Semantics & Pragmatics Volume 12, Article 23, 2019 https://doi.org/10.3765/sp.12.23

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Goldstein, Simon. 2019. Free choice and homogeneity. Semantics and Prag- matics 12(23). 1–47. https://doi.org/10.3765/sp.12.23.

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Free choice and homogeneity*

Simon Goldstein Australian Catholic University

Abstract This paper develops a semantic solution to the puzzle of Free Choice permission. The paper begins with a battery of impossibility results showing that Free Choice is in tension with a variety of classical principles, including Disjunction Introduction and the Law of Excluded Middle. Most interestingly, Free Choice appears incompatible with a principle concerning the behavior of Free Choice under negation, Double Prohibition, which says that Mary can’t have soup or salad implies Mary can’t have soup and Mary can’t have salad. Alonso-Ovalle 2006 and others have appealed to Double Prohibition to motivate pragmatic accounts of Free Choice. Aher 2012, Aloni 2018, and others have developed semantic accounts of Free Choice that also explain Double Prohibition. This paper offers a new semantic analysis of Free Choice designed to handle the full range of impossibility results involved in Free Choice. The paper develops the hypothesis that Free Choice is a homogeneity effect. The claim possibly A or B is defined only when A and B are homogenous with respect to their modal status, either both possible or both impossible. Paired with a notion of entailment that is sensitive to definedness conditions, this theory validates Free Choice while retaining a wide variety of classical principles except for the transitivity of entailment. The homogeneity hypothesis is implemented in two different ways, homogeneous alter- native semantics and homogeneous dynamic semantics, with interestingly different consequences.

Keywords: free choice, homogeneity, alternative semantics, dynamic semantics

1 Introduction

In the last few decades there has been a flurry of research into the apparent validity of the ‘Free Choice’ inference, where sentences like (1) and (3) appear to imply sentences like (2) and (4).1

(1) You may have soup or salad. * Thanks to David Boylan, Simon Charlow, Melissa Fusco, Jeff King, Matt Mandelkern, Paolo Santorio, and Dan Waxman. Special thanks to Fabrizio Cariani and Jacopo Romoli. 1 See von Wright 1968 and Kamp 1974, Kamp 1978. early access hogotIasm lsia nrdcinadeiiainrlsfrconjunction. for rules elimination and introduction classical assume I Throughout 3 2 ihteeasmtos reCoc mle h qiaec faytopossibility two any of equivalence the implies Choice Free assumptions, these With hr o aesu n nacsil ol hr o aesld o hsreason, this For salad. have you where world accessible an and soup have you that where On disjunction. for semantics Boolean a and modals, for semantics worlds fe all, After reCoc sasrrsn neec rmteprpcieo lsia possible classical a of perspective the from inference surprising a is Choice Angeles. Free Los in be might Mary and York New in be Angeles. might Los Mary or York New in be might salad. have Mary (4) may you and soup have may You (3) (2) 2005, Geurts 2015, 2000. Fusco Zimmermann 2015, and Charlow 2017a, Willer2009, 2016 , Aloni al. Starr 2012, et 2005, Aher Simons 2005 , Ciardelli Bonevac2013, 2010, & Roelofsen BarkerAsher others 2018, among Aloni see choice, 2007, free of accounts semantic For implies Choice Free by which tonicity, Explosion. imply Choice Free claims. Choice, modals: Free possibility of of validity Monotonicity the Upwards between the and tension Introduction, inferences. serious compelling Disjunction is other there some showed of validity 1974 the Kamp with incompatible is this it because to tion, revision of kind some offered all have semantics. Choice world classical Free accessible an of both analyses is semantic there that have imply you not either does where This world salad. accessible have some you is or there soup that says simply (1) theory, at1 Fact o ups h naletrlto stransitive: is relation entailment the suppose For RECHOICE FREE EXPLOSION INTRODUCTION DISJUNCTION TRANSITIVITY MONOTONICITY UPWARDS reCoc localne lsia oi o osblt oasaddisjunc- and modals possibility for logic classical a challenges also Choice Free 3 (Kamp) ♦ A ♦ implies . A ♦ If rniiiy ijnto nrdcin pad oooiiyand Monotonicity Upwards Introduction, Disjunction Transitivity, ( | = A A 2 ∨ ♦ | = B B ♦ ) B ( A | = and ∨ ♦ B If A B ) A ∧ yDsucinItouto n pad Mono- Upwards and Introduction Disjunction by | A = ♦ | = | = B C B then , A then , ∨ ♦ 2 B B ob Transitivity by So . A ♦ | = A C | = ♦ B ♦ A implies io Goldstein Simon ♦ B For . early access . ) A B ♦ and ∨ A A ♦ ( ¬ ♦ ¬ = | = | B B ♦ ♦ ¬ ¬ B ♦ . ∧ ¬ . So by Transitivity, A A A ¬ . 3 ♦ ♦ . So by Contraposition = B | ¬ ¬ B ♦ B ♦ = = | | ¬ = | ) ) = | B B A ) ♦ ∨ B ∨ , then A A ∨ ( ( B A ♦ ♦ ( = ¬ ¬ | ♦ A If A , by giving up either Disjunction Introduction or Upwards Monotonicity. ) ∨ ¬ B A Transitivity, Contraposition, Free Choice, and Double Prohibition imply By Free Choice, ∨ = | A ( But even giving up Contraposition isn’t enough to avoid further trouble. The There are several reasons to think Free Choice and Double Prohibition are A semantic account of Free Choice requires major revision to a classical logic ♦ LEM CONTRAPOSITION DOUBLEPROHIBITION 2016, Willer 2017a) have given up the rule of Contraposition. joint acceptance of Free Choiceclassicality and in Double the Prohibition logic also of requiresConstructive disjunction. further Dilemma: Consider non- the Law of Excluded Middle and By Double Prohibition, hence by Contraposition again For this reason, the few defenders of both Free Choice and Double Prohibition (Starr are equivalent. Fact 2. Explosion. Proof. salad, since you would have soup or salad in eitherincompatible. such First, world. suppose we accept Transitivity and Contraposition. Double Prohibition. If there isthere no cannot accessible be world an where accessible you world have where soup you or have salad, soup, or one where you have of (2) does not appearappears to to imply imply the: (6) negation of .(1) Quite the(5) opposite: (5) actually (6) You can’t have soup or salad. You can’t have soup and you can’t have salad. and semantics for disjunctionproblems and stemming modals. from negation. Any Alonso-Ovalle suchlargest2006 presented challenge account what for also may semantic faces be accountsFree the serious of Choice Free seems to Choice. disappear Like under scalar negation. implicatures, While (1) implies (2), the negation Free choice and homogeneity this reason, all existing accountsto of Free Choice have rejected the inference from Then Free Choice and Double Prohibition again imply that any two possibility claims While the classical semantics above does not validate Free Choice, it does validate early access 4 which aeofrdnwsmnisfor semantics new offered have 2018) Aloni 2017a; Willer 2016 ; Starr 2012; (Aher osetepolmifral,spoeITi as,s htteei iuto in situation a is there that so false, is IAT suppose informally, problem the see To Introduction. 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( | A ♦ = A B ∨ D and ) B . then , ) B | ( = r osbe hl h latter the while possible, are A ♦ ∨ A ♦ B ∧ ( A | ♦ = ∨ B C ) B ∨ ∨ ) nor D ( io Goldstein Simon ¬ ♦ ¬ A ♦ 4 ¬ ∧ ( Others, A ♦ ∨ B B ) ) . early access A ♦ . But these B ♦ implies would be false; so both ) ) B B ∨ ∨ A A ( , because this last inference has ( ♦ B must be possible. If neither were ♦ to be true, either both or neither of ♦ ¬ B ) B ∨ and A A ( 5 implies ♦ A ¬ ♦ . For the same reason, B would be false; so both must be true. Double Prohibition is ) and B otherwise. A ∨ , because to evaluate this inference we hold fixed the homogeneity A ) ( iff Mary can’t have soup and Mary can’t have salad. B ♦ iff Mary may have soup and Mary may have salad. ∨ are possible. A is true, either both or neither of ( B must be possible. If both were possible, ) ♦ TRUE FALSE UNDEFINED B B and ∨ On first glance, one might balk at giving up Transitivity in the face of these Once we incorporate homogeneity effects into our semantics, we can also intro- In this paper, I develop a new semantics for Free Choice that validates Double A A and ( of Strawson validity isple, not if an Strawson artifact validity is ofpresuppositions, adopted the transitivity to can semantics characterize fail pursued in entailmentdescriptions, garden here. in Strawson’s variety For a original inferences language exam- application. involving definite with Consider: (8) The King of France is either bald or not bald inferences together don’t require that no homogeneity effect. The failure ofincompatibility Transitivity results responds regarding in Double a Prohibition. similar way to the incompatibility results. One thing worth observing here is that the intransitivity Here, an argument is validsion only is if defined. it Crucially, is this truth notionclassical preserving of principles in entailment above, contexts allows except where us the the to transitivityimplies conclu- preserve of all entailment. of To foreshadow, the of the possibility of possible, then must be impossible. duce a definition of entailment that is sensitive to definedness: Strawson entailment. (7) Mary may have soup or salad. By relying on homogeneityboth in Free Choice the and right Double♦ Prohibition. way, Free we Choice can is valid straightforwardly because validate whenever idea is that disjunctive possibility claimsof are defined only when either both or neither Prohibition while avoiding all ofclassical the principles results above above remains in valid, a except unified for way. one: Every the one transitivity of ofeffects the entailment. (von Fintel 1997; Križ 2015a) into the semantics of Free Choice. The key Free choice and homogeneity Prohibition simultaneously. Any such semantics requiresthe some three response incompatibility to results each above. of valid for the same reason. In order for To achieve this solution, the main theoretical insight is to introduce homogeneity A early access 2016. Rooth see semantics, alternative of overview helpful a For 6 5 (9): implies Strawson and valid, Strawson is This aiaeDul rhbto n rsremc fcascllgcb sn single a using by logic classical of much preserve and Prohibition Double validate atclr hspprdvlp w e eatc:hmgnosatraieseman- alternative homogeneous semantics: In new results. two similar develops quite paper with this frameworks, particular, semantic different very into incorporated unique a is There (9) hr smc ro oko alrso rniiiy lhuhgnrlynti h otx fStrawson of 2015). context (2013, the Ripley in 1994), not (1992 , generally Tennant although (1958), Transitivity, Smiley of See failures entailment. on work prior much is There 1. 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To implement this idea, we let the K }} B B K ∨ A B J J ∨ A returns the singleton of the claim that no alternative in A J K ∈ A }} is true. is a usual Kripkean modal operator, which existentially ∈ A and and 1 } A K B K ¬ K | A A ♦ J B A A | J J ) = B J is true just in case each of the possibility claims p ∪ A S ∩ ( K are operators that take a set of alternatives and flatten them, pro- A ♦ w − A A · { ♦ | ¬ { J T occurs in the meta language rather than the object language, and sim- W w { { = = ♦ K K and {{ = = —the ‘proto-possibility’ operator—which maps a proposition to a new B B · K K ♦ . Then ♦ = , whose interpretation is parametric to a choice of possibility operator n ∧ ∨ A A K · ♦ · A presents both of p ¬ A A ♦ return the singleton of the claim that J J J J J K is true. To validate Free Choice, we let possibility modals universally quantify B Both For this proposal to work, though, we also need a story about the rest of our The crucial idea we’ll rely on from alternative semantics is that a disjunction . Summarizing, we reach the following first pass theory: A i. K K ,..., v. ii. ∨ iv. 1 · iii. ♦ A A conjunction of a series of proto-possibility claims, distributed over the prejacent’s Suppose the set of propositions in is true. In other words, the alternative sensitive possibility modal is a generalized quantifies over accessible worldsJ (Kripke 1963; Kratzer 2012).J Then we can let Definition 2. generality by deriving the meaningoperator of possibility modals ( proposition. ply denotes some function or other from propositions to propositions. For example, of worlds where the sentencethe is alternatives true. from ducing a singleton set. J over the alternatives in their scope. 