On the presuppositional strength of interrogative clauses


A central question in the study of presuppositions is how a presupposition trigger contributes to the meaning of a complex expression containing it. Two competing answers are found in the literature on quantificational expressions. According to the first, a quantificational expression presupposes that every member of its domain satisfies the presuppositions triggered in its scope, and according to the second, a quantificational expression presupposes that at least one member of its domain satisfies the presuppositions triggered in its scope. The former view implies that an interrogative clause, a kind of quantificational expression, presupposes all of its possible answers’ presuppositions, whereas the latter view implies that an interrogative clause presupposes that the presuppositions of at least one of is possible answers are satisfied. This paper contributes to the debate by showing that ‘alternative’ interrogatives, formed with or, project presuppositions in the same, distinctive manner that other disjunctive constructions do: generally, universally. A theory that treats disjunctive words as restricted variables, bindable by various quantificational operators, is extended to account for the presuppositions of ‘alternative’ interrogatives, disjoined declaratives, and disjoined conditional antecedents in a uniform manner. The paper then explores some ways to reconcile the proposal with two special cases where interrogatives have been claimed to have weaker presuppositions: (1) constituent interrogatives in presupposition-weakening contexts, and (2) polar interrogatives containing bias-inducing scalar particles like even.

This is a preview of subscription content, access via your institution.


  1. 1.

    Following the practice in Guerzoni and Sharvit (2014), we use ‘alternative’ to refer to the sort of interrogative in (4a) (with the meaning implied by (4b)). Alternative—without ‘’—has its standard use (as in an alternative solution).

  2. 2.

    With Karttunen and Peters (1979), we treat or as symmetrical (for example, we take Either Jack’s children are away or he has no children to be semantically equivalent to (6)). There is no consensus in the literature—or among speakers—regarding this. See Sharvit (2020) for an argument against asymmetric treatments of disjunction.

  3. 3.

    Type t is the type of truth values (elements of {1, 0}) and type e is the type of individuals. For any types σ and ρ: (σ, ρ)—sometimes abbreviated to σρ—is the type of functions from entities of type σ to entities of type ρ, and (s, ρ)—sometimes abbreviated to sρ—is the type of functions from possible worlds to entities of type ρ.

  4. 4.

    We assume that meaning composition proceeds generally along the lines of Heim and Kratzer (1998). See Appendix 1 for more detail.

  5. 5.

    The mismatch could, in principle, be overcome by type-shifting. The derived reading (for all (w’, Q) such that w’ ∈ Accw and Mary is swimming or dancing in w’: Sue is Qing in w’), which is false unless there is no accessible world where Mary is swimming or dancing, would be ruled out by general pragmatic principles.

  6. 6.

    We limit our discussion to canonical pronunciations. Variations on the canonical pronunciation of ‘alternative’ interrogatives are attested, but all pronunciations share a falling final boundary tone (see Bartels 1999; Biezma and Rawlins 2012, 2015; Pruitt and Roelofsen 2013).

  7. 7.

    It follows from this analysis, and from the assumption that or can take more than two disjuncts (see (13’), the generalized counterpart of (13), in Appendix 1), that Is Mary singing swimming or dancing, which can be pronounced with more than one ‘alternative’ prosodic pattern, supports more than one ‘alternative’ reading.

    (i) a. Is Mary singingH* (or) swimmingH* or dancingH*L-L%  
         {Mary is singing, Mary is swimming, Mary is dancing}
      b. Is Mary singing or swimmingH* or dancingH*L-L%  
        {Mary is singing or swimming, Mary is dancing}  
      c. Is Mary singingH* or swimming or dancingH*L-L%  
         {Mary is singing, Mary is swimming or dancing}

    In (i.a) the three disjuncts form an “open” disjunction. In (i.b), the first two disjuncts are -“closed”; the derived disjunct and the third form an “open” disjunction. In (i.c), the last two disjuncts are -“closed”; the derived disjunct and the first form an “open” disjunction. Similarly, “If Mary is singing or dancing or swimming then Sue is” can be pronounced with more than one prosodic pattern, supporting more than one sloppy VP reading.

  8. 8.

    See Guerzoni and Sharvit (2014) for an argument for the view that polar interrogatives are underlyingly ‘alternative’ interrogatives. Guerzoni and Sharvit’s theory of ‘alternative’ interrogatives is inspired by Larson (1985) and Han and Romero (2004); it assumes a designated ‘alternative’ question-forming word (whether) and takes questions to be functions from worlds to sets of true propositions (as in Karttunen 1977). In such a framework, the proposition-level disjunctive word and existential closure are as in (i)–(ii) (cf. Rooth and Partee 1982; see Appendix 2).

