 99 + 0 ⋅ 9 99 ≈ .91 8119 participants were recruited using Amazon's Mechanical Turk. Sample was 61 percent male, mean age 36. Only answers from participants who self-reported as native speakers of English were used. Raw data les for this experiment and the others discussed below can be found at https://osf.io/nyaqk/. 8 Disjunctive Antecedent Conditionals e primes are a way of recreating the kind of reasoning discussed above, in support ofAnswer 1/Answer 2 respectively. Aer reading their prime, participants were asked whether they agreed or disagreed with one of following statements, answering on a scale from 1 ("Completely Disagree") to 7 ("Completely Agree"): (D1) It is likely that: if you draw a gold coin or silver coin, you will win. (I1) It is likely that: if you draw a metal coin, you will win. (D1) and (I1) constitute a reasonable minimal pair-one is a DAC and one isn't, but given the background information, their antecedents are equivalent: to draw a gold or silver coin just is to draw a metal coin (the fact that gold and silver are metals and plastic is not a metal was stated explicitly in the prompt). erefore, if we nd that participants agree more with (D1) in Strong than in Weak, but we don't nd this pattern with (I1), this would be evidence that ordinary English speakers do nd bothAnswer 1 andAnswer 2 plausible judgments about probability of (4), and hence evidence that DACs have both a strong and weak interpretation. e results of the experiment are shown in the following graph: Figure 1: Mean responses for Experiment 2. Error bars show standard error of the mean. We can see from the graph that the results of the experiment do showwhat we expect if there really are two compelling answers to the question of how probable DACs like (4) are.e crucial result is that, while the dierence inmean agreement with (D1) is signicant between the Strong andWeak conditions,9 there is no dierence inmean 9Aplanned comparisonwas used to compare agreement that (D1) is likely in the Strong condition 9 Disjunctive Antecedent Conditionals agreement with (I1) between the Strong and Weak conditions.10 is suggests that there really are two interpretations of (D1): one on which it is likely true and one on which it is not likely true.11 1.1.2 Discussion We have seen both armchair reasoning and experimental evidence in support of the conclusion that both Answer 1 and Answer 2 are plausible answers to the question how probable (4) is, given the information in Coins. us, we now have two sources of evidence supporting the observation that DACs like (4) have two distinct interpretations-a strong interpretation on which they entail both of their simplications and a weak interpretation on which they entail only the disjunction of their simplications. How does this observation bear on the three kinds of theories discussed above? Start with Univocal Strong theories. Such theories predict that Answer 2 is correct and Answer 1 is incorrect: they predict that (4) is certainly false because it entails something (that if John draws a silver coin, he will win) that is certainly false. us, such theories must nd some way to explain why Answer 1 seems appealing, even though it is incorrect. As with specicational DACs like (2), a Univocal Strong theorist might appeal to context shiing here to explain why we are sometimes led to think that (4) is more likely than its simplication (5-b). However, unlike with specicational DACs, it seems that context-shiing will not help the Univocal Strong theory explain the appeal of Answer 1. With specicational DACs, there is a reasonable motivation for context-shiing. Recall that, on that strategy, the univocal strong theory predicts that (2) is true because the US spending more than half its budget on education is not an epistemic possibility in that context-in eect, that is what the conditional says according to this theory. As such, the theory still predicts that (2-b) is true-just trivially so. However, when we (M = 3.00, SD = 2.2) with agreement that (D1) is likely in theWeak condition (M = 4.67, SD = 1.9), t(59) = 3.2, p < .01, d = .836. 10A planned comparison was used to compare agreement that (I1) is likely in the Strong condition (M = 5.96, SD = 1.5) with agreement that (I1) is likely in the Weak condition (M = 6.00, SD = 1.3), t(56) = −0.1, p = .92. 11Speakers do show more resistance to the weak reading of DACs than indenite antecedent conditionals: in theWeak condition, the mean agreement with (D1) (M = 4.67, SD = 1.9) is signicantly lower than mean agreement with (I1) (M = 6.00, SD = 1.3), t(58) = 3.2, p < .05. However, we expect this result, given that we expect that our attempts to contextually prime the weak reading will not lead every participant to interpret (D1) in that way. As such, we expect that some participants will access the strong interpretation of (D1) even in the context where we tried to prime its weak interpretation (Weak); those participants should then judge (D1) to be false, even in theWeak condition. By contrast, we expect nothing similar from (I1), since it has only one interpretation, on which it is true. 10 Disjunctive Antecedent Conditionals come to assert (2-b), we expand the domain of possibilities in the context to include some where the US spends more than half of its budget on education. en, (2-b) will be false, as evaluated in this new context. However, this strategy will not predict that the truth of (4) is more likely than the truth of (5-b). e reason is that the context in which we evaluate (4) is already one in which it is epistemically possible that John will draw a silver coin-(4) is judged to be probable despite it being possible that John might draw silver (if there were no silver possibilities, (4) would have probability 1 on its weak reading, but that is clearly incorrect). So, no context shi should occur when we move from evaluating (4) to (5-b). erefore, it is unclear how the same context-shiing strategy invoked by Fine 2012a,b, Willer 2015 to handle specicational DACs will help the univocal strong theory make sense of the observation that (4) seems to have a weak interpretation on which it is probable. is is by itself not a knockdown objection to every Univocal Strong theory. It remains to be seen whether an alternative strategy might be able to help such a theory account for the observation about the probability of DACs. But the foregoing discussion does cast doubt on the future success of such theories. Turn next to UnivocalWeak theories. Since they predict that if A or B, C does not entail both of its simplications, such theories are in a position to predict that Answer 1 is correct. But how, then, do they account for the intuitions underlyingAnswer 2? It turns out that the strategy used to account for specicational DACs can also predict these conicting intuitions about the probabilities of DACs. Given the Lewisian Univocal Weak semantics from above, we can distinguish between antecedent underspecication and antecedent indeterminacy. If A or B, C is antecedent underspecied at w just if either all of the closest A ∪ B-worlds to w are A-worlds or all of the closest A ∪ B-worlds to w are B-worlds. By contrast, if A or B, C is antecedent indeterminate at w just if the closest A ∪ B-worlds to w contain some A-worlds and some B-worlds. How can we appeal to this dierence in interpretation of the closeness relation to account for the intuitions underlying bothAnswer 1 andAnswer 2? Suppose rst that