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Here is the problem: observe first that predications involving plural definites, Homogeneity effects have been used to explain apparent violations of excluded A i. For any operator ( ii. ! · like (12), plausibly license inferences to universal claims like (12) (13). The cherries in my yard are ripe. middle for both conditionalstreated as and a plurals. kind ofhomogeneity semantic is presupposition, also involving treated undefinedness. as In apattern.Križ kind For2015a of our undefinedness, results, but with we aon need different Strawson not projection entailment take (as a in stand onvon Fintel this1997). issue; but we need to rely In §2, we reviewed how alternativeaddition semantics to handles giving Free up Choice. the We Upwardsalso saw Monotonicity invalidated that of Double in possibility Prohibition. modals, Now the weeffects. semantics introduce our main idea: homogeneity (10) (11) Mary may have soup or salad, butMary I may don’t have know soup which. or3 salad, but I won’t Homogeneity tell you which. ♦ One last remark: the alternatives approachalways does go not in require that foroperation possibility Free which modals flattens Choice. alternatives. To avoid this result,Definition 4. we can introduce a closure Free choice and homogeneity Prohibition corresponds to the trivializing conditionany that any other. possibility Summarizing: claim implies Observation 1. (10) and (11). We can then allow this closure operator occur in the prejacent, generating the form 8 See Zimmermann 2000 among9 others For for discussion, discussion. see among others von Fintel 1997 and Križ 2015a. frasmlraayi fconditionals. of analysis similar a for 1997 Fintel von See 10 early access htta s rt lrldfiie r endol hnete the either when only defined are definites plural first, is: that what is (12) and satisfied is (12) of conditions definedness the if that is (13) entails (12) etimn i o icag h ooeet condition. homogeneity the discharge not did entailment (12)-(13) htsespzln:ddw utpoefo oia rtsadvldinferences valid and truths logical from prove just we did puzzling: seems That (12) . 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G sud atlgcli htit that is tautological sounds (14) which in sense The ’s. yes or no u utisedcaiywith clarify instead must but , 10 G s fti odto sstse,their satisfied, is condition this If ’s. 10 well io Goldstein Simon F saeeither are ’s : early access , (21-a) . By contrast, in the Free no 11 (unless metalinguistically) to deny Mary’s no . This has at least two upshots. First, in the case (21-b) Half of the professors smiled. Soup is permitted; salad is not. Soup is permitted; salad is not. b. Sue didn’t buy the boat or the car. A: TheB: professors smiled. Well / #yes / #no, half of them. A: MaryB: may have soup and sheWell / may #yes have salad. / no, she can have soupA: (but she can’t haveMary salad). B: may have soup or salad. Well / #yes / #no, she can have soup (but she can’t have salad). Finally, Tieu et al. 2019 has found further empirical evidence for a homogeneity totally wrong’, or ‘totallyparticipants overwhelmingly right’. rewarded the They medium found strawberry.case By of that contrast, ordinary in in disjunction, the participants thebut gave overwhelmingly a Free rejected mixed Choice pattern of vignette, of response Free to Choice Tieu et al. a2019 homogeneityfound account. exactly Second, the Free pattern Choice of patterned responses quite predicted by differently than the In each case, participants heardthen a had puppet to assert reward one the ofon puppet the with whether relevant a the sentences, small, and puppet medium, was or ‘totally large wrong’, strawberry depending ‘in between, not totally right but not In addition, they gave participantsboth a a related boat scenario and in a which car, and a asked character them(21) bought to consider: a. Sue bought the boat or the car. effect with Free Choice inthey particular. gaveTieu et participants al. a2019 scenarioaskedran in them an which to experiment Sue consider: in is which only allowed to buy(20) a boat, and (20) Sue is allowed to buy theSue boat is or not the allowed to car. buy the boat or the car. speaker can felicitously deny theChoice conjunction claim by(19) saying it ispermission harder to to have use soup orand salad the in other a is scenario not. where one of these items is permitted Free choice and homogeneity (17) Context: (18) Context: (19) Context: (18) is a conjunction of possibility claims. When one of those conjuncts is false, a We can observe a parallel effect in the case of Free Choice. 11 early access enwdvlpanwsmnisfrFe hieb nertn homogeneity integrating by Choice Free for semantics new a develop now We ol to world to worlds from functions total consider we o hc oei §10). in more which (of niiul hc raste lk.Antrlnx tpwudb oepoewhether explore to be would undefined step over next natural supervaluation A of alike. kind them treats special which a individuals involving projection, of pattern Choice way Free a of and accounts Choice, based Free implicature of and case homogeneity the the in distinguishing effect of homogeneity a of in evidence both display offers on implicature exclusivity of kind nwhether on ‘ 5. 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Free of semantics the into effects homogeneity incorporate now We f p p ( x ∩ − ) . q q # g x ( n ola obnto fta ucinwt nte ucinalso function another with function that of combination Boolean any , = = x is ) eoe h mletfnto apn any mapping function smallest the denotes ’ g λ λ Where n hc asayother any maps which and , w w : : p p ( ( 11 w w p ) ) and 6= 6= { # # q 0 6= 6= , r oa ucin rmwrd to worlds from functions total are # q q , ( 1 ( w w } espoeta hnvrafnto asa maps function a whenever that suppose We . ) ) . . # x p p o nta fcnieigst fworlds, of sets considering of instead Now . ( o#. to ( w w 12 = ) = ) { 0 A and 1 and 1 , (21-a) ∨ # , B 1 x } r osbe ralo hmare them of all or possible, are hc satisfies which q h enn fasnec is sentence a of meaning The . q 2019 al. et Tieu way, this In . ♦ ( · ( w w ( A = ) ) ∨ 6= B 1 1 ) sdfie nywhen only defined is { f 0 to , # 1 , io Goldstein Simon 1 or } : 0 depending , early access K A J } 1 ) = w ( A ♦ K A J ∈ map sentences to sets of 1 K A · J ∀ ) = . w v Let ( q . ) = w 1 or ( 1 if: A ♦ } ) = K ) = K 13 w B w ( A : J is defined only if either all alternatives in ( J p } K . 1 ∈ A ∈ C , ) · J ♦ # B A w , , S ( K ∀ 0 q { A } iff J 1 1 6= , C } ∈ 0 # 1 { } = | A ) = K K 6= | n ∈ w B A ) ) = ( J } A J v B We can implement our theory of possibility modals by injecting K w p 1 i ∃ ( ∪ ( S , ∩ : A p K 12 0 w J − : A { A ,..., . w { J S 1 w λ ∈ W w ] A { { = = λ ) (Homogeneous alternative semantics) n λ , K K w { = = = 1 ( B B [ K K K , except in letting our meanings be defined relative to these more complex = q ∧ ∨ C · A A ∈ ♦ K J ∪ · i A p ¬ A ♦ p J J J ∀ S J J i. i. v. ii. ii. iv. iii. iii. semantics validates Free Choice. The precise projection patterns abovethe aren’t same necessary for with our Strong results, Kleene which projection, would or go with through the just connectives in Heim 1992. the next section we turn to this task. 5 Results forms of sentences. Definition 7. Finally, to get our desired predictions,ing we candidate need a for definition languages of involving consequence. definedness One conditions lead- is Strawson-validity is valid just in casepremises the are conclusion true. is As true in whenever the §2, conclusion we is also defined assume and that the entailment is sensitive to the closed total functions from worlds to Now we can retain basicallytaining our semantics from before forBoolean operators. the fragment not con- them with definedness conditions. are possible, or all are impossible. Definition 6 Free choice and homogeneity (Strawson 1952 , von Fintel 1997, 1999, 2001). According to this notion, an argument With all of our assumptions in place, let’s turn to the principles from §1. First, the We now have all of the tools necessary in order to solve the problems from §1. In 12 frdiscussion. for 2005 Bonevac & Asher See 13 early access Whenever gi,we ecnie whether consider we when Again, defined. is conclusion the where worlds the to attention our restrict can we or smr nutvl oplig nfc,orsmniscnmk hsdistinction. this make can semantics our fact, In compelling. intuitively more is worry h endeso h ocuinrqie htete oho ete of neither or both either that requires conclusion the of definedness The aiae h neec from inference the validates nteatraiesmnisfo 2rmisvldwe nihdwt homogeneity with enriched when valid remains §2 from semantics alternative the in of one least at that ♦ 2. Observation hssmnisgvsu the up gives semantics this 1, Fact escape To Monotonicity. Upwards and duction, explained. be still so could preserving, truth out flat also and is It valid. Strawson whenever merely that not is Choice Free For of validity the premise the so of and truth the with Combined possible. 4. Observation theory Choice. this Free with valid, incompatible is Upwards Introduction validates Disjunction semantics since alternative particular, homogeneous In §2, Monotonicity. in theory the Unlike of form 3. Observation same the for ordinary and in valid, before. As as remains disjunction. reason Introduction of properties Disjunction classical semantics, the alternative concerned which 1, Fact to since semantics, alternative homogeneous Furthermore, in defined. valid is be sentence to the inference that an provided for §2, less in takes as it worlds same sentences the possibility in makes true semantics alternative homogeneous all, After effects. · B flosfo oegnrlpoet.Ayifrneta a valid was that inference Any property. general more a from follows 2 Observation ooeeu lentv eatc aiae reCoc,DsucinIntro- Disjunction Choice, Free validates semantics alternative Homogeneous distinguish to fails it because strong, too is result last this that worry might One semantics. alternative traditional from departure first our have we point, this At respond must it Choice, Free validates semantics alternative homogeneous Since r ohtu,o hyaebt as.Bttetuhcniin for conditions truth the But false. both are they or true, both are ♦ · A ♦ · A | = ( A ∨ ♦ ♦ · · ∨ B ( ( A B A ♦ ahee yapyn ) h olsweeoedsuc strue. is disjunct one where worlds the !), applying by (achieved ♦ · · ) ∨ ∨ ( A ♦ ♦ A strue. is A · · B B ( A ♦ | | = ∨ = · ) A ) A | = stu,i sdfie.I h etnei end either defined, is sentence the If defined. is it true, is B r ohSrwo ai,temria pelo reChoice Free of appeal marginal the valid, Strawson both are ∨ ♦ A · ) and ( B ♦ ∨ · strue, is A ) ( B A ∨ | = ♦ · ∨ B B ♦ ♦ · · ) B A stu;s oehrti l eursta ohaetrue. are both that requires all this together so true; is A 13 rmtevldt fFe hie hc n might one which Choice, Free of validity the from ♦ ) · to ∧ A A ♦ ∧ ♦ · implies · B ( ♦ · A B ∨ 14 sas re oatog ohFe Choice Free both although So true. also is B A ) hc oehv huh sdirectly is thought have some which , ∨ B ersrc teto oteclosed the to attention restrict we , ♦ · A hsmasthat means this , ♦ · ( io Goldstein Simon A ∨ A ♦ · B and B ) ♦ · require strue, is A B and are early access . A · ♦ to be ¬ ; so by B are both B · ♦ B and is a logical ) A , and this last ) B and · B ♦ , we are no longer A ) ∨ implies B ) ∧ ¬ A ( ∨ B A · ♦ A · ∨ ♦ ( are treated symmetrically. · ¬ A , we do not have the first ♦ ( ( . A B ¬ · B ♦ ∨ · ♦ · ) ♦ entails ¬ and B A · ♦ A · ♦ ∧ A . This is exactly parallel to our treatment · implies ♦ A B ( ) · · 15 ♦ ♦ , which by Double Prohibition implies B ¬ ) is false. This implies that ∨ B ∧ ¬ . ) A ) ∨ A ( B B · · A ♦ ♦ ∨ ( is defined. But this sentence appears nowhere in the ∨ ¬ ¬ · ♦ A ) A ( ¬ ( = ! B | does not by itself entail · ♦ ♦ ) and ∨ A , and so this latter argument remains invalid. satisfies Double Prohibition. B ¬ B ) · A ♦ B · ♦ ( · ∨ ♦ B = · | ♦ · , ♦ A = implies ∨ 6| B ( ♦ to · · is true, A B A ♦ ♦ ( A · · ) ♦ ¬ ¬ ♦ · · ♦ ♦ B ¬ ¬ imply the last conclusion ∨ B A , the premise · ( ♦ B · ♦ ¬ · ♦ ¬ implies Our semantics offers the same solution to this problem as with1 Facts and2. Now let’s turn to3. Fact This showed that Free Choice and Double Prohibition To avoid2, Fact homogeneous alternative semantics gives up Transitivity. While In §1, we saw that the joint validity of Free Choice and Double Prohibition We’ve now diagnosed how homogeneous alternative semantics escapes the first B · ♦ instance involving Free Choice: truth. First, LEM remains valid. In particular, we validate the potentially problematic premise of1 Fact above. When we removeentitled the to