    (i) ork〛w,g(p1)(p2) = 1 iff g(k)(w) = 1 ∧ (g(k) = p1 ∨ g(k) = p2)
    (ii) 〛 (X) = 1 iff {Z| X(Z) = 1} ≠ ∅
  9. 9.

    The Limit Assumption (Lewis 1973) guarantees that there is such a set. The context parameter that fixes the similarity relation is implicit. For the generalized counterpart of (31), which subsumes a presuppositional version of if2, see (31’) in Appendix 1.

  10. 10.

    For the presuppositional property-level disjunctive connective, as well as a generalized counterpart, see Appendix 2.

  11. 11.

    If the LF of the polar Did John eat the cake or the candyL*H-H% is (i) (see Sect. 2.1), the fact that it presupposes that there is cake and candy is also accounted for by the presuppositions of or and .



  12. 12.

    Note that (Either) 3 equals 3 or Jack’s children are away is correctly predicted to be infelicitous unless Jack has children because SIM(w)(∅) is undefined. Yet the conditional presupposition in (32) does not suffice to account for (i)–(ii). Explaining (i)–(ii), along with their corresponding ‘alternative’ interrogatives, conditionals, and related facts observed in Hurford (1974) requires reference to pragmatic constraints.

    (i) #Either Jack has children or his children are away.
    (ii) Jack has no daughters. Either he (also) has no sons, or his sons/#children are away.
  13. 13.

    According to Biezma and Rawlins (2012), the disjuncts in an ‘alternative’ interrogative are determined by the alternatives set up by the Question-under-Discussion (QUD). In the Ans-based system, the QUD requirement can be made part of (44a).

  14. 14.

    Greenberg (2018) argues that even’s scalar presupposition involves comparison along a contextually-supplied dimension (not necessarily likelihood). Most of our examples involve alternatives that are on contextual rather than logical scales, so the predictions could be reproduced with her lexical entry as well.

  15. 15.

    In (73), cake need not be pronounced with more prominence than in the even-less Did John eat the cakeL*H-H%. We take this to imply that in (73) the canonical polar pronunciation is sufficient to signal the prominence of the focus-associate of even.

  16. 16.

    A similar explanation would apply to the negative bias of polar interrogatives with minimizing NPIs such as lift a finger (as in Did John (even) lift a finger?; see (2) in Sect. 1).

  17. 17.

    In Krifka (2001), the speech act operator does not yield a truth value. In Sauerland and Yatsushiro (2017), the speech act operator is decomposed into several components. Ask is compatible with all other aspects of these proposals.

  18. 18.

    This includes treating even as an intervener in the sense of Beck (1996, 2006) and Beck and Kim (2006), or as a negative polarity item (NPI), or part of an NPI, along the lines of Lee and Horn (1994) and Crnič (2014), reducing the unacceptability of (85b) to the exclusion (observed in Higginbotham 1993) of NPIs from ‘alternative’ interrogatives. Importantly, declarative clauses such as those in (89) are neither intervention environments nor NPI environments.

  19. 19.

    Even may associate with an item outside the surface disjunction, given the right context.

    A: John took syntax and phonology. He is a very good student, but like everyone in his class he failed SOMEthing.
    B: Really? Did even HE fail syntaxH* or phonologyH*L-L%


  1. Abenina-Adar, Maayan and Yael Sharvit. 2018. Domain uniformity in questions. Presented at Semantics and Linguistic Theory 28, MIT, May 18, 2018.

  2. Abrusán, Márta. 2014. Weak island semantics. Oxford: Oxford University Press.

    Google Scholar 

  3. Abusch, Dorit. 2002. Lexical alternatives as a source of pragmatic presuppositions. In Proceedings of Semantics and Linguistic Theory 12, 1–19. Ithaca, NY: CLC Publications.

  4. Abusch, Dorit. 2010. Presupposition triggering from alternatives. Journal of Semantics 27: 37–80.

    Google Scholar 

  5. Bartels, Christine. 1999. The intonation of English statements and questions. New York: Garland.

    Google Scholar 

  6. Beaver, David. 2001. Presupposition and assertion in dynamic semantics. Stanford: CSLI Publications.

    Google Scholar 

  7. Beck, Sigrid. 1996. Wh-constructions and transparent logical form. Ph.D. dissertation, University of Tübingen.

  8. Beck, Sigrid. 2006. Intervention effects follow from focus interpretation. Natural Language Semantics 14: 1–56.

    Google Scholar 

  9. Beck, Sigrid, and Shin-Sook Kim. 2006. Intervention effects in alternative questions. The Journal of Comparative Germanic Linguistics 9(3): 165–208.