if A or B, C is antecedent underspecied at each epistemically possible world and that the probability that all of the closestA ∪ B-worlds areA-worlds just is the probability of A given A ∪ B (and likewise for the probability that all the closest A ∪ Bworlds areB-worlds). en, this theory will predict that the probability of (4) is .91.12 12Here is the calculation: (i) P(if G or S, W) = P(if G or S, W ∣ if G or S, G) ⋅ P(if G or S, G) + P(if G or S, W ∣ if G or S, S) ⋅ P(if G or S, S) = P(if G, W) ⋅ P(G∣G ∪ S) + P(if S, W) ⋅ P(S∣G ∪ S) 11 Disjunctive Antecedent Conditionals Next, suppose instead that if A or B, C is antecedent indeterminate at each epistemically possible world. en, if A or B, C will be true at any such world only if both if A, C and if B, C are both true at that world. us, if if A or B, C is antecedent indeterminate at each epistemically possible world, then the theory predicts that the probability of (4) is 0.13 us, a UnivocalWeak theory is in a position to account for the intuitions underlying both Answer 1 and Answer 2. To be clear, the theory achieves this result without positing ambiguity: rather, the theory does so by appealing to underdetermination of a context-dependent parameter-the closeness relation governing conditionals. Finally, turn to Ambiguity theories. Such theories predict that there is an interpretation of if A or B, C-its weak interpretation-whose probability may exceed the probability of one of its simplications. So, it is open to the Ambiguity theory to account for the intuitions underlying Answer 1 in exactly the way the Univocal Weak theory does above: on one interpretation of (4), its probability just is .91. But Ambiguity theories account for the intuitions underlying Answer 2 dierently, by appealing to the strong interpretation of if A or B, C. On its strong interpretation, the probability of (4) is 0, because on that interpretation (4) entails (5-b), and the probability of (5-b) is 0. Let us recap what has been covered so far. By looking at the probability of (4), rather than judgments about its truth value, we found evidence that DACs have both weak and strong interpretations. We also saw that the intuitions supporting this observation are hard to account for givenUnivocal Strong. Finally, we saw how both a Univocal Weak theory and an Ambiguity theory can predict this observation. erefore, wewill have to look to other data to distinguishUnivocalWeak theories from Ambiguity theories. 1.2 Probabilied DACs In the previous section, we looked at intuitions about the probabilities of DACs. In this section, we will look at intuitions about probabilied DACs. ese are instances of the schema if A or B, probably C, such as: (8) If John draws a gold coin or a silver coin, he will probably win. = 1 ⋅ 90 99 + 0 ⋅ 9 99 ≈ .91 13Notice that this assumption is incompatible with strong centering: the principle that states that if A is true at w, then the closest A-world to w are just {w}. us, someone adopting this kind of strategy to defend a UnivocalWeak theory must deny strong centering. 12 Disjunctive Antecedent Conditionals Before we get to the predictions, it will be helpful to make two independently motivated assumptions about probabilied conditionals. e rst assumption is about the semantics of probably (cf. Yalcin 2010): (9) Probably A is true i the probability of A is greater than 0.5. e second assumption is about probabilied conditionals: it is that the if-clause of a probabilied conditional updates the domain of the probability operator in its scope (cf. Kratzer 1986, 2012, Yalcin 2010), yielding the following truth conditions for probabilied conditionals without disjunctive antecedents: (10) If A, probably C is true i the probability of C given A is greater than 0.5. ese two assumptions are motivated by reection on the behavior of probabilied conditionals. Consider, for instance, the following case from Kratzer 1986: Yog and Zog played 100 games of chess last night. ere were no draws. Yog hadwhite for 90 of the games. Yogwon 80 of the 90 games he played as white. But Yog lost 10 of the 10 games he had as black. Focus on the last game Yog and Zog played, and consider the following two probabilied conditionals, both about this game: (11) If Yog was playing white, he probably won. (12) If Yog lost, he was probably playing black. Intuitively, (11) is true while (12) is false. Why? A plausible explanation appeals to our two assumptions above. (11) is true because the probability that Yog won given that he was playing white is greater than 0.5-indeed, this conditional probability value is around .89. And (12) seems false because the probability that Yog was playing black given that he lost is not greater than 0.5-in fact, this conditional probability value is equal to 0.5. us, given those assumptions, we account for the intuition that (11) is true and (12) is false; as such, these intuitions are evidence for those assumptions. With our two assumptions in hand, we turn next to consider probabilied DACs like (8). In particular, notice that the UnivocalWeak theory discussed in the previous section predicts that such probabilied DACs should only have one interpretation: ProbabiliedWeak: if A or B, probably C is true i the probability of C given A ∪ B is greater than 0.5. 13 Disjunctive Antecedent Conditionals By contrast, an Ambiguity theory predicts that probabilied DACs are ambiguous between a probabilied weak interpretation and a probabilied strong interpretation: Probabilied Strong: if A or B, probably C is true i if A, probably C and if B, probably C are both true (i.e., i the probability of C given A is greater than 0.5 and the probability of C given B is greater than 0.5). But wait, shouldn't also Univocal Weak theories predict that probabilied DACs can be interpreted in a way equivalent to the probabilied strong interpretation? Above, we showed how the semantically encoded weak truth conditions were sometimes strengthened-in particular, when if A or B, C is antecedent indeterminate at w, it is true at w only if both if A, C and if B, C are true at w. Does the same move yield similar results for probabilied DACs? It turns out, it does not.14 To see why, suppose the following: ▸ e probability of C given A ∪ B is greater than 0.5, ▸ e probability of A given A ∪ B is nonzero, and ▸ e probability of B given A ∪ B is nonzero. e rst supposition ensures if A or B, probably C is true on its probabilied weak interpretation, while the second and third ensure that the conditional is antecedent indeterminate-that there are someA-worlds and someB-worlds among the closest A ∪ B-worlds. Recall that it was the assumption that the conditional was antecedent indeterminate that allowed us to predict the strong (simplication-entailing) interpretation of a non-probabilied DAC within our Lewisian UnivocalWeak theory. However, even given these suppositions, it may be that if B, probably C is false, because it may still be the case that the probability of C given B is less than 0.5. is is because it is possible that the probability of C given A ∪ B is high because the probability of C given A is high, and because the probability of A given A ∪ B is high.15 14A similar point is observed in Alonso-Ovalle 2009, Santorio forthcoming. 15Coins is a situation with these properties. Here, we have G/S = that John draws gold / that John draws silver andW = that John wins. ▸ P(G∣G ∪ S) = .91 ▸ P(S∣G ∪ S) = .09 ▸ P(W   ∣G) = 1 ▸ P(W∣S) = 0. 