intermediate restrict step our attention to worlds where seem to require further revisionsof to Excluded the Middle logic or of Constructive disjunction, Dilemma. giving In up particular, either the the concern Law was that Observation 7. ¬ particular, given Transitivity and Contraposition, Free Choiceimmediately and Double imply Prohibition Explosion. ForContraposition by Free Choice, impossible. raises the danger of Explosion: that any two possibility claims are equivalent. In incompatibility result from §1. Nowinvolving let’s Double turn Prohibition. to the Here, other thepossibility incompatibility first operator results, point to notice is that for any proto- Crucially, the first inference is only valid because weargument restrict from attention to worlds Observation 6. Here again, the key istreated that symmetrically. the definedness of the conclusion requires Free choice and homogeneity transitivity of entailment. Inentails particular, while Observation 5. The semantics above accepts each of the relevant instances of Contraposition. without giving up one of these principles, we would be forced to accept the absurd where the conclusion When ‘Interconnectedness’ principle IAT, that 14 early access hr h E ntneaoei nend where undefined: is above instance LEM the where when esae gotcaotwehrhmgniyi rspoiin riseda instead or presupposition, a is homogeneity whether about agnostic stayed we h A rnil osntcnanadsucinudrtesoeo possibility a of scope the under disjunction a contain not does principle IAT The end h etnei urnedt etrue. defined be to only guaranteed is is above sentence instance the The defined, defined; defined. be whenever always to true need be the doesn’t to truth lowers needs logical considerably A simply entailment truth. it Strawson logical a that being is on consider standards to thing one Here, 8. Observation sapoc srgtaotpuas u o reCoc.O h te hand, other valid. the be On would Choice. LEM where Free validity, not Strawson but accepting plurals, from about 2015a salad? Križ right have prevents is can’t nothing approach I 2015a’s but Križ soup have perhaps can So I can’t. if you what ?Well, or salad, or soup have can (iv) you Either (iii) unsuccessful: books? is the case of our half in read argument he analogous if them. the what Interestingly, read Well, didn’t he or books the read either Adam (ii) 2015a invalid: Križ is following 47). the p. that (see (i) suggesting Middle Excluded grounds, of empirical Law on the LEM up of gives denial who the 2015a , motivates Križ from differ we Here Free of validity joint the Prohibition. deliver Double to Tran- and able of Choice is exception the semantics the result, the with importantly, each principles, Most With classical sitivity. §1. all from validate to results manages incompatibility semantics the worlds of the treatment exactly elegant at an fails it But world. every at for true order be in must So it effect. validity, homogeneity a a be trigger to not it does principle IAT the So modal. 10. Transitivity: Observation of conclusion failure the another truth, have logical we a is, is That above not. the is of premise the while however, Crucially, IAT. Choice 9. reach Free Observation we applying above, that disjuncts is two point the the to Here, Prohibition 3. Double Fact and of derivation the in used instance eoecniun,oent bu h ttso ooeet si re.Above, order. in is homogeneity of status the about note one continuing, Before provides semantics alternative homogeneous that seen have we section, this In the including Dilemma, Constructive validates above semantics the Likewise, A and B r ooeeu,ete ohpsil rbt mosbe u when But impossible. both or possible both either homogeneous, are | ♦ = · 6| ( = ( A ♦ · ( ∨ ♦ · A B A ∨ ) ∧ B ¬ ∨ ♦ ) · ¬ ∨ B ♦ · ) ( ∨ A ♦ · ( ( ∨ ¬ A B ♦ · ∨ ) A B | ( = 16 ¬ ∧ ) ♦ · ♦ · 14 A B ∧ ) ♦ · B A ) and ∨ ( ¬ B ♦ · ifri oa status. modal in differ A ¬ ∧ ♦ · B io Goldstein Simon ) early access . K A ! For J 18 by 0 does not (22-a) is mapped to K A J by # the team members look happy in In particular, 15 by giving a semantics for the conditional 16 17 (22-b) B to 17 ♦ ∧ A (22-a) ♦ = | 1 B ♦ ) = w ∨ ( K A A J ♦ . w λ = K Either Mary can have cake andhave Mary neither. can have ice cream, or she can A ! J map the same worlds to true; but any world mapped to is consistent with there being an ordinarily grumpy team member who is not happy in K b. A J and We’ve now developed a plausible strategy for validating Free Choice with ho- K A ! WIDEFREECHOICE this picture the picture. For recent work,Free see ChoiceMalamud effects2012 , do Križ not2015b , disappears exhibit non-maximality. Križ forKriž small &2015b pluralities Spector observes and2017. thatSee explicit Interestingly, non-maximalityZimmermann conjunctions. also 2000 for athat semantic apparent account instances of of Wide Wide Freeet Free Choice, al. Choice Simons 2018 , actually2005 Meyer involvefor narrow & an scope Sauerland argument disjunction2016 (and forHoeks a response), and Willer 2017b for a pragmatic account of Another aspect of homogeneity is non-maximality: the truth of Homogeneous alternative semantics invalidates this inference,Choice because never Wide allows Free an alternative-denoting sentence to scope below a possibility Definition 8. J homogeneity effects are expected to disappear. mogeneity effects. But the strategyChoice developed so occurs far not has only one when significant possibility deficit. modals Free scope over example, disjunction,(23) butseems also to imply (24): (23) (24) Mary might be in New York orMary she might might be be in in New Los York Angeles. and she might be in Los Angeles. erations above by treating cases where the antecedent is undefined in the same phenomenon as presuppositional. Inseem particular, to the project homogeneity from the effectseem antecedent does to of not imply conditionals. : (22-b) (22) a. If Mary can have cake or ice cream, we will have a blast. Free choice and homogeneity different kind of undefinedness conditionChoice, as there in isKriž at2015a. least But some in pressure the to case side of with Free Križ 2015a against treating the Then we can invalidate the inference from which applies ! to the antecedent. To predict this, we could rely on a conditional that differs from our Boolean op- when disjunction at least appears to take wide scope to possibility modals. way as cases where the antecedent is false. Under any such flattening operation, Wide Free Choice. 17 18 15 Thanks to several independent16 referees The for relevant help flattening here. operation is defined as follows: frohripeettoso hsidea. this of 21 implementations other 2017. for Cariani 2005 See Geurts 20 and 2000 Zimmermann See 19 early access hr sa niaecneto ewe oa ijnto n h aiiyo Free of validity the and Disjunction Modal between connection intimate an is There aiae ieFe Choice. Free Wide validates ooeeu yai semantics dynamic Homogeneous 6 that requiring disjunction effect, let homogeneity can the we contribute case, modals this possibility than In rather semantics. can itself dynamic effects into homogeneity incorporated semantics, be alternative also to addition In employ. can Choice accessible Choice some Free Wide is by So triggered system. are effects this homogeneity in no reason, this For modal. 2010; Gillies & Fintel von 1991; Kratzer 1972; 2016. Karttunen Lassiter see and modals epistemic with discussion For is possible possibly anything that so Free axiom, implies 4 possible: the Disjunction itself accept Modal we assumptions suppose more For Choice. few a given Similarly, tions. that implies implies immediately axiom this T Choice, the Free all, After entailment. of transitivity  of duality the Given anything that says which axiom, T the possible: accept is we true that suppose why, see To Choice. disjunct valid. each is require inference’ disjunctions dynamic idea: simple within possible. one be Prohibition on to Double rely we’ll and so, Choice do Free To semantics. validate we section, this In It concept. of proof a merely not is This possible. are disjuncts neither or both either 4 T DISJUNCTION MODAL A ♦♦ A enwso ooeet fet r olta ait ftere fFree of theories of variety a that tool a are effects homogeneity show now We ejs a htFe hieipisMdlDsucin ie e assump- few a given Disjunction, Modal implies Choice Free that saw just We the and axiom T the fixed holding Choice, Free from follows Disjunction Modal implies | = A ♦ | = A ♦ A A . 21 19 hti,w edasmniso hc h olwn ‘ignorance following the which on semantics a need we is, That ♦ 20 A A might ol rsm accessible some or world ∨ ♦ A B ∨ and sawy end n rea ol uti aethere case in just world a at true and defined, always is B | = must ♦ A ∧ h xo asthat says axiom T the , ♦ ♦ B 18 A and ♦ B B world. . must A ∨ B ssrn,s that so strong, is | = io Goldstein Simon ♦ ( A ∨ B ) By . early access . , and p # . So, sum- supports a B s ♦ . . So by Upwards B , Modal Disjunction ♦ ♦ . S . Support figures in our s . As before, we rely on a to # S implies ] = A B [ and the undefined state #. s ∨ W A context change potential , so that s 19 , which by the 4 axiom implies . s B According to dynamic semantics, the meaning of a total function from ♦♦ ] = 23 A A [ assigning truth values to every atomic sentence s w ) iff implies contains every subset of By Modal Disjunction, has no effect on A ) S B 22 A = | ∨ s ( A ( A ♦ just in case Suppose Transitivity, Upwards Monotonicity, T and 4. Then Free Choice is is the set of worlds assigns every sentence supports ] A · [ s W To give an update semantics, we need two things: a definition of information i. ii. The set of states iv. iii. over states to include sensitivity to the undefined state See Stalnaker 1973; Karttunen 1974; Heim 1982; Heim 1983; Veltman 1985; Groenendijk & Stokhof semantics for disjunction and in our definition of entailment. Definition 9. context change potential—a function from follow setsHeim of1992 worldsby to relying setscontinuity on of with definedness worlds. the conditions We above also on we updating. modellet However, definedness for our with context a change special potentials undefined remain state total functions on theclaim set of states. Finally, to change the context in which it is said—its states (or contexts), and anpotential interpretation to function each which sentence assigns in a ouras context language. change aVeltman 1996 setmodels of an possible information state worlds. Then an interpretation function assigns every sentence a both disjuncts to be possible. Now we implement this idea withina dynamic sentence semantics is not its truth conditions. Rather, the meaning of a sentence is its ability Monotonicity, ming up, if our possibilityand operators Modal satisfy Disjunction the are T equivalent. and 4 axioms, then FreeFact 4. Choice Free choice and homogeneity If we combine the 4implies axiom Free with Choice. the Upwards Monotonicity of valid iff Modal Disjunction is valid. (in particular, update semantics). weak Kleene notion of undefinedness in the metalanguage. Analogous with the previous section, we can extend the usual Boolean operations we define an analogue of truth in a dynamic setting. Say that a state We’ve now given some justification for our motivating idea, that disjunctions require 1990; Groenendijk & Stokhof 1991; and many others. 22 See Zimmermann 2000 for23 a similar observation. 