    Google Scholar 

  10. Biezma, Maria, and Kyle Rawlins. 2012. Responding to alternative and polar questions. Linguistics and Philosophy 35: 361–406.

    Google Scholar 

  11. Biezma, Maria, and Kyle Rawlins. 2015. Alternative questions. Language and Linguistics Compass 9: 450–468.

    Google Scholar 

  12. Borkin, Ann. 1971. Polarity items in questions. In Proceedings of CLS 7, 53-62. Chicago: The Chicago Linguistic Society.

  13. Crnič, Luka. 2014. Non-monotonicity in NPI licensing. Natural Language Semantics 22: 169–217.

    Google Scholar 

  14. Dayal, Veneeta. 1996. Locality in WH-quantification. Dordrecht: Kluwer.

    Google Scholar 

  15. Fodor, Janet Dean. 1970. The linguistic description of opaque contexts. PhD dissertation, MIT.

  16. Geurts, Bart. 1996. Local satisfaction guaranteed: A presupposition theory and its problems. Linguistics and Philosophy 19: 259–294.

    Google Scholar 

  17. Geurts, Bart. 1999. Presuppositions and pronouns. Leiden: Brill.

    Google Scholar 

  18. Greenberg, Yael. 2018. A revised, gradability-based semantics of even. Natural Language Semantics 26(1): 51–83.

    Google Scholar 

  19. Guerzoni, Elena. 2003. Why even ask? On the pragmatics of questions and the semantics of answers. PhD dissertation, MIT.

  20. Guerzoni, Elena. 2004. Even-NPIs in yes/no questions. Natural Language Semantics 12: 319–343.

    Google Scholar 

  21. Guerzoni, Elena, and Yael Sharvit. 2014. ‘Whether or not anything’ but not ‘whether anything or not’. In The art and craft of semantics: A festschrift for Irene Heim, ed. L. Crnič and U. Sauerland, 1, 199–224. Cambridge, MA: MIT Working Papers in Linguistics.

  22. Halliday, M.A.K. 1967. Notes on transitivity and theme in English. Part 1. Journal of Linguistics 3: 37–81.

    Google Scholar 

  23. Hamblin, C.L. 1973. Questions in Montague English. Foundations of Language 10: 41–53.

    Google Scholar 

  24. Han, Chung-hye, and Maribel Romero. 2004. The syntax of whether/Q…or questions: Ellipsis combined with movement. Natural Language & Linguistic Theory 22: 527–564.

    Google Scholar 

  25. Hausser, Roland. 1976. Presuppositions in Montague grammar. Theoretical Linguistics 3: 245–280.

    Google Scholar 

  26. Heim, Irene. 1982. The semantics of definite and indefinite noun phrases. PhD dissertation, University of Massachusetts at Amherst.

  27. Heim, Irene. 1983. On the projection problem for presuppositions. In Proceedings of WCCFL 2, ed. D. Flickinger, M. Barlow and M. Wescoat, 114–125. Stanford: Stanford University Press.

  28. Heim, Irene. 1984. A note on polarity sensitivity and downward entailingness. In Proceedings of NELS 14, ed. Charles Jones and Peter Sells, 98–107. Amherst: GLSA Publications.

  29. Heim, Irene. 1992. Presupposition projection and the semantics of attitude verbs. Journal of Semantics 9: 183–221.

    Google Scholar 

  30. Heim, Irene. 1999. Notes on superlatives. Manuscript. MIT.

  31. Heim, Irene, and Angelika Kratzer. 1998. Semantics in generative grammar. Malden: Blackwell.

    Google Scholar 

  32. Higgins, Francis Roger. 1973. The pseudo-cleft construction in English. PhD dissertation, MIT.

  33. Higginbotham, James. 1993. Interrogatives. In The view from building 20: Essays in linguistics in honor of Sylvain Bromberger, ed. K. Hale and S.J. Keyser, 195–227. Cambridge, MA: MIT Press.