14 Disjunctive Antecedent Conditionals erefore, if A or B, probably C being antecedent indeterminate at each epistemically possible world is not sucient to predict the probabilied strong reading of probabilied DACs on a Lewisian Univocal Weak semantics. us, probabilied conditionals provide a point at which the predictions of our Lewisian UnivocalWeak theory diverge from those of Ambiguity theories. It remains to be seen, though, what kinds of readings probabilied DACs really do have. To explore this issue, I conducted an experiment modeled along the same lines as Experiment 1 above. 1.2.1 Experiment 2: Probabilied DACs Participants for this experiment were recruited through AmazonMechanical Turk.16 Aswith Experiment 1, each participant was presentedwith a second personal version of theCoins scenario, and thenwas told to keep inmind one of the following primes: Weak: Since there are mostly gold coins, if you draw one of the gold or silver coins, you will very likely draw a gold coin, and hence win. Strong: If you draw a gold, you will win; if you draw a silver coin, you will lose. en, participants were again divided into two groups. ose in the DAC group were asked whether they agreed or disagree with the following probabilied DAC, as before answering on a scale from 1 ("Completely Disagree") to 4 ("In Between") to 7 ("Completely Agree"): (D2) If you draw a gold coin or silver coin, you will likely win. ose in the Indenite group were asked whether they agreed or disagreed with the following probabilied non-DAC: (I2) If you draw a metal coin, you will likely win. Given the information presented, on the probabilied weak interpretation, (D2) is true (since it is likely that you will win given that you draw one of the gold or silver coins), while on the probabilied strong interpretation, (D2) is false (since it is not likely that you will win given that you draw a silver coin). us, if we nd that ▸ en P(W∣G ∪ S) = P(W∣G)  111111111111 111111111111¶ 1 ⋅ P(G∣G ∪ S)  111111111111111111111111 11111111111111111111111¶ 0.91 + P(W∣S)  1111111111 1111111111¶ 0 ⋅ P(S∣G ∪ S)  1111111111111111111111 111111111111111111111¶ 0.09 = .91 + 0 = .91 16137 participants were recruited using Amazon's Mechanical Turk. Sample was 57 percent male, mean age 34. Only responses from participants who self-reported as native speakers of English were used. 15 Disjunctive Antecedent Conditionals participants agreed more with (D2) in Strong than in Weak, but do not nd this pattern for (I2), this would be evidence that probabilied DACs have both a probabilied weak and a probabilied strong reading. By contrast, if we nd instead that participants agree to the same degree with (D2) across the Strong/Weak conditions, this would be evidence that probabilied DACs have only either the probabilied weak or the probabilied strong reading (depending on whether they overall agree or overall disagree with (D2)). e results of the experiment are shown in the following graph: Figure 2: Mean responses for Experiment 2. Error bars show standard error of the mean. We can see from the graph that the results of the experiment are evidence that probabilied DACs do have both probabilied weak and probabilied strong readings. e crucial result is that, while the dierence in mean agreement with (D2) is signicant between the Strong andWeak conditions and crosses the midpoint,17 there is no signicant dierence in mean agreement with (I2) between the Strong and Weak conditions.18 us, the results of Experiment 2 are evidence that probabilied DACs do have both probabilied strong and probabilied weak interpretations, in line with the predictions of Ambiguity but not our Lewisian UnivocalWeak theory (or Univocal Strong theories for that matter). 17A planned comparison was used to compare agreement with (D2) in the Strong condition (M = 3.30, SD = 2.1) with agreement with (D2) in the Weak condition (M = 5.28, SD = 1.9), t(63) = −4.0, p < .001, d = −.99. 18A planned comparison was used to compare agreement with (I2) in the Strong condition (M = 5.49, SD = 1.6) with agreement with (I2) in the Weak condition (M = 6.14, SD = 1.2), t(70) = −1.9, p = .06. 16 Disjunctive Antecedent Conditionals Summary Here is where things stand. In §2.1, we found evidence that DACs like (4) have both a strong and a weak interpretation. is observation made trouble for Univocal Strong theories, but was compatible with our Lewisian Univocal Weak theory and Ambiguity theories. In §1.2, we found evidence that probabilied DACs like (8) have both a probabilied strong and probabilied weak interpretation, an observation that is not predicted by our Lewisian Univocal Weak theory, but which is predicted by an Ambiguity theory. us, Ambiguity theories seem to be in the best shape, given the foregoing evidence. Of course, the discussion here is still preliminary-it could turn out that a more sophisticated univocality theory may offer a better explanation for the foregoing data, andmore evidencemay surface which pushes against Ambiguity in favor of some kind of univocality theory. Rather than attempt to anticipate every possible response onbehalf ofUnivocal Strong/Univocal Weak theories, in the rest of the paper, I precede under the assumption that DACs are indeed ambiguous between a weak and strong interpretation. In what follows, I will explore the question of what kind of ambiguity theory of DACs is correct. 2 e role of focus In this section, I discuss a new observation about DACs, which is that focal stress on or tends to disambiguate in favor of the strong interpretation, in contrast with pronouncing the disjunctive antecedent "as a block," running the two disjuncts together with at intonation. For quasi-prosodic notation, let capital letters indicate focal stress and 'x-y-z' indicate pronouncing 'x,' 'y,' 'z' without pauses or intonational dierentiation: (13) a. If John draws a gold coin OR a silver coin, he will win. . [Or-emphasis] b. If John draws a gold-coin-or-a-silver-coin, he will win. . [Flat intonation] My intuition is that uttering (13-a) most naturally has the strong interpretation, in contrast with uttering (13-b). Call this the Focus Observation: FocusObservation: Uttering aDACwhile emphasizing the or in its antecedent tends to bias its strong interpretation. Another piece of evidence supporting the Focus Observation comes from comparing specicational DACs with and without or-emphasis: 17 Disjunctive Antecedent Conditionals (14) a. #If John draws a gold coin OR a silver coin, he will draw a gold coin. b. If John draws a gold-coin-or-a-silver-coin, he will draw a gold coin. Given that John will only draw one coin, my intuition is that (14-a) is quite odd, in contrast with (14-b), which is ne. is is further evidence supporting the Focus Observation. I pause to note that my statement of the Focus Observation is purposefully vague. Focal stress has many uses. Below, I suggest that focal stress plays a role in mediating transformations between surface and logical form, but focal stress is also used to indicate focus marking (F-marking), or contrast (contrastive focus). Given these other uses of focal stress, it is not surprising that dierential focal stress on or may be interpreted in a variety of ways. My claim is not that every utterance of (4) with or-emphasis will result in its being read with the strong interpretation; rather, my claim is that there is a use of focal stress that has this eect.19 Before we move on to thinking about what might account for the Focus Observation, I want to provide some additional empirical support for it. I turn to this support in the next section. 