24 early access iharltvl ttccneto fcnucin hc nescstersl of result the intersects which conjunction, of conception static relatively a with gi,hwvr ealwsm pae nti ytmt eudfie.T e the with get To updating undefined. when be to system this in updates some allow we however, Again, operations: 10. Definition ntefia case, final the In We homogeneity. Proof: to appeal disjunction again can Disjunction the we Modal that where that say is so Here can valid. possible, are be Choice to Free disjunct and each require disjunctions that that s condition definedness with extra update the usual imposes our union simply we above, from with updating that assume we Choice, Free about state predictions absurd right the Otherwise, updating. by unchanged is state properties initial exploring the tests, so, are they state. new Instead, current us in. give the are don’t of we sentences world modal contrast, what By about state. information information an in worlds the whenever undefined is this above, intersection of either with updating of account returns smooth above a complement to of leads negation. above for complement conditions set definedness of the definition Boolean our information §2, fixed an in updating hold proposal, can this we Next, with On negation. true. state for are they semantics dynamic where worlds usual narrow the the simply to sentences state atomic information that an assume down we First, properly. behaves state fined [ a. iii. ♦ a. ii. a. i. A l hti eti ogv eatc o ijnto,wihipeet h idea the implements which disjunction, for semantics a give to is left is that All down narrowing of business the in are far so considered we’ve operations The unde- the that so connectives the for semantics dynamic usual a extend now We = ] s b. b. b. [ A ] s ∀ ∀ ∀ ∀ ∀ ∀ seither is ∩ ¬ s s s s s s , , , A # : # : # : { s s s w 0 0 0 eun xcl h olsta ol o uvv paigwith updating survive not would that worlds the exactly returns s 6= 6= 6= ∩ − ∪ [ | Let ♦ # s s s A : # : # : # s , [ = = = ] A = /0 rsmtiges.I h rtcase, first the In else. something or , − s s s ] A ♦ s s s A 6= − ∩ ∪ s , A ∩ ∪ . or − ∩ s s sudfie.T oe hsuigorBoenoperations Boolean our using this model To undefined. is s /0 # # 0 0 xlrswhether explores 0 and , # B } ======othat so , ∪ { { { # # s # A w w w [ A ∪ # ∨ | | | ] whenever . w w ewa leeetnin fteuulBoolean usual the of extensions Kleene weak be w 24 B s ∈ ∈ ∈ [ sdfie nywhen only defined is A s s s , ∧ , or w s w B [ 20 ¬ ∈ w 6∈ = ] s A s s ∈ s a ecnitnl pae with updated consistently be can ] 0 or 0 } } sjust is s s s [ 0 A [ s } A [ ] A s s ] ∩ [ [ s ] ♦ 6= A [ s s are A A [ ] − B # ] ] or is hsgvsu h meaning the us gives This . hc nti etn merely setting this in which , ] s gi,gvntedefinition the given Again, . # [ # s A o ipiiy ework we simplicity, For . ntescn case, second the In . [ A B ] hc ytedefinition the by which , ] and are. B ♦ hr h same the share A io Goldstein Simon sundefined is s /0 [ ♦ results. A A A = ] As . If . /0 . early access A 25 worlds A . When B whenever: and C A B = | ♦ s then narrows down the ∧ ¬ ] A B , or it can be consistently ♦ ∨ B ¬ A [ = | and . s ) just in case A or C B = | ♦ n ∧ worlds. Summarizing all we have said, we A narrows down the state to the worlds where 21 A B B ♦ ] ,... is true. This theory of disjunction incorporates ∨ 1 A = [ nor | A A B s ( s A ∪ C or if } /0 } A ] 1 6= B ] [ ] entail . s ), and introduce a dynamic version of Strawson entailment, B ) = n A n [ [ p ∪ s = | A s A ( ] ] | ∩ is only defined at information states that contain both w A = ] | [ A | w [ # otherwise s B ,... A s s { [ 1 w ( s ∨ are true. Now, however, we’ve enriched this idea with a homogeneity A ∩ − { (Homogeneous dynamic semantics) A s s ∩ B unions together the result of updating with each of ] = ] = #. ;...; s 1 B B B ] = ] = 6= A and ∨ ∧ ∨ A A ] ] = = A ♦ C p ¬ A A A | worlds, or contain neither are non-modal, this results in classical truth conditions for disjunction, so that [ [ [ [ [ [ s s s s s s s B B i. i. v. ii. ii. iv. iii. the notion of support ( supported and the conclusion is defined is a stateDefinition which 12. supports the conclusion. updating an information state with one of effect, so that and reach the following definition: Definition 11 updated with neither of them.state Finally, when to defined, worlds where onehomogeneity of effects into the normal dynamic theory of disjunction,and according to Free choice and homogeneity modal status. Given our dynamicstate framework, this can means be that consistently either updated the information with each of which says that an argument is valid just in case any state where the premises are which With our theory in place, we can now turn to the puzzles from §1. With our semantics in place, we need a definition of entailment. Here, we rely on 25 See Veltman 1996 and van Benthem 1996 139-41 for other options. early access Whenever gi,tefis neec sol ai eas ersrc teto oinformation to attention restrict we because valid only is inference first the Again, requires conclusion the of definedness the semantics, alternative homogeneous in As aiae h ntneo pad oooiiyta ssDsucinIntroduction. Disjunction uses that Monotonicity Upwards of instance the validates bevto 13. Observation both supports 11. Observation semantics. dynamic homogeneous in valid is Choice Free Results 7 nteagmn from argument the in conclusion the where states 14. Observation the up gives premise also semantics While this entailment. 1, of Fact transitivity escape To Monotonicity. Upwards and tion, support out flat also is it valid; Strawson merely preserving. not is Choice Free compelling. more of validity the distinguish of premise neither or both either that theory this 1: Fact for interest particular Of valid. is Monotonicity Upwards tics: attention our supports restrict also to which us in allows states which validity, information Strawson to on relies proof the Here, 12. Observation ho- in As a disjunction. for of although valid, reason. properties different is classical Introduction Disjunction the semantics, concerning alternative mogeneous 1 , Fact to respond that ooeeu yai eatc aiae reCoc,DsucinIntroduc- Disjunction Choice, Free validates semantics dynamic Homogeneous to us allows also theory this semantics alternative homogeneous in as Again, seman- alternative homogeneous with property another shares also system This also must it Choice, Free validates semantics dynamic homogeneous Since s [ A ] ♦ ♦ ∪ A A s s [ hsmasthat means this , supports B osntb tefimply itself by not does A ] ♦ snnepy So non-empty. is ∨ A B ♦ ♦ ♦ A and ic ecninr h osblt that possibility the ignore can we since , A ( A | = A ♦ ♦ | 6| = = ♦ ∨ ( A A A B B ♦ ♦ ∨ ♦ ∨ to rnihr u since But neither. or , ) ( B B A A B | ♦ = A ♦ ♦ ∨ ) | ( = B , B and ♦ A A B s n oti atragmn ean invalid. remains argument latter this so and , A ♦ [ ∨ ♦ sspotd n so and supported, is ∨ ) s A ( ∧ A uprsboth supports B B B A ∨ ) ♦ implies ♦ ∨ r osbe obndwt h upr fthe of support the with Combined possible. are sdfie.We n uhsaesupports state such any When defined. is B sdfie.Btti etneapasnowhere appears sentence this But defined. is B B B ] . sdfie.Ti ntr eursta either that requires turn in This defined. is ) 22 rmtevldt fFe hie hc is which Choice, Free of validity the from ♦ ( A s ♦ ∨ supports A B and ) ♦ n hsls implies last this and , ( A ♦ ∨ B ♦ B s . ( A ) otisno contains ssupported. is ∨ B ) eas know also we , io Goldstein Simon B ♦ worlds. B the , A it , s early access . . , ) A A A A B ♦ ∨ to be ¬ A ; so by B ( B ♦ ♦ , we do not ¬ and A A ♦ ¬ does not imply implies A is supported, both ) B ) ♦ implies B . This is exactly parallel ∨ ) A ∨ A B ( ♦ A ), but ( ∨ ♦ ¬ ♦ ♦ A ( ¬ ♦ ¬ and B ) ♦ B 23 , which by Double Prohibition implies ) ∨ be the dual of ∧ ¬ B A ) . ( A ∨  B B ♦ ♦ A ♦ ∨ ¬ ¬ ( A ∧ ♦ = ( | ¬ A ♦ ) imply the last conclusion ♦ (letting ¬ B B implies = ∨ A | ♦ = | is supported, it is also defined. When it is defined, either A  ¬ B B B ( ) implies ♦ ∨ ♦ ♦ B ¬ B ¬ ¬ A ∨ ♦ A ¬ ( implies ♦ A ¬ are both possible or both impossible. So if are impossible. B B As in homogeneous alternative semantics, this theory avoids2 Fact by giving up A first natural reaction to this result is to note that homogeneous dynamic In §1, we saw that the joint validity of Free Choice and Double Prohibition We’ve now diagnosed how homogeneous dynamic semantics escapes the first Homogeneous dynamic semantics has so far had the same results as homoge- have the first premise to our treatment of1 Fact above. When we remove the intermediate step result. The semantics above acceptsin each that of result. the relevant instances of Contraposition Observation 17. and immediately imply Explosion. ForContraposition by Free Choice, semantics gives up Contrapositionsemantics, in general. As inNonetheless, ordinary this versions failure of of Contraposition update is not relevant to our second impossibility and increases the danger of Explosion, whereparticular, any given two Transitivity possibility and claims Contraposition, are Free equivalent. Choice In and Double Prohibition involving Double Prohibition. First, one ofthat the Double signature Prohibition properties is of valid: this system is Observation 16. Here again, the key is that the definedness of the conclusion requires Upwards Monotonicity, the T axiom, thehold 4 in axiom, this and system Transitivity. except Allparticular Transitivity. of But instances the while relevant above to Transitivity4 fails Fact do in not general, fail. the incompatibility result from §1. Now let’s turn to the other incompatibility results, bare disjunctions. Homogeneous dynamic semanticsbut validates also not Modal only Disjunction: Free Choice Observation 15. treated symmetrically. Free choice and homogeneity neous alternative semantics. The two frameworks differ, however, in the features of Transitivity. While Whenever We saw with4 Fact that Modal Disjunction is equivalent to Free Choice given early access si ooeeu lentv eatc,tekyosraini htStrawson that is observation key the semantics, alternative homogeneous in As where states to attention our restrict to entitled longer no are we hnvri sdfie,isngto ssupported. is negation its defined, is it whenever hti,w gi aeaohrfiueo Transitivity: of failure another have again we is, That ¬ rbe a htaltgte hs rnilsipidIT that IAT, the implied Again, principles these Dilemma. together Constructive all and that Middle was Excluded problem of Law the with tension symmetrically. h eatc aae ovldt l lsia rnils ihteecpinof exception the with principles, classical result, all each validate With to §1. from manages results semantics incompatibility the logical the of treatment elegant obtains. condition former the that asserts defined when and impossible, claim deontic like consistent. modals, is IAT of next flavors that the other predicts In for generalization consistent. account This be to modals. should system it the validity, generalize a we be section, not should IAT Although fashion. true. a is premise the and defined is conclusion the where state is: That inconsistent. but invalid, only not is it system: this in spectacularly fails IAT not. is 20. conclusion Observation the truth, logical a is above the of premise the while Crucially, 19. Observation we above, disjuncts two is the point to the Prohibition Here, IAT. Double 3 . reach and Fact by Choice of imposed derivation Free test the Constructive applying validates in that the above used state, semantics instance the such the Likewise, including any passed. Dilemma, At is worlds. disjuncts The the of defined. of kind one is both neither it contain contain which either which in which states states or at the defined we at only states, supported is all state be at above to support sentence than the Rather truths. require logical only of less much requires entailment 18. Observation instance: following the including valid, remains LEM ♦ B First, 2. and 1 Facts with as strategy solution same the uses semantics our Here, in Prohibition Double and Choice Free placed which 3, Fact to turn let’s Now nti eto,w aese hthmgnosdnmcsmnispoie an provides semantics dynamic homogeneous that seen have we section, this In The disjunctions. nested to smoothly generalizes above semantics the Finally, strong too in IAT rejects semantics dynamic homogeneous that worry no might is One there because vacuously, holds only 19 Observation reason, same the For ) salgcltruth. logical a is ♦ ( A ∨ ( B ∨ 6| ♦ | ( = = C ( )) A ♦ ♦ ( ∨ sdfie nyi ihralof all either if only defined is A A B ∧ ∨ ) ♦ B ¬ ∨ ) B ¬ ∨ ) ♦ ∨ ( A ♦ ( ¬ ( ∨ ♦ A B A ∨ ) ¬ ∧ 24 B | ( = ) ♦ ♦ B A ) ∧ ♦ B ) ∨ A - ( C ¬ ♦ r osbeo l are all or possible are A A ( olsand worlds ♦ ¬ ∧ A A and ∧ ♦ ♦ B io Goldstein Simon B ) B ) r treated are ∨ B ( worlds, ¬ ♦ A ∧ early access /0 6= . We ] . A A , which [ B /0 w  R ∩ = ] | A s [ is also defined. . But given the w /0 ] narrow down an R A and [ 6= , so that A w ] A B  R B  ♦ ∨ = | = | A [ s w ) w /0 R R ∩ , denotes the information state ] s to the worlds that are consistent w and A [ in R w A w . Then we let w R ♦ w ( into this system that is parameterized = | ∪ can be consistently updated with  } w w /0 , where R 25 R R , this implies that 6= s B ]  A [ ∧ w where A R w |  . w # and otherwise produces #. = | { w ) 6= ∩ ] B only if for every world is arbitrary in s , which relates any world A ) ∨ d [ w B R w A ] = ( R ∨ A  A  [ ( , this in turn requires that s . But we can allow this dynamic disjunction operator to embed under  ∨ /0.Since ♦ 6= ] B [ w supports R s Now we can retain our earlier semantics for disjunction, defined in terms of the We introduce a possible worlds modal Above, we appealed to possibility operators in two separate ways. First, Free . The resulting theory validates Free Choice: For semantics for example, we can modelaccessibility Free