    Google Scholar 

  34. Hirsch, Aron. 2017. Disjoining questions. Manuscript, McGill University.

    Google Scholar 

  35. Hoeks, Morwenna. 2018. Coordinating questions. MA thesis, University of Amsterdam.

  36. Hoeks, Morwenna, and Floris Roelofsen. 2019. Disjoining questions: The scope puzzle. In Proceedings of SALT 29, ed. K. Blake et al., 562–581. Washington, D.C.: LSA.

  37. Horn, Laurence. 1989. A natural history of negation. Chicago: The University of Chicago Press.

    Google Scholar 

  38. Horn, Laurence. 1996. Exclusive-company only and the dynamics of vertical inference. Journal of Semantics 13: 1–40.

    Google Scholar 

  39. Howard, Edwin. 2014. Superlative degree clauses: Evidence from NPI licensing. MSc thesis, MIT.

  40. Hurford, James. 1974. Exclusive or inclusive disjunction. Foundations of Language 11: 409–411.

    Google Scholar 

  41. Iatridou, Sabine, and Sergei Tatevosov. 2016. Our even. Linguistics and Philosophy 39: 295–331.

    Google Scholar 

  42. Kamp, Hans. 1981. A theory of truth and semantic representation. In Formal methods in the study of language, ed. J. Groenendijk, T. Janssen, and M. Stokhof, 277–322. Amsterdam: Mathematical Centre.

    Google Scholar 

  43. Karttunen, Lauri. 1973. Presuppositions of compound sentences. Linguistic Inquiry 4: 169–193.

    Google Scholar 

  44. Karttunen, Lauri. 1974. Presuppositions and linguistic context. Theoretical Linguistics 1: 181–193.

    Google Scholar 

  45. Karttunen, Lauri. 1977. Syntax and semantics of questions. Linguistics and Philosophy 1(1): 3–44.

    Google Scholar 

  46. Karttunen, Lauri, and Stanley Peters. 1976. What indirect questions conventionally implicate. In Papers from the 12th Regional Meeting. Chicago: The Chicago Linguistic Society.

  47. Karttunen, Lauri, and Stanley Peters. 1979. Conventional implicature. In Syntax and semantics 11, ed. Oh Choon-Kyu and David Dinneen, 1–56. New York: Academic Press.

    Google Scholar 

  48. Kratzer, Angelika and Junko Shimoyama. 2002. Indeterminate pronouns: the view from Japanese. In The proceedings of the third Tokyo conference on psycholinguistics (TCP 2002), ed. Y. Otsu, 1–25. Tokyo: Hituzi Syobo.

  49. Krifka, Manfred. 2001. Quantifying into question acts. Natural Language Semantics 9(1): 1–40.

    Google Scholar 

  50. Ladusaw, William. 1979. Negative polarity as inherent scope. PhD dissertation, University of Texas at Austin.

  51. Lahiri, Utpal. 1998. Focus and negative polarity in Hindi. Natural Language Semantics 6: 57–123.

    Google Scholar 

  52. Larson, Richard. 1985. On the syntax of disjunction scope. Natural Language & Linguistic Theory 3(2): 217–264.

    Google Scholar 

  53. Lee, Young-Suk, and Laurence Horn. 1994. Any as indefinite plus even. Manuscript, Yale University.

  54. Lewis, David. 1973. Counterfactuals. Cambridge, MA: Harvard University Press.

    Google Scholar 

  55. Pruitt, Kathryn, and Floris Roelofsen. 2013. The interpretation of prosody in disjunctive questions. Linguistic Inquiry 44(4): 632–650.

    Google Scholar 

  56. Rooth, Mats. 1992. A theory of focus interpretation. Natural Language Semantics 1(1): 75–116.

    Google Scholar 

  57. Rooth, Mats, and Barbara Partee. 1982. Conjunction, type ambiguity and wide scope or. In Proceedings of WCCFL 1, ed. D. P. Flickinger, M. Macken and N. Wiegand, 353–362. Stanford, CA: Stanford Linguistics Association.

  58. Sauerland, Uli, and Kazuko Yatushiro. 2017. Remind-me presuppositions and speech-act decomposition: evidence from particles in questions. Linguistic Inquiry 48(4): 651–678.

    Google Scholar 

  59. Schlenker, Philippe. 2008. Presupposition projection: Explanatory strategies. Theoretical Linguistics 34(3): 287–316.

    Google Scholar 

  60. Schwarz, Bernhard, and Alexandra Simonenko. 2017. Decomposing universal projection in questions. In Proceedings of Sinn und Bedeutung 22, ed. U. Sauerland and S. Solt. Berlin: ZAS.