2.1 Experiment 3: Focal stress As before, participants for this experimentwere recruited throughAmazonMechanical Turk.20 Participants were divided into two groups: Emphasis and Flat. Each participant read the Coins vignette, and then the weak prime: Weak: Since there are mostly gold coins, if you draw one of the gold or silver coins, you will very likely draw a gold coin, and hence win. ose in the Emphasis group then listened to a recording of someone saying (E), with the indicated or-emphasis: 19For instance, DACs with contrastive focus on or do not fall into the generalization reported in the Focus Observation. Here is an example: (i) I'm not sure if John will draw a gold coin or a silver coin, or a gold coin AND a silver coin. But even if he draws a gold coin OR a silver coin, he'll win (since he'll draw a gold coin). Here, or is stressed, but the stress is used to indicate a contrast between or and and (appearing in the previous sentence). Although this contrastive use of focal stress is denitely possible, it requires prior linguistic material to be contrasted with. erefore, I will set aside such cases in our discussion of the Focus Observation. anks to an anonymous reviewer for bringing this kind of example to my attention. 2081 participants were recruited; the sample was 47 percent male, mean age 34. Only answers from participants who self-reported as native speakers of English were used. 18 Disjunctive Antecedent Conditionals (E) It is likely that: if John draws a gold coin OR a silver coin, he will win. ose in the Flat group listened to a recording of someone saying (F), with the indicated at intonation: (F) It is likely that: if John draws a gold-coin-or-a-silver-coin, he will win. Each participant was then asked whether they agreed or disagreed with the statement, rating their answers on a scale from 1 ("Completely disagree") to 7 ("Completely agree"). Given that each participant read the weak prime, we expect that participants will tend to agree with (F), in line with the results of previous experiments. But if the Focus Observation is correct, we expect participants to tend to disagree with (E). In particular, we expect there to be signicantly less agreement with (E) than (F), in line with the previous experiments.21 Indeed, the pattern of responses matches the previous experiments: participants were more inclined to disagree with (E) (M = 3.20, SD = 2.3) than with (F) (M = 4.35, SD = 2.3), t(79) = 2.3, p = .026, d = .5. Figure 3: Mean responses for Experiment 3. Error bars show standard error of the mean. 21Why do I compare at vs. or-emphasis across conditions biasing the weak interpretation? e reason is that I think that the default interpretation of DACs is the strong interpretation (I discuss this below on page 33). So, without the weak prime, we would expect strong interpretations of the DACs in both conditions (at vs. or-emphasis). So, the ideal way to test the Focus Observation is to see whether, in a context in which we would otherwise expect the weak interpretation of some DAC to be salient, emphasis on or biases its strong interpretation. 19 Disjunctive Antecedent Conditionals us, I conclude that the results of this experiment are evidence that focal stress on or biases the strong reading of DACs. 2.2 e focus hypothesis Existing ambiguity theories of DACs do not predict the Focus Observation. I will briey discuss two such theories. Alonso-Ovalle 2009 proposes that the weak reading of DACs is due to the presence of a covert existential closure operator which is posited as a last resort strategy to avoid interpreting the DAC as necessarily false, while Santorio forthcoming proposes that the strong reading of DACs is due to the presence of a covert distributivity operator (similar to one oen posited to account for distributed readings of plural descriptions). However, neither of these accounts naturally predicts theFocusObservation. To do so, Alonso-Ovalle would have to say that focal stress on or indicates the absence of a covert existential closure operator; but there is no independent motivation for this kind of proposal. And Santorio would have to say that focal stress on or indicates the presence of a distributivity operator, but I cannot think of any independent evidence for this claim, either. For instance, Santorio appeals to an analogy between plural denites and conditional antecedents, so we might try to look for evidence there. Take a disjunctive plural denite that most naturally has the collective reading: (15) e boys or girls carried a piano. Does adding focal stress to or change this to a distributive reading (one in which each boy or each girl carried a piano)? (16) e boys OR girls carried a piano. It seems it does not. To my ear, (15) and (16) sound equivalent.22 A more promising strategy, I think, is to connect the Focus Observation with another observation about doubled disjunctive antecedent conditionals (DDACs)- instances of the schema if A or if B, C-due to Starr 2014. Notice rst that the most 22Here is another way to make this point. Notice that we can force a distributive reading for (15) by adding each: (i) e boys or girls each carried a piano. is means that either each of the boys carried a piano or each of the girls carried a piano. However, notice that merely adding focal stress to or is not sucient to yield this interpretation: (16) does not mean the same thing as (i). anks to an anonymous reviewer for suggesting this strategy. 20 Disjunctive Antecedent Conditionals natural interpretation of the following DDAC is that it entails both of its simplications: (17) If John draws a gold coin or if John draws a silver coin, he will win. Secondly, notice that specicational DDACs like (18) are infelicitous: (18) #If John draws a gold coin or if John draws a silver coin, he will draw a gold coin. is conditional seems infelicitous because it seems to entail something we know to be false (given the information in Coins): that if John draws a silver coin, he will draw a gold coin. So, it seems that DDACs lack the weak reading on which specicational DACs are felicitous. is suggests the possibility of accounting for the ambiguity in DACs as a syntactic ambiguity: when the conditional's LF contains two if-clauses coordinated by or, it has the strong reading, and when its LF contains a single, disjunctive, if-clause, it has the weak reading. Call the former the Double-if LF and the latter the Single-if LF: ▸ Double-if: [ [ if A or if B ] C ] ▸ Single-if: [ [ if A or B ] C ] e semantic claim is that the two LFs are assigned dierent interpretations, as follows: Strong Double-if Disjunctive conditionals with a Double-if logical form univocally have the strong (simplication-entailing) interpretation. Weak Single-if Disjunctive conditionals with a Single-if logical form univocally have the weak interpretation. On this explanatory strategy, DDACs are most naturally read as simplifying because they are most naturally interpreted as having a Double-if LF (whether they univocally have such an LF is a questionwe'll return to shortly).is strategy also promises a way of accounting for the Focus Observation, by appealing to the following hypothesis: The FocusHypothesis Dierential focal stress on a sentential connective '∗' appearing in surface form 21 Disjunctive Antecedent Conditionals under the scope of