relation Choice for deontic modals by appeal to a deontic test operator  Observation 21. has no effect when Definition 13. and to an underlying accessibility relation made up of the accessibleinformation state possibilities to from the worlds impose an extra condition so that this update is defined only if Choice involved embedding disjunction under someond, sort our semantics of had possibility disjunctions modal. themselves Sec- require that each disjunct be possible. for disjunction in a way that validates Free Choice for deontic modals. So far, we’ve focused our discussionresults on for epistemic other modals. flavors Here, of weconditional modals. way achieve The similar and upshot still is reap thatthemselves the we use benefits can a of treat dynamic the modals kind above, in provided of a that possibility truth disjunctions operator. Homogeneous dynamic semantics seems tailor-madeChoice. for But epistemic how versions can of it Free extendnext to section Free we Choice achieve this for result. other flavors of modality? In the 8 Extension to other modals Free choice and homogeneity Free Choice and Double Prohibition. So far, however, the reader may have a concern. To implement this last condition, we union the overall update with This semantics for modals allows us to model arbitrary flavors of modality. For Transitivity. Most importantly, the semantics is able to deliver the joint validity of with the normative rules at We’ll now see that these twoWe can appeals integrate to a possibility static modals semantics can for be deontic treated modals separately. with our dynamic semantics 26 early access a o nyivld u nosset hssee o tog ic o example for since strong, too seemed This inconsistent. but invalid, only not was when contrast, By say. knowledge, common is what with seslk tcudb true: be could it like seems (25) A h esnta eaeal ovldt hspicpergrls ftesrcueof structure the of regardless principle this validate to able are we that reason The hr psei oasaesniientol oteiptifrainsaebtas oawrdof world a to also but state information input the to only not sensitive are modals epistemic where valid: (25) that 2018, Goldstein in semantics dynamic evaluation. the use could we modals, epistemic for results similar reach To worlds. two the of one at holds world, supports first neither state the world, This At second permissible. worlds. the two At of permissible. up deontically made both state a consider example, For 22. Observation is but invalid, still is IAT prediction. right consistent: the also just makes semantics new our Here, one ( IAT with that section, predicted previous we the section, in previous that the as In results exception. same More the Prohibition. validates has Double semantics also validates this that generally, semantics way revised a this in Similarly, disjunction Choice. for Free semantics dynamic our into integrated dynamic of hallmark a is context local semantics. to sensitivity This unembedded. when says like operator, shifting information an if So state occurs. information it whole which the in environment the on depending information, because UE OIIEINTROSPECTION POSITIVE OUTER scnitn with consistent is osmaie h rca rprynee eei httefloigpicpeis principle following the that is here needed property crucial the summarize, To ev o enta rt odtoa eatc o osblt oascnbe can modals possibility for semantics conditional truth a that seen now We’ve ♦ sadnmcoperator. dynamic a is ♦ aymyhv opadMr a aesld rMr a o aesoup have not may salad. Mary have or not salad; may have Mary may and Mary and soup have may Mary A sdnmc tcni rnil rdct osblt fdfeetbde of bodies different of possibility predicate principle in can it dynamic, is (  A R w ∧ htis, That .  B s ) , ♦ ∨ ♦ A 26 ( A ¬ ( rdctspsiiiyo oyo nomto.But information. of body a of possibility predicates ♦  rdctseitmcpsiiiyof possibility epistemic predicates  A A astesm hn ne h cp of scope the under thing same the says ♦ ∧ ¬ ∧   thsadfeetefc.Under effect. different a has it , A 26 B  ) | = B ∨ )  ( 6| = ¬ A  . ⊥ A ¬ ∧  [ ♦ B A ) A ♦ eas ahdisjunct each because , ∧ A nor ♦ sebde under embedded is B B ] ∨ A r deontically are [ io Goldstein Simon —consistency ♦ ¬ A ♦ A  A ple to applies , and ♦ ∧¬  A B ♦ as says R are B  is ] ) early access s ∨ 28 A , and  } . But u /0 , w = { is accessible } 6 = v w is itself defined , s { w B ♦ ] = ∨ A  A [ . s ♦ v . Now let u satisfies the 4 axiom, Wide ♦ . In particular, the inference is .  27 , since ♦ A but not w . Since ♦ B is accessible from any v ♦♦ B 27 ♦ , which cannot see u ∧ supports s A ♦ supports = s | B contains is accessible from s ♦ and v uniquely. ∨ A v A ♦ ♦♦ , since A  . Unlike the narrow scope version, this inference’s validity turns on is universal, so that whenever B . For suppose supports  R w s ∧ You make take an apple or you may take a pear, but I won’t tell you which. A  be a claim true at only if to For this reason, homogeneous dynamic semantics allows that Wide Free Choice Things are a bit more complicated for versions of Wide Free Choice with other A s B WIDEFREECHOICE accessibility relation. Here our accounttested makes in theHoeks kinds et ofChoice al. predictions2018, is that which accepted were found fordeontic experimentally facts, deontic that while modals narrow wide scope scope independently Free Free of Choice only speaker seems knowledge to of hold when the speaker is In (26) and ,(27) Freemodals Choice under is the cancelled: scope these ofcase sentences disjunction. with seem In this to form homogeneous involves have dynamic a deontic semantics, modal any being interpreted relative to a non-universal  let does not support is cancellable for deontic, but not epistemic modals. (26) (27) You may take an apple or you may take a pear, but I don’t know which. modals. In homogeneous dynamic semantics, this involves the inference from properties of the accessibility relation associated with from every is valid in homogeneous dynamic semantics.homogeneity effects In are homogeneous contributed dynamic by semantics, disjunction itself.in So Free Choice is valid for the epistemic test operator possibility modals. Choice and Double Prohibition. Thisbetween section these explores two two systems. factors that might decide 9.1 Wide free choice Free choice and homogeneity 9 Comparing the two accounts valid only if As we saw above, Free Choice also occurs when disjunction takes wide scope to While this inference is invalid in homogeneous alternative semantics, the inference We’ve now developed two different semantics that use homogeneity to validate Free 27 A similar prediction is28 achieved See in Kamp Zimmermann 1978 2000. and Willer 2017b for discussion. early access fo 2 r qal oplig ecnte xli hs ae by cases these explain then can We compelling. equally are §2) from (11) and (10) cet rdefo reCoc oWd reCoc.I we If Choice. Free Wide to Choice Free from bridge a create 5 and 4 Facts Together, eb like verbs (h sentences (the (27) and of (26) choice of the analogues of scope regardless narrow the valid But are relation. These accessibility inferences. Choice Free scope narrow not is modal epistemic can We relevant modals. the operator epistemic sometimes test for that the even proposing a cancelled by be be cases may can these there Choice explain suggests Free 2005 which Simons Finally, in rules. cases the few about expert an as treated ooeeu yai eatc ade hsdt ihes.Asaesupports state A ease. with data this ¬ handles semantics dynamic Homogeneous (29) (28) (29): attitude negative entail under to Choice appears Free Wide of instances embed can we effect, Free this Wide for and Choice Free 5, and 4 Facts equivalent. from are assumptions Choice the of all fixed hold Transitivity, by ♦ Choice, Dilemma Constructive Proof. valid. is Disjunction Modal iff valid is Choice Free two these 5. Fact explored, already have we Modal to principles equivalent. connected are of intimately principles variety is Choice a dis- Free Given of Wide theory that Disjunction. modal out a turns it is But semantics junction. dynamic homogeneous that know We Choice. scope wide taking disjunction have they form logical to of level the at that suggesting IEDUL PROHIBITION DOUBLE WIDE ( A  ♦ ∨ ial,i ooeeu yai eatc hr sn nlgu a ocancel to way analogous no is there semantics dynamic homogeneous in Finally, utlk reCoc,Wd reCoc lodsper ne eain otest To negation. under disappears also Choice Free Wide Choice, Free like Just Free Wide validates semantics dynamic homogeneous that coincidence no is It ihannuieslacsiiiyrelation. accessibility non-universal a with , A ♦ ∨ B o h ett ih ieto:b T, by direction: right to left the For A ups rniiiy ,te4aimadCntutv iem.Te Wide Then Dilemma. Constructive and axiom 4 the T, Transitivity, Suppose ♦ Angeles. Angeles. lxblee htBli a’ ei e okadBli a’ ei Los in be can’t Billie and York New in Los be can’t in Billie be that might believes Billie Alex or York New in be might Billie that doubts Alex implies B ∨ doubts ) B nyi ihrboth either if only implies ♦ ♦ hc r nutvl qiaetto equivalent intuitively are which , A ♦♦ u rather but , ∨ A ♦ ♦ B ∧ A A ♦♦ implies ∧ ∨ ♦ B B B  implies yte4axiom, 4 the By . o h ih olf ieto:b oa Disjunction, Modal by direction: left to right the For . ¬ ne neitmcacsiiiyrelation. accessibility epistemic an under A ( ♦ ♦ and A A ∧ ∨ ♦ ♦ B ♦ A 28 B B r osbe rbt r mosbe The impossible. are both or possible, are ∨ . ) A ♦ | = B implies ¬ ob rniiiyadWd Free Wide and Transitivity by So . ♦♦ ♦ A A ¬ ∧ eivsnot believes ♦ ∧ A ♦ ♦♦ B and B implies B (28) example, For . implies io Goldstein Simon ♦ A ♦ ∧ B ♦ oby So . B So . early access - - + + + + + + + + HDS - - - - + + + + + + HAS - - - - - + + + + + AS x is a philosopher or ) C A A B B B B B B B ¬ ♦ ♦ ♦ ♦ ♦ ∨ = ♦ | ∨ ¬ ∧ ∧ ∧ A = = | | A ∧ ¬ A A A A ∧ ¬ = B A | A ♦ ♦ ♦ ⇒ = A ♦ | A ♦ = = = = ♦ | | | ¬ ⇒ ¬ ⇒ C ) ¬ B B ( = = = B | ♦ = ∨ | ∨ ) B B ∨ ) ∨ A B B A B = = | | A ( ∨ 29 & ♦ ♦ ♦ A A A ∧ B ( ♦ A = | ¬ ♦ rules out the former condition; so both must be A ) = ( | B ♦ ∨ A ♦ ( ¬ difficulty when disjunction is embedded under quantifiers. For should be defined only if either both claims are possible, or both are 29 prima facie Of course, to understand this prediction we have to integrate the dynamic se- The following table summarizes the pros and cons of the theories above. Double Prohibition Wide Free Choice LEM IAT Modal Disjunction Transitivity Contraposition Disjunction Introduction UM Free Choice mantics above with a semantics forso, quantification. the I’ll problem now above argue disappears. that Ifcan when we predict we extend that do our the semantics disjunctiveall in restrictor linguists. the in right(33) way,quantifies we over all philosophers and be a linguist. After all,x on is the a account linguist above theimpossible. open Intuitively, formula however, (33) is falsenot if a someone philosopher who did must not be go a to linguist the and party. face a example, consider (33): (30) EveryOne philosopher might or worry linguist that went homogeneous dynamic to semantics the makes party. a bad prediction about between disjunction and quantifiers. 