  61. Sharvit, Yael. 2020. Disjunction and (A)symmetry. Presented at Sinn und Bedeutung 25, 2020.

  62. Soames, Scott. 1986. Incomplete definite descriptions. Notre Dame Journal of Formal Logic 27(3): 349–375.

    Google Scholar 

  63. Stalnaker, Robert. 1975. Indicative conditionals. Philosophia 5: 269–286.

    Google Scholar 

  64. Szabolcsi, Anna. 1986. Comparative superlatives. In MIT working papers in linguistics 8, ed. N. Fukui et al., 245–265. Cambridge, MA: MIT Press.

  65. Szabolcsi, Anna. 1995. Can questions be directly disjoined? In Proceedings of CLS 51. Chicago: The Chicago Linguistic Society.

  66. von Fintel, Kai. 1994. Restrictions on quantifier domains. PhD dissertation, University of Massachusetts at Amherst.

  67. Westerståhl, Dag. 1985. Determiners and context sets. In Generalized quantifiers in natural language, ed. A. ter Meulen and J. van Benthem, 45–71. Dordrecht: Foris.

    Google Scholar 

  68. Wilkinson, Karina. 1996. The scope of even. Natural Language Semantics 4(3): 193–215.

    Google Scholar 

  69. Zehr, Jeremy, Aron Hirsch, Hezekiah Akiva Bacovcin, and Florian Schwarz. 2017. Priming local accommodation of hard triggers in disjunction. In Proceedings of NELS 47, ed. A. Lamont and K. Tetzloff, 285–298. Amherst, MA: GLSA.

Download references


We thank Christine Bartels and Brice Roberts for helping us make sense of some key facts concerning intonation. For their questions, critical comments, suggestions and native speaker judgments, we thank the audiences at SALT 28 (2018, MIT), the UCLA Syntax-Semantics Seminar (2018), the Workshop on Asymmetries in Language (2019, ZAS Berlin), and the Cornell Linguistics Colloquium (2019), as well as the reviewers and editors of Natural Language Semantics. All errors are ours.

Author information



Corresponding author

Correspondence to Yael Sharvit.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Appendix 1: Some rules and conventions

Generally following Heim and Kratzer (1998), we assume that for any possible world w and variable assignment g, (i)–(v), (10’), (13’), and (31’) hold.

(i) If γ is a branching node whose daughters are α and β and 〚α〛w,g ∈ Dom(〚β〛w,g), then:    
  〚γ〛w,g = 〚β〛w,g(〚α〛w,g)    
(ii) If γ = [ β], then: 〚γ〛w,g = [λw’: β ∈ Dom(〚〛w’,g). 〚β〛w’,g]    
  If γ = [ β], then: 〚γ〛w,g = 〚β〛w,g(w)    
(iii) If γ = [k β] and k is a numerical index, then:    
  〚γ〛w,g = [λx: β ∈ Dom(〚〛w,g[k→x]). 〚β〛w,g[k→x]],    
   where Dom(g[k→x]) = (Dom(g) ∪ {k}), g[k→x](k) = x, and for all m ∈ Dom(g[k→x]) such that m ≠ k: g[k→x](m) = g(m)   
(iv) If γ is a pronoun or a trace and k is a numerical index, then:    
  〚γkw,g is defined only if g(k) is defined. If defined, 〚γkw,g = g(k).    
(v) a.   〚γ〛g is defined iff for all w’ and w”, 〚γ〛w’,g and 〚γ〛w”,g are defined and [[γ]]w’,g = 〚γ〛w”,g;  
    if 〚γ〛g is defined, then for all w’: 〚γ〛g = 〚γ〛w’,g.  
  b.   〚γ〛w is defined iff for all g’ and g”, 〚γ〛w,g’ and 〚γ〛w,g” are defined and 〚γ〛w,g’ = 〚γ〛w,g”;  
    if 〚γ〛w is defined, then for all g’: 〚γ〛w = 〚γ〛w,g’.  
  c.   〚γ〛 is defined iff for all w’, w”, g’ and g”, 〚γ〛w’,g’ and 〚γ〛w”,g” are defined and  
    〚γ〛w’,g’ = 〚γ〛w”,g”;  
    if 〚γ〛 is defined, then for all w’ and g’: 〚γ〛 = 〚γ〛w’,g’.  
(10’) For any, n ≥ 2, sequence S, and any P1, P2, …, and Pn such that P1(w)(S) is of type t and P2, …, and Pn are of the same type as P1:    
 orw(P1)…(Pn)(S) = 1 iff {Q| Q(w)(S) = 1 ∧ (Q = P1 ∨ … ∨ Q = Pn)} ≠ ∅    
     (cf. (10), Sect. 2)
(13’) For any k ∈ Dom(g), any n ≥ 2, and any P1, P2, …, Pn of the same type:    
 orkg(P1)…(Pn) = 1 iff g(k) = P1 ∨ … ∨ g(k) = Pn    
     (cf. (13), Sect. 2)
(31’) For any, \( {\fancyscript{Q}} \) and \( {\fancyscript{P}} \), 〚if〛w(\( {\fancyscript{Q}} \))(\( {\fancyscript{P}} \)) ∈ {1, 0} only if:    
   (i) {S| S is a sequence and \( {\fancyscript{Q}} \)(w)(S) ∈ {1, 0}} ≠ ∅, and  
   (ii) for all S such that S is a sequence and \( {\fancyscript{Q}} \)(w)(S) ∈ {1, 0}:  
    SIM(w)({w’| \( {\fancyscript{Q}} \)(w’)(S) = 1}) ⊆ {w’| \( {\fancyscript{P}} \)(w’)(S) ∈ {1, 0}}.  
  If ifw(\( {\fancyscript{Q}} \))(\( {\fancyscript{P}} \)) ∈ {1, 0}, 〚ifw(\( {\fancyscript{Q}} \))(\( {\fancyscript{P}} \)) = 1 iff:   
   for all S such that S is a sequence and \( {\fancyscript{Q}} \)(w)(S) ∈ {1, 0}:   
   SIM(w)({w’| \( {\fancyscript{Q}} \)(w’)(S) = 1}) ⊆ {w’| \( {\fancyscript{P}} \)(w’)(S) = 1}.   
     (cf. (31), Sect. 2)