an operator 'O' can indicate the presence (at LF) of two occurrences of that operator taking narrow scope with respect to the connective. e hypothesis states that when you have a sentence with the following surface form: (19) O(φ ∗ ψ) in which the connective receives focal stress, this can indicate that the logical form of the sentence is in fact: (20) Oφ ∗Oψ How does this hypothesis connect our two observations above? Well, for DACs, the operator is if (understood not as a two-place operator as in standard logic textbooks, but as a one-place operator that maps a clause to a clause) and the connective is or. The FocusHypothesis predicts that focal stress on themain or of its antecedent can be used to indicate that the DAC has the Double-if LF.en, by StrongDouble-if, we predict that it would then have the strong interpretation. us, adopting The FocusHypothesis, Strong Double-If, and Weak Single-If has the potential to open up space for a new kind of ambiguity theory of DACs, one which can account for all of the data discussed so far in this paper. However, is The Focus Hypothesis even plausible? Start with the following example: (21) Everyone was inside or outside. ere is an interpretation of (21) in which it is trivially true: of course, everyone has the property of being inside or outside. However, notice that when you emphasize or, you get a dierent interpretation: (22) Everyone was inside OR outside. e natural reading of (22) is non-trivial-it is equivalent to:23 (23) Everyone was inside or everyone was outside. is pattern supports The Focus Hypothesis: here, the operator is everyone and the connective is or. Consider another example. Suppose you know that a particular die is weighted, but you cannot remember how it is weighted: either it is weighted to land only on '5', 23Just as with DACs, (21) has both the trivial and non-trivial interpretation. My claim is that emphasis on or helps to bias the non-trivial interpretation. 22 Disjunctive Antecedent Conditionals or it is weighted to only land on '6'. Now consider: (24) e die always lands on '5' OR on '6'. is sentence seems equivalent to "the die always lands on '5' or always lands on '6'." Notice, though, that if you were to say: (25) e die always lands on '5' or on '6'. this has the interpretation that is true if on every roll, the die lands either '5' or '6'. us, (25) is weaker than (24); (25) is compatible with the possibility that the die sometimes lands '5' and sometimes lands '6'. Since you know this is not the case, you shouldn't utter (25); to do so risks misleading your audience. is example also supports The FocusHypothesis: in this case, the operator is always. Turn nally to a case involving epistemic modals. Since epistemic modals and disjunction lead to free choice eects which I want to control for, I use the connective and. Start with: (26) John might be inside and outside. Supposing that no one can be both inside and outside, this sentence is prominently read as trivially false. However, pausing slightly between the conjuncts and emphasizing and brings out a non-trivial reading:24 (27) John might be inside . . . AND outside. is sentence is naturally read as equivalent to: (28) John might be inside and John might be outside. is nal example also supports The FocusHypothesis: here, the operator ismight and the connective is and.25 24Why does pausing slightly help to bring out the non-trivial reading here? One hypothesis is that this helps to distinguish the use of focal stress from its contrastive use, which we nd in: (i) A: e disease might be infectious but it is not harmful. B: No, the disease might be infectious AND harmful. 25Szabolcsi & Haddican 2004 observes a related eect of focus and conjunction: (i) Mary didn't take English and Algebra. e most natural reading of (i) is that Mary took neither English nor Algebra. But when and is focused, the natural interpretation is that Mary didn't take both: 23 Disjunctive Antecedent Conditionals But what exactly is this phenomenon-why are these readings triggered by focal stress on the main connective? I do not yet have a settled answer to this question. Perhaps the phenomena is an instance of across-the-board (ATB) movement of the modals/quantiers out of their clauses,26 or perhaps it is the result of eliding the doubledmaterial at LF.My goal in this section is not to explain The FocusHypothesis, but rather merely argue that it is a plausible generalization.27 Regardless of what accounts for these focus eects, the fact that modals and quantiers exhibit analogous behavior with respect to focus is, I think, a compelling reason to adopt The Focus Hypothesis and pursue a unied explanation of the Focus Observation and our observation about strong DDACs, in line with the Double-if and Single-if conjectures. 2.3 Ignorance and specicational DDACs I close this sectionwith a brief discussion about some tricky data surroundingDDACs, which I draw on to motivate my syntactic approach to the ambiguity in DACs.28 We saw above that, unlike DACs, DDACs seem to lack a felicitous specicational reading, as seen by the contrast between (29) and (18): (ii) Mary didn't take English AND Algebra. is data may initially seem to be a counterexample to The Focus Hypothesis, since here things seem to go in the opposite direction to what the hypothesis predicts. In response, let me point out that this data merely reveals a new complication in the dialectic: in addition to explaining the data supporting The Focus Hypothesis, we must also now account for the negation data here that tells against it. is suggests that there is a more explanatory generalization which we have yet to uncover. Unfortunately, I do not know what that generalization is. Following a conjecture of Szabolcsi & Haddican 2004, I suspect that a promising account of these cases will appeal to a homogeneity presupposition in conjunctive clauses which is suspended (for some reason) by focal stress. 26ATB movement (see also Ross 1967, Williams 1978, Postal 1974) happens when a constituent moves out of several places of a coordinate structure at once, as in: (i) e person whoi [Mary loves t i] and [John hates t i] was present. According to this explanation, in the cases above, we are seeing two occurrences of the quanti- er/modal moving out of low scope position to occupy a single position above the connective at surface form: (22) ≈ [Everyone]i [t i is inside] or [t i (is) outside]. 27I concede that a full defense of The FocusHypothesiswould involve experimental work testing a wide range of cases. I lack the space to fully defend the hypothesis here. 28anks to an anonymous reviewer for bringing this data to my attention. 