9.2 Quantifiers Homogeneous dynamic semantics, and modal theories of disjunction in general, So far, we’ve considered one advantagehomogeneous of alternative homogeneous semantics: dynamic that semantics the over former but not the latter can validate Free choice and homogeneity ordinary update of impossible. (33): that it only quantifies over people who might be a philosopher and also might Wide Free Choice. Now we turn to one potential disadvantage: the interaction 29 This relates to observations in Büring 1998. early access el oe nvra unicto sn eeaie quantifier generalized a using quantification universal model We’ll hr h eeetof referent the where Eeydne hti etni sad. is beaten is that donkey sad. Every is it beaten, is donkey a If (32) donkey about predictions right (31) the equivalent: are make (32) to 2017. and is Goldstein (31) semantics & that Carter this so in anaphora, of found goals be the can details of implementation many full One omit a we’ll brevity, but For semantics; 1996. the al. of et Groenendijk in semantics dynamic ento 14. Definition g of terms in directly that so B semantics, test usual their retain anypair that so context the expanding Then worlds. of sets like indefinites model ∃ we’ll and input, as sentences REDUCTION vii. iii. 0 x vi. iv. ii. v. fe en pae with updated being after ifr from differs hsgvsus: gives This (31). in conditional the of antecedent the within occurs which , i. eew’lcnie eatc o uniesta sasmlfiaino the of simplification a is that quantifiers for semantics a consider we’ll Here u eatc escnet est fasgmn-ol ar ahrta just than rather pairs assignment-world of sets be contexts lets semantics Our h c c c c c c c g [ [ [ [ [ [ [ 0 A A ♦ ∀ ∃ ∃ Fx , x xA x w A ∨ → ( = ] = ] i = ] A B = ] where )( B = ] {h {h = ] ∃ ( C c g xFx g g ]= )] [ c 0otherwise /0 ( ∃ , 0 na ottevleo variable of value the most at in , ( w x w c otherwise # g ][ ∈ i [ → ∃ c if 0otherwise /0 | i A c A x and x [ [ ] c Fx ∃ ] Gx and g ∪ satisfies [ c if xA A [ ] x | c ] c g arw ontecnett h sinetwrdpairs assignment-world the to context the down narrows ] [ g g 6= [ B → 0 → ( A A 0 = | | x ifrol ntevleof value the in only differ ] = ] & ial,w a ipydfietesentence the define simply can we Finally, . ) oaheeRdcinb rt oc.Where force. brute by Reduction achieve to , /0 C | = ∀ ∈ F ] if h x g Crucially, . B V ( c , h Fx w ( g | = F , ∈ i )( )( w ♦ Gx i A w A c 30 ari h urn otx sjie yany by joined is context current the in pair ) → } ) ∧ } ♦ [ B ∃ B x et httecneti nhne by unchanged is context the that tests ] or x efrsanneiiaieupdate, non-eliminative a performs : c donkey a | = ¬ x ♦ h operators The . A ¬ ∧ ihaseiloperator special a with ♦ B ∀ x io Goldstein Simon httkstwo takes that → ∀ g x [ ( x and A ] g )( 0 B iff ♦ ) early access F is is → )] ) , and to the w Gx Gx ] x might are each ∨ ∨ at ∃ [ )] c F } Fx Fx c ( ( Gx is x x to refer to any ][ ∃ ∃ ) x x [ [ i ∈ , which updates x 0 c c ∃ # ( [ w g c ♦ , g h . This sentence has the ) : and test remain in the state, . Now 0 ] ] to allow where narrows down Hx w ♦ c Fx includes any possible value i Hx )( )] [ ] ][ | ∃ w x x ] , Gx Gx ∃ ∃ g A [ [ h ∨ c c ∨ ♦ [ ; and (ii) every individual who is } expands Fx . Summarizing, the whole sentence G Fx ( g ] G x x ][( ∀ = ∃ x or [ 0 . Since include some world where something is ∃ 0 c g [ . In other words, the disjunctive restrictor F c . This is a perfectly reasonable prediction, . c w | 31 above, then we make some bad predictions. G H )] c at ♦ is unchanged by include some Gx G . The key property here is that )] ] i ∈ to each cell. The result is that update proceeds ∨ x is w Hx ∃ is either Gx , ♦ ) [ 0 Fx x x c g ∨ → ( h 0 ][( ) x g is then defined only if Fx {{ ∃ ( [ Gx )] x S c ∃ ∨ [ Gx c ] = where Fx ∨ ( A i 0 # x Fx w ∃ ♦ , [ 0 ][( c g x , and a world where something is h ∃ F [ c at any world in the context is also explores whether , this reduces to the requirement that The crucial idea, building on a proposal in Büring ,1998 is too let overt epistemic This theory makes the right predictions about the interaction between quantifiers ] x G , and some world where something is a state by first partitioningthen it applying into the cells test which operator agree on assignment functions, and Definition 15. (33) (34) Someone who might be a linguist wentEveryone toIn who the a might party. context be where linguist both wenttions linguists to result. and the To philosophers party. make are the under rightdisjunction, discussion, predictions but bad here, offer predic- a we different can semantics keep for the overt same epistemic semantics possibility for claims. possibility function as a kind of ‘quasi-distributive’ operator, For example, (33) and (34) quantify over everyone in the domain; not simply those and disjunction. But without sayingabout more, the the interaction theory between makes quantifiers theGroenendijk and wrong et epistemic prediction al. modality. Beaver 1996 1994 noteand simply that understood if as the the overt epistemic test possibility operator modal assignment-world pairs where is then supported by asomething context is just in caseor (i) the context contains aand world avoids the where concerns above.philosopher When fails someone to who go must to be the party, a condition linguist (ii) and fails not and a (33) is false. include some of F only applies a possibility teston to each the global individual one context; it at does a not time. place Finally, this requirement same meaning as in turn decomposed into individual. non-empty. This requires that Free choice and homogeneity meaning of disjunctive quantified claims like Hx while cells which fail the test are eliminated. who might be linguists. With a simple dynamic theory of quantifiers and modals in place, let’s return to the ‘cell-wise’ rather than point-wise: cells which pass the early access hte htcl supports cell that whether hc ijnto mesudrquantifiers. under embeds disjunction which hsscinbifl oprsti ae’ hoiswt oeeitn semantic existing some with theories paper’s this compares briefly section This ♦ a When not. is disjunction in work at possibility of kind the quasi-distributive, are basic more 2017.) into Goldstein decomposed & be Carter might see it how operations, and modals, possibility quantification, for of meaning domain the maps over function quantify restrictedly restrictor the can because (34) and (33) Now ahacutdfesi oeitrsigwy rmm own. my from ways semantics. interesting dynamic some homogeneous in resemble differs instead account 2016 Each Starr seman- and alternative Willer homogeneous 2018 and resemble Aloni 2017a 2016, tics, Willer Starr and 2018, 2012 Aloni Aher 2012, While Aher 2017a. including Choice, Free of accounts accounts Semantic 10.1 accounts existing with Comparison 10 in way the about predictions bad making without modal be to disjunction allow ignored. can the be on can tested subtlety is this Choice variables, without Free language Wide the when of course, on fragment Of requirement quantification. definedness itself of homogeneous could domain of requirement kind the this another whether imposing is by research, achieved further be for left question, One of can. value the for candidate every either ♦ be, cannot for whether candidates is to question contains crucial the Here, Choice. ♦♦ Free Wide guarantee doesn’t conclusion the itself by required is what exactly whether is question ♦ The modals. possibility overt of meaning the application an has still it possibility, disjunction. epistemic of overt case of the model in good a not be is might operator who individuals to restricted not is disjunction # # # A Fx ♦ # h rca bevto o u upssi htwieoetpsiiiyclaims possibility overt while that is purposes our for observation crucial The fw aeaspitctdteteto unicto ndnmcsmnis we semantics, dynamic in quantification of treatment sophisticated a have we If enrich we once valid still is Choice Free that verify to is remains that All A A ∧ ∨ | partitions = ♦ ♦ # # ♦ B Gx # oseta hsi o oethat note so, is this that see To . ♦♦ Fx A hscniinfiseatywhen exactly fails condition This . implies ∨ # x Fx Gx c oa niiulta salnus tsm ol.(o oeo this on more (For world. some at linguist a is that individual an to noclsta ge ntevleo l aibe,adte checks then and variables, all of value the on agree that cells into leaves cusi h etitro unie,tersligquantification resulting the quantifier, a of restrictor the in occurs ♦ ♦ Fx # Lx c ∧ x ♦♦ nhne,while unchanged, , ♦ updates d Gx A and tde ojs ncs h elas supports also cell the case in just so does It . u nyimplies only but , d c 0 x hr h omrmgtbe might former the where , oteuino el hr h assignment the where cells of union the to ihrcnbe can either 32 ♦ F # n lomgtbe might also and ♦ A ♦ # nteohrhn,ti con by account this hand, other the On . # ♦ ♦ Fx A and ♦ | removes F = # Fx n a be can and ♦ ♦ # # ∧ A oeaat ftecontext the If apart. come ♦ fe l,tepremise the all, After . # d Gx 0 G sa pin Thus option. an as oee ftetest the if even So . ncnet where contexts in G rn candidate no or , F io Goldstein Simon ♦ n h latter the and # ( A ∨ B ) ♦ | = A , early access . . ) A B ♦ ∨ A . ( B ♦ = | = , we do not | . Then every supports s ) s A s B ♦ | . Deontic modals ¬ = ∨ . So , then A = v | n ( ) A ♦ and B = | ∨ = | s A , suppose that . ( rejects A A ♦ s ♦ , true at a world just in case the ♦ also rejects ¬ v = | A ) v v | , . . . , and . Crucially, any state which supports = B 1 v = | atom ∨ A rejects 0 0 fails despite the validity of Disjunction s s s A = | ( ) s 33 ♦ B rejects . So then then ∨ B B v B | : if = A A A B B violation | s = ( ∨ = | 1 0 ∀ = = which supports = ♦ | | | s s A 0 0 s s s s s iff = | ) = ) = ⊆ p p and or and or B A ( ( supports 0 : if : if s ♦ A A A A w w = | | s s | B = = . For this very reason, possibility modals are not upward : : ) n A A = = ∨ | | s s | ⊆ ⊆ = B A 0 0 s s s s A = ; ∈ ∈ | s s ∨ ∀ ∀ s s iff iff iff iff w w A ... ∀ ∀ ( ; which supports B B B B iff iff iff iff 1 . So every ♦ s A ∨ ∧ ∧ ∨ iff iff B A A A A = | ⊆ ♦ ♦ ¬ A A A p p A ¬ ∨ 30 | | | | | 0 = = = = = s A A = = = = = | | | | | ♦ s s s s s s s s s s which supports b. b. b. b. b. s Since preservation of support is transitive, we know from our earlier results Aher 2012 gives an inquisitive semantics for deontic modals, in which sentences i. a. v. a. supports ⊆ ii. a. iv. a. iii. a. 0 have that monotonic. The inference Introduction. That said, we could enrich anyAher premise 2012 thatwith supports Strawson the entailment, premises where does an not inference reject is the valid when conclusion. s that it doesn’t contrapose. In particular, although is preservation of support. Definition 17. Here, Aher 2012 is similarDouble to Prohibition homogeneous are alternative valid. semantics: To see Free why Choice and are interpreted in terms ofrules a are special violated there (andevery supported[/rejected] world by in a it.) state just in case true[/false]Definition at 16. Free choice and homogeneity are recursively assigned supporting and rejecting states by Then every To see why Double Prohibition is valid, suppose that A Within this framework, the relevant notion of entailment for validating Free Choice 30 frmr nteitrcinbtendotcadeitmcmodals. epistemic and deontic between interaction the on more for 2013 32 Groenendijk & Aher See 31 early access ssmnisfrpsiiiymdl losta oesae ohaccept both states some that allows modals possibility for semantics 2012’s Aher n reject and Non-Contradiction. of Law the concerns difference One adeeitmccnrdcin.Frsmlct,Iinr hs set fteview. the of to aspects state) these information ignore designated I a simplicity, example, For (for contradictions. complexities epistemic further handle with enriched is framework The 18. Definition modals. possibility for semantics Kripke ordinary an on relies exhaust by supported is requirement non-emptiness a with enriched proposal, 2018’s Aloni On language modals. natural possibility and disjunction for semantics different Disjunction Choice. from Modal Free invalidates Wide differs account and account the the since hand, semantics, dynamic other homogeneous the On semantics. alternative homogeneous of domain. notion epistemic a the in on focused analysis the with possibility, and LNC a. iii. a. vi. a. iv. a. ii. a. v. a. i. dfesfo ooeeu lentv eatc nsvrlrespects. several in semantics alternative homogeneous from differs 2012 Aher u iha with but 2012, Aher as framework semantic same the within works 2018 Aloni ¬ A ♦ b. b. b. b. b. ; A ¬ s s s s s s s s s s s s fcsso deontic on focuses 2012 Aher Finally, fails. LNC So supported. both are A hr n ustsupports subset one where | | | | | | ======♦ | ======A | | | | | A ¬ ¬ p p ♦ ♦ A NE A A osdraysaewihrejects which state any Consider . ⊥ A A A A iff iff ∨ ∨ ∧ ∧ Where s iff iff iff iff iff B B B B or ∀ ∀ uti aeteeaetonnepysbesof subsets non-empty two are there case in just w w iff iff iff iff s s s ∀ ∀ ( ∨ w w 6= | ∈ ∈ = s ∃ ∃ s R + = ∈ ∈ t t | s s | w = sdcmoe noa neligdsucinoperator disjunction underlying an into decomposed is ) /0 A A , , 31 : : t t = s s = | 0 0 w w A A are nmn epcswith respects many in agrees 2012 Aher Summarizing, : : : : ( ( { R ∃ t t and and p p v ∪ ∪ t w = ) = ) ⊆ | t t wRv 0 0 s s = = R = | 0 1 | A = w s s = | } : B B A & & : t n h te supports other the and 6= 34 t t | = /0 & NE = | A A ntersligsemantics, resulting the On . t & & | = t t p 0 0 violation A | = n rejects and = | B B hthsn counterpart no has that 32 B v then 2018 Aloni . nta state that In . s io Goldstein Simon hc jointly which A ∨ + ♦ ∨ B A early access . s B for ∈ and + . So B ∨ w w , with which which world, + NE A t s > have the . A supports ∨ ∧ s s + of s A | B 0 = ∈ | = u 0 also supports , for all > ∨ is indisputable w s w B w R and R + R or neither of them u . This immediately ∨ can see an B B A s . , and . Then B for every and that ) and ♦ | B B = NE B A + ♦ ∧ , we know that s ∨ + A w | = A ∨ R ( when any two worlds in A and ♦ ) ) | = w s ♦ A , we know there are two worlds R which supports ♦ NE NE B | is consistent with s = w ∧ ∧ ♦ v R R . So + B B s 35 ) ( ( ⊆ sentence (itself definable as ∨ supports t ∨ ∨ , there must be non-empty subsets of A NE and and consistent with s ) ) B . So ♦ NE when either both of s ∧ A A + s NE NE world. ∈ B ∨ are also subsets of ( ∧ ∧ 0 B A w , there must be non-empty subsets is indisputable at ∨ u A A B ( ) | R = + = ( | and NE ∨ . But this means that every world in supporting /0 B s s u ∧ B s A = = supports | iff iff A can see a , for each . ( s s | s s = A B B B and | = iff + + A is consistent with is consistent with . Since . From ∧ ¬ ∨ ∨ w 0 is only defined in ). Interestingly, this sentence is not accepted by the absurd state. This s just in case it is either supported or rejected there. In the semantics B w w ) w R A NE A A s p B R | | R = = R to support just in case there is some . + t = | and ) B ∨ ∧ ¬ s s s . So is indisputable. For suppose and B ♦ A s p A ( R + A b. b. where | ∈ = ¬ ∨ s and While homogeneous dynamic semantics and Aloni 2018’s system embody a It is not coincidental that Aloni ’s2018 semantics has so many similar predictions Next, Aloni 2018’s semantics has similar properties to our own with respect to Double Prohibition is valid, for suppose Modal Disjunction is valid because epistemic modals quantify over worlds in the Aloni 2018 shares many of the predictions of homogeneous dynamic semantics. w A = in ( A w vii. a. EXPLOSION concerns the logical status> of the gives the logic a curious property: the principle of explosion is invalid. is defined at above, are consistent with similar idea of disjunction, they do so in different ways. The most striking difference v implies that to those homogeneous dynamicdisjunctions semantics. are homogeneous Both with theories respect embody to the modal idea status. For that say that R same accessible possibilities. Wide Free Choice holds at any states with respect to relevant state. So if support each of and every world in all Free Choice, Modal Disjunction, and♦ Double Prohibition are allFor valid. such a support ♦ Free choice and homogeneity As in Aher , 2012 Aloni 2018 defines entailment as preservation of support. which with respect to Wide Free Choice. Say that early access u u w nltrlsse srce hnthe than richer is system unilateral own our But . 2019 Santorio & Romoli by work same the are conjunctions for alternatives negative the that stipulate simply can we with A A hsivldt ssrkn,snei en htcnucin r o lasa strong as always not are conjunctions that means it since striking, is invalidity This consadoron nteacut bv,tepoete fFe hieunder Choice Free of properties the above, accounts the In own. our and accounts disjunction. corresponding the as example, for down: percolates Explosion of failure similar a particular, In vldtsodnr reCoc naohrwy ygnrtn lentvsout alternatives generating letting by and way, updating, another in of Choice Free ordinary validates updates 2016 two of union alone. the update setting, each non-eliminative takes than a stronger which In is disjunction updates. for both meaning of dynamic union the ordinary an preference ranked with current top a happily a taking include interacts of to only way it not any refining include also and to but ordering expands orderings, context preference the current Otherwise, the results. all state some absurd allows the ordering failed, preference every is that Double sure Updating with make orderings. claim to along preference possibility tested of deontic Choice, set a Free a with contain context and context a Choice a lets Free 2016 Wide Starr Prohibition. both 2016 validate Starr update semantics, to dynamically dynamic modals seeks homogeneous which in on As Choice possibilities. Free over deontic preferences of theory a offers 2016 which data, this explain to with. us allows struggle richness accounts extra unilateral As trivalent. this other themselves section, are next alternatives the our in because discussed 2018, Aloni in recent discussed of basis ones the on weaker systems a unilateral is against conjunctions argues negated 2018 may with Aloni this Choice Second, section, Free effect. next since the prediction, in desirable discussed a As available. be alternative not homogeneous is In strategy disjunction. this relevant semantics, the for alternatives positive system, the bilateral as a In conjunctions. negated with occur not prediction do strong effects fairly Choice the Free makes that each it but semantics First, alternatives: alternative upshots. of homogeneous empirical set that has single means difference a architectural of This terms the trivalent. in is semantics modeled alternative alternative is sentence homogenous a in of positive contrast, meaning alternatives: of By sets alternatives. independent negative two and to appealing by derived are negation ∨ ¬ ∧ ial,teesa neetn rhtcua ifrnebtenbt fteabove the of both between difference architectural interesting an there’s Finally, Starr Choice. Free of accounts semantic offer also 2017a Willer and 2016 Starr B ♦ hc include which , A A ihu supporting without ∨ ♦ B A novsnneiiaieyudtn ihboth with updating non-eliminatively involves ¬ ∧ A A osntimply not does and ♦ nvral uniyoe h lentvsgnrtdby generated alternatives the over quantify universally NE B vldtsDul rhbto by Prohibition Double validates 2016 Starr Finally, . ic h ijnto operator disjunction the Since . ♦ NE A ∨ 36 ♦ ic hr sasae( state a is there since , ♦ A B novstoses is,tecnetis context the First, steps. two involves implies A ol.Ti o-lmntv update non-eliminative This world. ♦ A ∧ ♦ B ∨ A A eas nupdate an because + ♦ /0 ¬ ∧ hc supports which ) ol.I hstest this If world. A incorporates and A io Goldstein Simon 6| = ♦ A B ∨ Starr . + NE ¬ A , . early access ] B ♦ [ S , while ] A ♦ [ . This is just S B is evaluated at coincides with is not permitted ) B ] = B B ♦ B ∨ ♦ ∨ are both weaker than A ∨ A ( A ♦ ♦ ¬ A ♦  [ , and the result of updating ↑ ¬ works normally while ¬ S A ] s is consistent only if the union A ) and ♦ B [ therefore requires that the positive , the result is a new state consisting A S ∨ ) B , ♦ B A S ∨ ( ∨ A ¬¬ ♦ A ↑ ( 37 which can reverse the preference orderings , and is also consistent with s ♦ A A ¬ ↑ ¬ fails. In that case, s . Like Aher 2012 and Aloni 2018,Willer 2017a B B ♦ ♦ produces absurdity. So . Possibility modals test the state to ensure that updating ∧ B B . However, as we discuss below, Ciardelli et al. 2018 argue A , which is the absurd state. The result is that Wide Free ] B ♦ . On this proposal, B and ♦ ↑ ♦ ∧ s A ][ may is consistent with A A is permitted by all preference orderings, while s ♦ . Willer 2017a seeks to validate Free Choice even for this, so that ♦ [ ) A and . S = | B B . ) ♦ are both satisfied. A B ] = ∧ ¬ must B B A ∧ ¬ ♦ ∧¬ ♦ ¬ A ( A ∧ ♦ ¬ ¬ and Willer 2017a generates a similar empirical profile to homogeneous alterna- On Willer ’s2017a proposal, the meaning of a sentence is a pair of a positive and This theory differs in several respects from our own. First, the analysis only A ( . Finally, the theory may face a technical issue when ♦ ¬ A A ↑ ¬ [ architectural choice leads to further differences in the logic. For example, Willer lent ♦ against exactly this. At anysemantics. rate, Another this difference inference is is that Willer invalidarchitecture,2017a inrelies folding homogeneous on some alternative a kind quite of different alternative semantic semantics into a dynamic system. This s tive semantics. While FreeChoice Choice is and invalid. Double However, one Prohibitionhomogeneous significant are alternative difference valid, semantics between concerns Wide Willer the Free 2017a treatmentand of the DeMorgan equiva- explicit definition of positive andpossibility modal negative (or updating. the The positive negative updaterequires update that with updating the with the negation a of uniononly a of produce possibility alternatives with absurdity. modal) Crucially, the prejacentupdate is guaranteed with to both of every input alternative with the union of alternatives in the state with the prejacentof does not alternatives produce in absurdity. Choice may not actually be♦ valid, since it only holds in contexts where the testsnegative for update on states, themselves sets of alternatives or classical propositions. of the result of updating every input alternative with ♦ contexts where by all preference orderings. Incrashes, such a because context the testS for covers Free Choice inSecond, deontic the modals; analysis it is particular, does Starr less2016 not classicalseeks extend than to invalidateduality our to both of double own epistemic negation with modals. elimination respect and the to negation. In Free choice and homogeneity appealing to a nonclassical negation induced by validates Double Prohibition by appeal to bilateralism, in this case by offering an The result is that Free Choice is valid: what is required by When a state is updated with the disjunction early access hc a hi nust h riaytuooyadcnrdcin hssaegen- scale This contradiction. and tautology ordinary the to inputs their map which A hsscincmae hspprstere ihtoacut fFe Choice, Free of accounts two with theories paper’s this compares section This 02Ipiauebsdaccounts based Implicature 10.2 validity. Strawson of on accounts existing rely addition, not In do Non-Contradiction. Choice respect of Free with Law including the ways, and various Explosion in to differ theories extant But semantics. dynamic the including contradiction theories, epistemic dynamic the existing of of inconsistency properties signature preserves 2017a ossetwt h rmr mlctrsadtetuhcniin.Frexample, For conditions. truth the and implicatures primary alternatives similar the any with for consistent that require we step, X secondary this At implicatures. result. B assert who speakers why explains rule this addition, believe In would absurd. they bare principle, a of similarity assert case the who the speaker: speakers By In the why member. by explains no same first believe this the they disjunction, treated or be member, must every believe alternatives they of either set alternatives: a of of set member third every a tives: us gives This the connective. keeping the Any from of result split. side that connective right alternatives or generation: the left alternative generates connective of a method with second sentence a on relies 2008 than weaker alternatives: and of stronger sets those two into partitioned then are alternatives for alternatives erates members The scale strength. linear a by on relies 2008 Chemla natives. believes she when other. similarly alternatives the believes two she treats iff agent relevantone An way. replacing by similar derived a are in alternatives they items similar when treat similar principle: are In- alternatives similarity alternatives. Two of pragmatic of similarly. sets a sets by to of governed variety appeals is a which terpretation to Choice relative implicature. Free interpreted scalar of are a Utterances account alternatives. resembles pragmatic inference a the gives where 2008 Chemla 2007 , Fox and 2008 Chemla u fte eivdbt,te ol lobelieve also would they both, believed they if but ; n ontbelieve not do and and oeeitn eatcacut oecoet ooeeu lentv and alternative homogeneous to close come accounts semantic existing Some srntestesmlrt rnil odrv secondary derive to principle similarity the strengthens 2008 Chemla Finally, alterna- of sets these to interpretation of rule similarity a applies 2008 Chemla like disjunction a Take Y h pae believes: speaker the , B A h iiaiypicpesy hybelieve they says principle similarity the : { ∨ > B A } > and , ∨ yrpaeetof replacement by and B X hssnec eeae w eeatst falter- of sets relevant two generates sentence This . { iff ⊥ A ∧ r uewa n uesrn connectives, superstrong and superweak are Y B hssrnteigi nyapidwhen applied only is strengthening This . 