We use the following conventions:

(I) [λx: A. B] is shorthand for “the smallest function f such that f maps every x such that A to B” or “the smallest function f such that f maps every x such that A to 1 if B and to 0 otherwise” (whichever makes sense).
(II) For any Z whose type ends in t, m ≥ 0, and m-long sequence S:
  if m = 0: Z(S) ≡ Z;
  if m > 0 and S = (X1, …, Xm): Z(S) ≡ Z(X1)…(Xm).

Appendix 2: Presuppositions of disjunctive conditionals

Disjunctive conditionals, including those with ellipsis in the consequent, are subject to the K–P effect. Thus, in utterance contexts with more or less “normal” laws, both readings of (112a) intuitively presuppose that Mary has a son and a daughter and Sue has a son and a daughter, neither reading of (112b) presupposes that either woman has children, and the sloppy reading of (112b) presupposes that if Mary is a parent, Sue is a parent (we set aside the presuppositions of strict pronoun readings, where the consequent implies that Sue stands in some relation to Mary’s children).

(112) a. If Mary avoids her son or her daughter, then Sue does.  
    Strict VP reading:
    ‘Mary avoids Mary’s son or daughter’ → ‘Sue avoids Sue’s son or daughter’
    Sloppy VP reading:
    (‘Mary avoids Mary’s son’ → ‘Sue avoids Sue’s son’) ∧
    (‘Mary avoids Mary’s daughter’ → ‘Sue avoids Sue’s daughter’)
  b. If Mary is (either) childless or abusive with her children, then Sue is.  
    Strict VP reading:
    ‘Mary is childless or abusive with Mary’s children’ → 
    ‘Sue is childless or abusive with Sue’s children’
    Sloppy VP reading:
    (‘Mary is childless’ → ‘Sue is childless’) ∧
    (‘Mary is abusive with Mary’s children’ → ‘Sue is abusive with Sue’s children’)

This is predicted if the LF of the strict VP reading contains presuppositional if1 (see (31) in Sect. 2), proposition-level presuppositional ork and presuppositional (see (32) and (38)), and the LF of the sloppy VP reading contains presuppositional if2 in (113) below and property-level presuppositional ORk in (114) below. The predictions are as in (115)–(116).