24 Disjunctive Antecedent Conditionals (29) If John draws a gold coin or a silver coin, he will draw a gold coin. (18) #If John draws a gold coin or if John draws a silver coin, he will draw a gold coin. However, it is also true that both DACs and DDACs have felicitous ignorance interpretations, as seen by the fact that both of the following are felicitous: (30) John will win if he draws a gold coin, or a silver coin; I am not sure which, though. (31) John will win if he draws a gold coin, or if he draws a silver coin; I am not sure which, though. e fact that there are felicitous ignorance DDACs is evidence that DDACs do not always have the strong interpretation. Yet, the fact that there are not felicitous specicational DDACs is evidence that DDACs lack the weak interpretation of DACs. I think a version ofmy syntactic strategy for DACs can be extended to account for this dierence between DACs and DDACs. In particular, my proposal is that DACs and DDACs dier in the range of LFs they are ambiguous between. DACs are ambiguous between a Double-if and Single-if LF, while DDACs are ambiguous between a Double-if and Disjoined LF: ▸ Disjoined: [ [ if A ] C ] or [ [ if B ] C ] Roughly, on its Disjoined LF, the second consequent of (31) is elided: (32) John will win if he draws a gold coin, or John will win if he draws a silver coin. It should be clear why, if it has a Disjoined LF, an ignorance DDAC like (31) would be felicitous. But it may be less clear why a specicational DDAC with a Disjoined LF should be infelicitous. To show that it would be infelicitous, suppose that the specicational DDAC (18) has a disjoined logical form, making it equivalent to (33): (18) #If John draws a gold coin or if John draws a silver coin, he will draw a gold coin. (33) If John draws a gold coin, he will draw a gold coin or if John draws a silver coin, he will draw a gold coin. is disjunction is trivially true simply because one of its disjuncts is trivially true: 25 Disjunctive Antecedent Conditionals namely, that if John draws a gold coin, he will draw a gold coin.29 And, in general, it is infelicitous to assert trivialities.30 us, given that DDACs can have only Double-If or Disjoined LFs, we expect specicational DDACs to be infelicitous: they are false if interpreted with a Double-if LF (since, by Strong Double-If they would then have the strong interpretation), and they are trivial if interpreted with a Disjoined LF. By contrast, interpreted with the Single-if LF, a specicational DAC should have the weak interpretation (by Weak Single-If) and hence be possibly non-trivially true. Let's pause to summarize the results of this section. I have argued that we should explain the ambiguity of DACs by the fact that their surface forms underdetermine their logical forms in the following ways: LF-Ambiguity: a. DACs can have either a Double-if or Single-if logical form.31 29e or in (33) is intended to be truth functional disjunction. Geurts 2004, drawing on an example from Johnson-Laird & Savary 1999 observes data suggesting that certain conditionals coordinated by or behave like conjunctions of conditionals. In particular, when it's common ground that the antecedents of the disjoined conditionals are disjoint and exhaustive, their disjunction has a reading in which it entails both disjuncts: (i) Either he will stay in America if he is oered tenure or he will return to Europe if he isn't. Geurts 2004 extends his modal analysis of disjunction to handle this kind of case. My theory does not generate this prediction without additional supplementation. 30Consider the oddity of: (i) #If John draws a gold coin, he will draw a gold coin. where antecedent and consequent describe the same coin draw. ere are, of course, exceptions to this generalization: (ii) If he lied, he lied. But (ii) seems to be acceptable because there is a plausible alternative reason someone might assert it: perhaps as a concession, or to implicate that lying is inexcusable. Once we have that use in mind, (i) is more felicitous. But notice that this doesn't rescue (18) from infelicity. Why? e reason seems to be that felicitous concessive trivial conditionals are limited to those of the form if A, A. Notice, for instance, that (iii) is almost impossible to hear as a concession, even though it is equivalent to (ii) (which can be heard concessively). (iii) #If he lied, he either lied or danced. I think something similar is happening with (18): it is trivial, but cannot be heard as a concession (and thus as felicitous) because of its form. 31I leave open the possibility that DACs also have the Disjoined LF. 26 Disjunctive Antecedent Conditionals b. DDACs can have either a Double-if or Disjoined logical form. I combined this syntactic conjecturewith two semantic hypotheses: StrongDoubleIf, which states thatDouble-if conditionals univocally have the strong interpretation, and Weak Single-If, which states that Single-if conditionals univocally have the weak interpretation. Together, these principles promise a reasonable explanation of the Focus Observation (that focal stress on or biases the strong interpretation of a DAC), by appealing to the independentlymotivated FocusHypothesis (which predicts that focal stress on or in a DAC indicates that it has a Double-if LF).e principles also account for the complex data surrounding DDACs, in particular that there are no felicitous specicational DDACs but there are felicitous ignorance DDACs. I should emphasize that, although I have narrowed down a strategy for how one might build a theory of disjunctive antecedent conditionals, I have not yet provided such a theory. For one, I have not oered a syntactic theory that predicts LFAmbiguity or the FocusHypothesis. And secondly, I have not oered a semantic theory that predicts Strong Double-If and Weak Single-If. In the next and nal section, I aim to ll the latter lacuna by oering a semantic theory that predicts Strong Double-If and Weak Single-If. I set aside lling the syntactic lacuna for future work. 3 e semantics of DACs In the previous section, I provided an independent reason to adopt The FocusHypothesis, which proposes that focal stress on or in an DAC defeasibly indicates the presence of two if-clauses coordinated by that connective at LF. Now, I state a semantic theory that predicts the Double-if and Single-if hypotheses (restated here): Strong Double-if Disjunctive conditionals with a Double-if logical form univocally have the strong (simplication-entailing) interpretation. Weak Single-if Disjunctive conditionals with a Single-if logical form univocally have the weak interpretation. To do this, the semantics must allow if-clauses to be coordinated by connectives like or. But it is not obvious how such a semantics might work. For instance, existing operator and restrictor theories do not allow such coordination. According to operator theories, if denotes a two-place connective which maps pairs of propositions to propositions (standard representatives of such theories include ?Lewis 1973, Gillies 27 Disjunctive Antecedent Conditionals 2010). According to restrictor theories, ifmerely marks the restriction of a local operator (as in Lewis 1975, Farkas & Sugioka 1983, Kratzer 1986, von Fintel 1997, 2004). Neither of these theories allow if-clauses to be coordinated by sentential connectives like or, at least not in any straightforward way. An alternative theory that could account for coordinated if-clauses is one that treats them instead as one-place operators, that map clauses to clauses (cf. Schlenker 2004, Santorio forthcoming). My theory is of this kind. Very roughly, the idea is that if maps a proposition to its singleton, which is the argument of a modal operator beneath it.32 Interpretation on my theory happens in two stages. First, the natural language expression is translated into a formal language that states its logical form. Second, an evaluation function ⟦ ⟧ applies to logical form expressions, assigning them an extension relative to a context c and world w. Abstracting on the world parameter gives us the intension of the logical form of the expression, which is a function from worlds to extensions: e intension of e at c = ⟦e⟧c = λw . ⟦e⟧c,w On my semantics, if denotes a function from propositions to the