38 , ⊥ } A h> Chemla scale, linear a to addition In . ∧ , ∨ A ♦ ∨ ¬ ∧ , ihasaeae h resulting The scalemate. a with ∧ A B , . A ⊥i f hybelieve they iff ∧ A hc resconnectives orders which ∨ B otaitn h first the contradicting , B ontbelieve not do A ∨ A B ∨ f hybelieve they iff hsgvsus gives This . io Goldstein Simon ontbelieve not do ⊥ hc is which , { A A , B ∧ } B . . early access . . } is B ⊥ > and and B ♦ iff ∧ ¬ ♦ A A { B A ♦ ♦ ∨ . But this ¬ generates ∧ is absurd. with their A B A ♦ ∨ ⊥ ⊥i iff , iff she believes ∧ A and ; the second step , A B ∨ ♦ , . So, to summarize, , since ∨ . The similarity prin- B B } A h> or believing ♦ ♦ B B ∧ ♦ , ∧ A and A A ♦ . Now consider each disjunct A ♦ ) { ♦ B ∧ is absurd. Likewise, it implies the B A generates the alternative sets ( ⊥  ¬ ♦ grants permission for both items even . The similarity principle first implies , or believes neither. This primary impli- is also associated with three sets of al- } ⊥i via replacement and ordering by strength. is consistent with the primary implicature ∧ ¬ , B ) B . } 39 B ) ∧ A ♦ B ♦ , ⊥ ♦ B ,  , ∨ ∨ , since A ∧ , ¬ ) B iff A ♦ B ♦ A ( { = | A ( h> ♦ ∧ ∧ ) ♦ ♦ A A B ( ♦ ∧ ♦ { and believes A ( A  and ♦ , this implies belief in ¬ } via replacement and ordering by strength. Second, connective > } ¬⊥ { implies the speaker doesn’t believe ⊥ , we produce the secondary implicature that the speaker believes ♦ B , : ) ∨ ∧ B , or believes neither. When exclusivity is cancelled, strengthening can be A , since the belief that: B ∧ . For the secondary step this gives us that the speaker believes: B ♦ and ) A ♦ ( B  prima facie ♦ ∧ { Another difference between a homogeneous semantics for Free Choice and One significant empirical point of difference between Chemla 2008 and homo- Now consider Free Choice. A ( . The secondary step would strengthen this inference to believing: DUALFREECHOICE (35) a. You are not required to eat an apple and a banana. Chemla 2008 concerns the status ofduals Free Choice when we replace of disjunction is absent. ByChoice contrast, is in guaranteed. homogeneous This dynamic prediction semantics favors Wide homogeneous Free dynamic semantics, since exactly when this primary exclusivitynective implicature split is alternatives. cancelled. Similarity Now implies turn thatbelieves to the the speaker con- either believes applied to reach the result that the speaker believes geneous dynamic semantics concerns theassociated status with of three Wide sets Free of Choice. alternatives.the First, alternative the sets linear scale Second, connective split produces theciple alternative set the speaker doesn’t believe speaker either believes cature can be strengthened intobelieve the secondary implicature that theythat believe the speaker doesn’t believe ternatives. First, the linear scale and split produces the alternative set Since they believe individually. The similarity principle implied theB speaker believes step is blocked. For this step requires either believing Free choice and homogeneity applied to ¬ This is where the problem arises. Chemla 2008 suggests Wide Free Choice arises The former state would contradict the primary implicatures of when it suggests you can’t have both soup and salad. would contradict its truth conditions. Wide Free Choice is predicted to hold exactly when the wide scope exclusive reading You may have soup or you may have salad early access (37-a). from follow (37-b) or (38-b) of which determine could we 2018 al. et Ciardelli example, For controversial. is Choice Free Dual of status The adhmgnosdnmcsemantics, dynamic homogeneous and 2008 Chemla between between adjudicate To eatc aesc rdcin nec rmwr,cnucini rae classi- by treated contributed is sensitivity conjunction alternative framework, any By each So In Choice. cally. dynamic prediction. Free a homogeneous as such nor status make semantics same semantics alternative the homogeneous has neither Choice Free contrast, Dual predicts 2008 Chemla Iaeteck rteice-cream. the or cake the ate I (40) (40): LF the with (39) interpreted is (39) like disjunction simple a a of example, availability For the around revolves operator Choice exhaustification In Free covert exhaustification. of pragmatic theory of 2007 ’s version Fox syntactic particular, a through Choice Free explains B. exercise or A exercise skip will students Some a. above: (38) inference the of semantics weakening dynamic a validates a This with quantifiers. semantics dynamic for homogeneous extended B. we exercise §9.2, or In A exercise skip will students Some a. (37) Choice: Free of analogue an quanti- existential generate that general in predicts fiers theory 2008’s Chemla quantifiers. under disjunction Choice. Free ordinary some as with strong inferred as is not Choice is Free Bengali. effect Dual and the while regularity, Arabic that both found speak 2009 not Chemla might Similarly, Mary a. (36) invalid: is principle the of duality since the Choice, given Free equivalent Dual logically in implicit principle: is related principle closely this a of validity the against argue which , 2007 Fox in theory the to account 2008 ’s Chemla from turn we’ll Now concerns semantics homogeneity and 2008 Chemla between difference third A S o rntrqie oeta apple. an eat to required aren’t you So b. exh b. b. Arabic. speak not might she So, # b. ( C oesuet ih kpeecs n oesuet ih skip might students some and B. A exercise exercise skip might students Some exercise skip will students some B. and A exercise skip will students Some )( cake ∨ ice-cream exh ) hc ucin ongt e falternatives. of set a negate to functions which , 40 ♦ and  suggest 2018 al. et Ciardelli But .  ♦ ilb ofie odisjunction. to confined be will ¬ ♦ ( ¬ A ( A ∧ ∧ B B ) ) and | = ♦ ¬ io Goldstein Simon ¬  A ( A Precisely . ∧ B ) are . } ∧ A

early access ¬ ∨ ( A . .  , : ) ( ice-cream } } B ¬ ∨ , ♦ B ♦ just )( )( ♦ C C ( cake ∧ ( , and ) has the A exh A B ♦ exh ice-cream are false. It ice-cream ♦ )( ¬  , 0 )( . ∨ 0 , C ) ice-cream C C B ( B ( ∧ ¬ ∨ ♦ ∨ ice-cream cake A exh A exh ∧ )(  ( ∧ ¬ where negating every cake C ¬ ♦ ( A ice-cream C ♦ = ∧ cake | , , exh ) . There are two such max- ) . ) B , the Free Choice validating B } cake ∧ ∧ C { A A ( ∈ ( ♦ p  and ice-cream | ice-cream ¬ } ∨ ice-cream ) ∨ ∧ ¬ are innocently excludable: their nega- p ∧ ) ). While that inference is valid in homo- B is false, not both. If this is an available )( B cake B cake ♦ C , cake B ∨ 41 ( { ♦ ∧ cake . The result is that ♦ A ( ∧ = } ( A exh A or ♦ C ♦ { ∧ ¬ { ♦ is true, but that some alternatives in ¬ A )) ) = ♦ = 0 B | is innocently excludable given ice-cream C ∨ and B ∧ C ice-cream when combined with the truth of A ♦ B ( , the latter being constant functions to the left or right in- ∧ ∨ ♦ B R ♦ cake ♦ A ice-cream ice-cream )( { ♦ ∧ ¬ is false, because it is the unique alternative that is innocently . Where ∨ C ∨ cake ( ∨ A { and , and ♦ L A exh , cake cake and ♦ ( ∧ )( 0 ♦ C ( ice-cream ∧ exh says that has the same truth conditions as (41), and in particular guarantees cake simply requires that either One question for Fox 2007 concerns the status of Free Choice under negation (see This account of Free Choice makes a variety of empirical predictions that depart Validating Free Choice requires double exhaustification. Hold fixed a classical exh )) includes just three alternatives: B )) 0 for example, perhaps only applyingstronger exhaustification meaning, when or it only produces whenstrategy a it is avoids logically in extra ignorance principle implicatures. promising, Whileprinciple one this challenge does is not to make rule sure out such an too economy many logical forms. For example, Simons 2005 logical form for (42), then Double Prohibition is not(42) explained. You can’t have soup or salad. ¬ Second, the account differs fromof homogeneous Wide dynamic Free semantics Choice in ( thegeneous treatment dynamic semantics, the inference Fox is2007 predictedmakes to the same fail prediction in as Fox homogeneous2007. alternative (Here semantics). B tions guarantee from homogeneous alternative semanticsFirst, and this homogeneous account predicts dynamic Dual semantics. same Free status Choice as ( ordinary Free Choice. In particular, the logical form semantics for logical form is (41): (41) Crucially, each of member of that subset isimally consistent strong with subsets: their intersection is put. This produces the alternative set says excludable. An alternative in in case it is in the intersection of the set of maximal subsets of C Free choice and homogeneity is equivalent to To rule out this sort of logical form, Fox 2007 could appeal to an economy principle; Aloni 2018 for discussion). In principle, negating the logical form Alternatives are determined recursively from awhich set includes of alternative connectives to early access i hst emtbt obyadqarpyexhaustified quadruply and doubly both permit to thus is 2007 Fox for challenge A i h eaino reCoc li,adtesecond the and claim, Choice Free a of negation the is (45) of disjunct first The Where form. logical complex more a to appeals 2007 Fox reading, this explain To hspeupsto stiilol if only trivial is presupposition This aemyd both. do may Jane (44) C dance. or sing may Jane and (43) dance, may Jane sing, may Jane that compossible. allows are (43) options of both reading when one even example, occur For can effects Choice Free that observes ontdul xasiy u fdul xasicto sntpeeti oia form, logical in present not is sentences exhaustification Choice double if Free But negated exhaustify. that doubly predicts not 2007 do Fox in the account contrast, By based semantics. implicature dynamic homogeneous and semantics alternative geneous (47) is: (46) of sug- of presupposition This accounts the standard assumption, disjunct. on disjunction first For a semantically. the for projection, by accounted presupposition filtered be must is Choice Maria presupposition Free that this gests presuppose Japan; not does in (45) study that can is observation 2019’s Santorio & Romoli (46) Where Japan. in study sentence to a permitted is Mary that presupposes disjunct (45) between (45): interaction Consider the projection. explore presupposition 2019 and Santorio Choice from & Free Romoli comes 2019. semantics, Santorio dynamic & homogeneous Romoli or semantics alternative geneous under exhaustification negation. out of ruling scope while the and either, requiring without forms logical 00 = ovdb ihrhomo- either by solved 2007, Fox and 2008 Chemla both for problem final A { A , exh ¬¬ ¬ our in first the is Japan. she in or study Boston, go or can Tokyo who in family study go can’t Maria Either B ♦ C } ♦ ( ( h eeatlgclfr is: form logical relevant the , A C ihtepresupposition the with ( A 0 ∨ )( ∨ B exh ) B ∨ ) ( C → C )( ♦ A ♦ ♦ A ( exh ( C 00 )( ¬¬ A ♦ ) A ♦ ∨ 42 hstefloiglgclform: logical following the has (45) , ( A exh A ∨ ∨ ( B B C C ) 00 )( presupposes | = B ♦ ))) A hsi rdce yhomo- by predicted is This . ¬ A → C io Goldstein Simon C ♦ ie this Given . A represents early access , Se- Lecture Notes : University of Natural Language Proceedings of salt 4 conflict with Conditional . CSLI Publications. prima facie ). Then using homogeneous alternative ) 43 B ¬ ) is in > ) C A ( > ∨ Disjunction in alternative semantics ) B 3(10). 1–38. . https://doi.org/10.3765/sp.3.10 ( Exploring logical dynamics B 33 ∧ > 7218. 22–31. ) A C > = ( | 145. 303–323. https://doi.org/10.1007/s11229-005-6196-z. A 15(1). 65–94. https://doi.org/https://doi.org/10.1007/s11050-007- = ( | C Synthese > ) B ∨ 35–60. sion. mantics and Pragmatics 9010-2. Massachusetts Amherst dissertation. in Computer Science deontic modals in inqs. Semantics A ( van Benthem, Johan. 1996. Another place to look forsuggesting concerns Free about Choice various differs pragmatic from analysesBott scalar is implicature2014) a with and growing respect acquisition body to (Tieu of et processing literature al. time2016). (Chemla & Asher, Nicholas & Daniel Bonevac. 2005. Free choice permission isBarker, strong Chris. permis- 2010. Free choice permission as resource-sensitive reasoning. Beaver, David. 1994. When variables don’t vary enough. In Aloni, Maria. 2007. Free choice, modals, and imperatives. Aloni, Maria. 2018. Free choiceAlonso-Ovalle, disjunction Luis. in state-based 2006. semantics. Aher, Martin. 2012. Free choice in deontic inquisitive semantics (dis). Aher, Martin & Jeroen Groenendijk. 2013. Searching for directions: Epistemic and dynamics) provides a powerful tool withthat which within to classical combat logic various various results constellationsincompatible. showing of plausible principles are jointly References bination of principles thatincompatibility appear jointly results, inconsistent. showing They that develop Simplification a battery of of DisjunctiveExcluded Antecedents Middle ( semantics, they validate both principles, byPutting giving these up papers the transitivity together, of the entailment. synthesis of homogeneity and alternatives (or Building on Santorio 2017, Carianigeneous & Goldstein alternative2018 semanticsdevelop for a version conditionals of in homo- order to validate another com- Free choice and homogeneity a doubly exhaustified logical formthe cannot second provide disjunct the of local(45). context for interpreting 11 Conclusion ( 33 early access hma maul 08 iiaiy oad nfidacuto clrimplica- scalar of account unified a Towards Similarity: 2008. 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