(113) if2 w(\( {\fancyscript{Q}} \))(\( {\fancyscript{P}} \)) ∈ {1, 0} only if:  
   (i) {P| \( {\fancyscript{Q}} \)(w)(P) ∈ {1, 0}} ≠ ∅, and
   (ii) for all P such that \( {\fancyscript{Q}} \)(w)(P) ∈ {1, 0}:
    SIM(w)({w’| \( {\fancyscript{Q}} \)(w’)(P) = 1}) ⊆ {w’| \( {\fancyscript{P}} \) (w’)(P) ∈ {1, 0}}.
  If 〚if 2w(\( {\fancyscript{Q}} \))(\( {\fancyscript{P}} \)) ∈ {1, 0}, 〚if 2w(\( {\fancyscript{Q}} \))(\( {\fancyscript{P}} \)) = 1 iff:   
   for all P such that \( {\fancyscript{Q}} \)(w)(P) ∈ {1, 0}:  
   SIM(w)({w’| \( {\fancyscript{Q}} \)(w’)(P) = 1}) ⊆ {w’| \( {\fancyscript{P}} \) (w’)(P) = 1}.  
(114) For any x of type e, and P1 and P2 of type (s, et), 〚ORkw,g(P1)(P2)(x) ∈ {1, 0} iff:  
  a. g(k)(w)(x) ∈ {1, 0},
  b. g(k) = P1 ∨ g(k) = P2, and
  c. (P1(w)(x) ∈ {1, 0} ∨ SIM(w)({w’| P2(w’)(x) = 0}) ⊆ {w’| P1(w’)(x) ∈ {1, 0}}) ∧
   (P2(w)(x) ∈ {1, 0} ∨ SIM(w)({w’| P1(w’)(x) = 0}) ⊆ {w’| P2(w’)(x) ∈ {1, 0}})
  If 〚ORkw,g(P1)(P2)(x) ∈ {1, 0}, 〚ORkw,g(P1)(P2)(x) = 1 iff g(k)(w)(x) = 1.  

This analysis comes at a cost. While if1 and if2 can be merged into one item (see their generalized counterpart (31’) in Appendix 1), proposition-level presuppositional ork and property-level presuppositional ORk cannot be merged into one item: both ork and ORk have a (generalizable) conditional presupposition, but ork asserts ‘g(k) = p1 ∨ g(k) = p2’, whereas ORk presupposes ‘g(k) = P1 ∨ g(k) = P2’ and asserts ‘g(k)(w)(x) = 1’. It is worth noting that the generalized presuppositional disjunctive word in (117), with the necessary adjustment of existential closure in (118), covers both property-level and proposition-level disjunction and makes the same predictions as ork and ORk (see Sharvit 2020, for a similar proposal).

(117) For any n ≥ 2, P1, P2, …, and Pn, sequence S, and numerical index k:   
 orkw,g(P1)…(Pn)(S) ∈ {1, 0} iff:   
  a. g(k)(w)(S) ∈ {1, 0},  
  b. g(k) = P1 ∨ … ∨ g(k) = Pn, and  
  c. for all P ∈ {P1, …, Pn}: P(w)(S) ∈ {1, 0} ∨  
    {D| D ⊆ {P1, …, Pn} ∧ {w’| D ⊆ {Q| Q(w’)(S) ∈ {1, 0}} ⊈ {w’| P(w’)(S) ∈ {1, 0}} ∧
    SIM(w)({w’| D ⊆ {Q| Q(w’)(S) = 0}}) ⊆ {w’| P(w’)(S) ∈ {1, 0}}} ≠ ∅.
  If 〚or kw,g(P1)…(Pn)(S) ∈ {1, 0}, 〚orkw,g(P1)…(Pn)(S) = 1 iff g(k)(w)(S) = 1.   
(118) 〛(X) ∈ {1, 0} iff Dom(X) ≠ ∅.
  If 〚〛(X) ∈ {1, 0}, 〚〛(X) = 1 iff {Z ∈ Dom(X)| X(Z) = 1} ≠ ∅.

Adopting (117) requires adopting the view that the extension of an interrogative is a set of true possible answers as in Karttunen (1977), and not a set of merely possible answers as in Hamblin (1973). Presupposition projection in polar and ‘alternative’ interrogatives are still accounted for under (117), as questions are still required to have satisfied presuppositions in order to be issued. However, the semantic presuppositions of wh-interrogatives and their pragmatics (see Sect. 3.3) would need to be reworked if Karttunen (1977) is adopted.

Appendix 3: Functional restrictors

It has been proposed (see, for example, von Fintel 1994) that quantificational restrictors can be “functional”. For example, in (119), where the subject of the prejacent of even varies across possible answers, the restrictor of even is the complex f2-pro4. Who binds its trace, t4, and the co-indexed pro4, which is the argument of the free pronoun f2, whose value is determined by the context.