singleton set containing that proposition: ⟦if A⟧c,w = {A}; where A = λw . ⟦A⟧c,w To streamline the presentation of the theory, I will ignorewill in our target sentences. For now, I will focus on bare indicative conditionals like: (34) If John came to the party, Sue came to the party. Following Kratzer 1986, I propose that bare consequent clauses contain a covert modal element ◻ in their logical forms, and that this modal's domain is updated by if-clauses via restriction. We will model this by letting f be a function from a world and a proposition to a set of worlds (intuitively, the domain of the modal). So, on my semantics, the extension of a modal operator maps a proposition to a function from a set of propositionsR to a truth value, as follows:33,34 32My theory is similar to Starr 2014, on which if highlights its complement, and then clauses subordinated to an if-clause test to see whether every highlighted proposition contextually entails that clause. However, I implement my theory in a very dierent semantic framework than Starr's. I set aside a full comparison between our theories for another day. 33I use subscripts on lambda expressions to indicate the type of variable: t is the type of truth value, e of individual, s of possible world. ⟨st⟩ is the type of a function from worlds to truth values (or possible worlds proposition), and ⟨⟨st⟩, t⟩ is the type of a set of possible-worlds propositions. 34is semantic entry for the covert modal,◻, is provisional. Going this way results in an epistemic 28 Disjunctive Antecedent Conditionals ⟦◻ f C⟧c,w = λR⟨⟨st⟩,t⟩.∀X ∈R ∶ ∀w′ ∈ fc(X,w): ⟦C⟧c,w ′ = 1. Notice that there are two layers of quantication here: one over the propositions inR and one over each modal domain generated (pointwise) by each proposition inR. I assume that every modal has these two layers of quantication.35 If-clauses will provide the value forR, but when the modal appears unembedded, I will suppose that context supplies the set containing the universal proposition (true at every world): {W}.is ensures that the quantication overR becomes vacuous, and thus ensures that bare modals have their usual semantics.36 Putting the ifand consequent clauses together yields the following truth conditions for simple conditionals if A, C: ⟦[ if A ] ◻ f C⟧c,w = 1 i ∀X ∈ {A} ∶ ∀w′ ∈ fc(X,w): ⟦C⟧c,w ′ = 1 is reduces to the condition that every world in fc(A,w) is aC-world; I submit that these are reasonable truth conditions for simple conditionals, modulo the provisions set aside in footnote 34. By dividing up the compositional details of conditionals in this way, our semantics can now, in principle, handle coordinated if-clauses. However, when we think about natural candidates for the semantic value of or, we run into an immediate problem: ⟦or⟧c,w = λutλvt . u = 1 ∨ v = 1 is natural semantic value for ormaps pairs of truth values (the extensions of sentences) to 1 i either input truth value is 1, and to 0 otherwise. is works well for coordinating simple sentences, but not for if-clauses, whose extensions are not truth values, but sets of propositions. In response to this type-clash, I propose that or strict conditional semantics that allows for violations of Conditional Excluded Middle. For reasons I do not motivate in this paper, I prefer an epistemic variably strict conditional semantics that validates Conditional ExcludedMiddle, as in ?Stalnaker 1980. We could achieve this result by letting the covert modal be a selection function modal, as in Cariani & Santorio forthcoming. 35For independent support that modals have two layers of quantication, see Mandelkern et al. 2017. 36In eect, R is a variable, which can be bound by if-clauses, and when le unbound defaults to the value assigned to it by an assignment function initialized by context (which, as above, I suppose is the set containing the universal proposition). I am suppressing this extra complication to streamline the presentation of the theory. I also want to allow for the possibility of "long distance" binding of this value across sentences (cf. Groenendijk & Stokhof 1991, Kratzer 2012): (i) A tree might have fallen on the driveway. It would have destroyed your car. 29 Disjunctive Antecedent Conditionals type-shis to a function from pairs of sets of propositions to a set of propositions as follows:37 Type-shifting Disjunction: ⟦or↑⟧c,w = λP⟨⟨st⟩,t⟩λQ⟨⟨st⟩,t⟩ . P ∪Q is corresponds to generalized disjunction (cf. Partee & Rooth 1983). Combining if A, if B and or yields the following: ⟦if A or↑ if B⟧c,w = {A,B} en, as desired, our semantics predicts both the Double-if and Single-if hypotheses: Double-if ⟦ [ if A or↑ if B ] ◻ f C⟧c,w = 1 i ∀X ∈ {A,B} ∶ ∀w′ ∈ fc(X,w): ⟦C⟧c,w ′ = 1. . . . i ∀w′ ∈ fc(A,w): ⟦C⟧c,w ′ = 1 and ∀w′ ∈ fc(B,w): ⟦C⟧c,w ′ = 1 Single-if ⟦ [ if A or B ] ◻ f C⟧c,w = 1 i ∀X ∈ {A ∪ B} ∶ ∀w′ ∈ fc(X,w): ⟦C⟧c,w ′ = 1. . . . i ∀w′ ∈ fc(A ∪ B,w): ⟦C⟧c,w ′ = 1 is is the main prediction of the account. In the rest of the section, I review how the theory, together with other motivated assumptions, predicts the rest of the data discussed above. Recall that our syntactic assumptions are: Ambiguous LF: a. DACs can have either a Double-if or Single-if logical form. b. DDACs can have either a Double-if or Disjoined logical form. and The FocusHypothesis Dierential focal stress on a sentential connective '∗' appearing in surface form under the scope of an operator 'O' can indicate the presence (at LF) of two occurrences of that operator taking narrow scope with respect to the connective. ese, together with Double-If and Single-If, predict the following: Prediction 1: Simple DACs are ambiguous between weak and strong readings. 37I want to remain neutral for now about whether this type-shiing is carried out by a covert type shiing operator, exible types, or a distinct rule of interpretation. 30 Disjunctive Antecedent Conditionals On its Double-if logical form, if A or B, C has the strong (simplication-entailing) interpretation, while on its Single-if logical form, it has the weak interpretation. Prediction 2: Uttering aDACwhile emphasizing the or in its antecedent tends to bias its strong interpretation. (Focus Observation) Given The Focus Hypothesis, focal stress on or can signal that if A or B, C has a Double-if logical form, and hence (we predict) a strong interpretation. Prediction 3:erewill be felicitous ignoranceDDACs but not felicitous specicational DDACs. is is because we predict that DDACs lack the Single-if LF, and instead are only ambiguous between a Double-If LF or a Disjoined LF (recall the reasoning in §2.3). Prediction4: ProbabiliedDACs are ambiguous between a probabiliedweak and probabilied strong interpretation. Probabilied Weak: if A or B, probably C is true i the probability of C given A ∪ B is greater than 0.5. Probabilied Strong: if A or B, probably C is true i if A, probably C and if B, probably C are both true (i.e., i the probability of C given A is greater than 0.5 and the probability of C given B is greater than 0.5). e probabilied weak interpretation arises from the possibility of interpreting if A or B, probably C as having the following logical form: ▸ Single-if: [ [ if A or B ] probably C ] while the probabilied strong interpretation arises from interpreting the probabili- ed DAC as having the following LF: ▸ Double-if: [ [ if A or↑ if B ] probably C ] I end with a response to a worry. Since my view is an ambiguity theory, I face a concern about overgenerating readings