(119) a. Who even ate the CAKE?   
  b. who-C1g(λxλp. p = 〚[even-f2-pro4 [[t4 ate the cakeF] ~ f2-pro4]]〛g[4→x])   
   Which x ∈ C1 is such that ‘x ate the cake’ is least likely in f2(x)?   
  c. C1 ⊇ {j, b, k}   
   f2(j) {[John ate the cake]〛,[John ate the candy]〛, 〚[John ate the chips]〛, …}
   f2(b) {[Bill ate the cake]〛,[Bill ate the candy]〛, 〚[Bill ate the chips]〛, …}
   f2(k) {〚[Kat ate the cake]〛,[Kat ate the candy]〛, 〚[Kat ate the chips]〛, …}

This raises the possibility that the ‘alternative’ interrogative in (120a) below (= (85b) in Sect. 4) could, in principle, have a satisfiable presupposition: f3 in (120b) has a different silent pronominal argument in each of the disjuncts: the first is anaphoric to the cake and the second to the candy.


We claim that (120b) is still excluded by DU, since the functional restrictors f3-pro1 and f3-pro2 are not identical.

Yet DU alone does not suffice to rule out (120a) once we acknowledge the availability of functional restrictors, because functional restrictors are not generally banned from constructions that involve ellipsis. For example, thanks to the assumption that Q(uantifier) R(aising) is available, the elided quantifier in (121) is licit.


Nevertheless, the QR option is not available for (120a), for independent reasons. In other words, an independent constraint bans (122), where the focus-associates of even are traces bound from above even by the cake and the candy. That constraint does not ban either (119b) or (121b), which do not contain illicit traces.

(122) *[5 [ [the cake 1 [even-f3-pro1 [[John ate [t1]F] ~ f3-pro1]]] or5
   [the candy 1 [even-f3-pro1 [[John ate [t1]F] ~ f3-pro1]]]]]

That such a constraint is indeed at play is corroborated by the fact that a QR-ed phrase cannot bind a trace that is the focus-associate of the focus-sensitive superlative operator est, despite the fact that a wh-phrase can bind such a trace. While (123a,b) are both acceptable (Szabolcsi 1986; Heim 1999), the reading of (124b) where every student binds a trace that is the focus-associate of est is not available.

(123) a. Who gave the most books to Joe? (cf. ANN gave the most books to Joe)
  who-C1g(λzλp. p = 
[est-f3-pro1 [[2 [t1]F gave d2-many books to Joe] ~ f3-pro1]]〛g[1→z])
   Which z ∈ C1 is such that ∀x ≠ z[x ∈ f3(z)]: z gave Joe more books than x gave Joe?
  b. Who did Ann give the most books to? (cf. Ann gave the most books to JOE)
  who-C1g(λzλp. p = 
[est-f3-pro1 [[2 Ann gave d2-many books to [t1]F] ~ f3-pro1]]〛g[1→z])
   Which z ∈ C1 is such that ∀x ≠ z[x ∈ f3(z)]: Ann gave z more books than Ann gave x?
(124) a. JANE gave the most books to every student.  
   every student [1 est-f3-pro1 [[2 JaneF gave d2-many books to t1] ~ f3-pro1]]  
   For every student x and every z in f3(x) such that z ≠ Jane:  
    Jane gave x more books than z gave x.
  b. Jane gave the most books to EVERY STUDENT/every student.  
   *every student [1 est-f3-pro1 [[2 Jane gave d2-many books to [t1]F] ~ f3-pro1]]  
   For every student x and every z in f3(x) such that z ≠ x:  
    Jane gave x more books than Jane gave z.
(125) Where C is a set of degree-properties, 〚estw(C)(P) ∈ {1, 0} only if P ∈ C and {d| P(w)(d) = 1} ≠ ∅.
  If 〚estw(C)(P) ∈ {1, 0}, 〚estw(C)(P) = 1 iff {d| {Q ∈ C| Q(w)(d) = 1} = {P}} ≠ ∅.
  (Howard 2014; cf. Heim 1999)

Finally, it is worth noting that (100a) in Sect. 4If Mary is even dancing or swimming, then Sue is—has a narrow-scope-even strict VP reading. Its DU-compliant LF in (126) below contains functional restrictors but does not contain illicit focused traces.


Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abenina-Adar, M., Sharvit, Y. On the presuppositional strength of interrogative clauses. Nat Lang Semantics 29, 47–90 (2021). https://doi.org/10.1007/s11050-020-09169-5

Download citation


  • Disjunction
  • Presuppositions
  • Questions
  • Alternative questions