of conditionals. I hope to have partially mitigated that concern here in my discussion of the empirical data in §1, where we found evidence that DACs do seem to have two distinct interpretations, which can be brought out in dierent contexts. Nonetheless, two questions remain. e rst is whymany people gravitate towards the strong interpretation of DACs by default. Here, I appeal to the fact that the strong interpretation is logically stronger 31 Disjunctive Antecedent Conditionals than the weak interpretation (it asymmetrically entails the latter), and the observation that we generally aim to interpret speakers as making stronger claims when there are multiple possible interpretations (bounded from above by what we think they could plausibly know). Compare (Taylor 2001, Wilson & Sperber 2012): (35) a. I haven't had breakfast. b. I haven't been to Paris. Tense is context-dependent, so past tensed clauses could, in principle, be interpreted as about any past time interval. Despite this contextual exibility, the most natural interpretation of (35-a) is that the speaker has not had breakfast today, while the most natural interpretation of (35-b) is that the speaker has never been to Paris. is can be explained by the fact that we generally aim to interpret speakers as making stronger claims (as constrained bywhatwe think they could plausibly know). In both cases, an interpretation about a larger past interval of time will be stronger than one about a shorter past interval of time. However, the default stereotypes about eating breakfast and going to Paris are dierent: it is commonly assumed that people eat breakfast once per day, but there is no general assumption about how many times a person goes to Paris. e strongest possible interpretation of (35-b) is that the speaker has never been to Paris. Furthermore, this interpretation is not defeated by the background assumptions in a default context, which is why it is the most natural in such a context. Similarly, the strongest possible interpretation of (35-a) is that the speaker has never had breakfast. However, this interpretation is defeated by the default assumption that people eat breakfast once per day, and so we instead opt for the strongest interpretation compatible with that default assumption, which is that the speaker has not had breakfast today.38 Given the observation that we tend to interpret speakers as making stronger claims by default, I claim that this accounts for whymany people by default interpret if A or B, C as having the strong interpretation (via its Double-if logical form). e second question is why we don't overgenerate strong readings for specicational DACs like (2). Here, I appeal to a strategy similar to the one invoked by Alonso-Ovalle 2009. On its strong reading, (2) would be necessarily false, so we expect sensible speakers to avoid this interpretation, and this accounts for why we don't typically read (2) as having the strong interpretation. However, my theory differs from Alonso-Ovalle 2009 in that I do not advocate that avoiding contradictions is the only condition under which the weak reading is made available. 38Further evidence that we prefer stronger interpretations comes from the fact that it would be unnatural (without signicant additional context) to interpret (35-a) as meaning the weaker claim that the speaker hasn't had breakfast in the last hour, or last minute, and so on. 32 Disjunctive Antecedent Conditionals is concludes the presentation of my theory. I conclude in the next section with possible extensions and avenues for future work. 4 Future directions Here is a brief summary of the paper. I provided new experimental evidence that DACs are ambiguous between a strong (simplication-entailing) interpretation and a weak (non-simplication-entailing) interpretation. Second, I discussed a new observation about DACs (the Focus Observation)-that or-emphasis can be used to disambiguate them in favor of the strong interpretation. I argued that this observation makes trouble for existing ambiguity theories (in particular, the theories of Alonso-Ovalle 2009 and Santorio forthcoming). I then motivated The Focus Hypothesis, which I used to connect the Focus Observation with an observation aboutDDACs. Finally, I developed a new theory ofDACs around these observations. According to my theory, the strong interpretation of a disjunctive conditional is the result of it having a Double-if logical form, while the weak interpretation is the result of it having a Single-if logical form. I showed how such a theory was in a position to predict these observations. us, on my view, simplification of disjunctive antecedents can be thought of as subdividing into two distinct principles: one which is about DACs with Double-if logical forms (which is valid) and one which is about DACs with Single-if logical forms (which is invalid). ere are many avenues of future research that grow out of this project. One question is how my theory relates to that of Starr 2014. Of particular interest here is how our theories compare when it comes to doubled conjunctive antecedent conditionals like: (36) If John draws gold and if John draws silver, he will win. Prima facie, (36) seems to entail both of its simplications, unlike its normal conjunctive cousin, which does not: (37) If John draws gold and John draws silver, he will win. However, not all doubled conjunctive antecedent conditionals entail both of their simplications. Suppose that it is not known which types of coins are winners, but that gold coins might be winners. en (38) could be true even though it is not true that if all the gold coins are winners, John will win (since there's no guarantee that he will draw gold): (38) If John draws gold and if all the gold coins are winners, he will win. 33 Disjunctive Antecedent Conditionals I leave sorting out this issue and the comparison with Starr 2014 as a topic for future work. Another potential upshot of the theory sketched here is that it is well-positioned to account for the behavior of other kinds of quantied DACs. In particular, I have in mind both adverbial and deontic DACs/DDACs like: (39) If Smith dances or (if Smith) tells a joke, Sue usually laughs. a. If Smith dances, Sue usually laughs. b. If Smith tells a joke, Sue usually laughs. (40) If John draws a gold coin or (if John draws) a silver coin, he has to put it back. a. If John draws a gold coin, he has to put it back. b. If John draws a silver coin, he has to put it back. Both adverbial and deontic DACs/DDACs seem to exhibit the same range of behavior as ordinary DACs/DDACs. is suggests that a unied account ought to be pursued for all of these conditionals. e theory given in §3 can be extended to handle these conditionals, much in the same way it could probabilied DACs. I leave spelling out the compositional details of these extensions for future work. Finally, as noted in §2, more work is needed to understand the role of focus in transformations between surface and logical form. In particular, it remains to be seen what the best explanation of The Focus Hypothesis is. Although more work remains, I hope to have at least made a case in favor of a new way of thinking about disjunction, conditionals, and the principle simplification of disjunctive antecedents.39 References Adams, Ernest. 1975. e Logic of Conditionals. Synthese Library, vol. 86. Boston: D. Reidel. Alonso-Ovalle, Luis. 2009. 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