Structured propositions and the logical form of predication


Jeffrey King, Scott Soames, and others have recently challenged the familiar identification of a Russellian proposition, such as the proposition that Brutus stabbed Caesar, with an ordered sequence constructed out of objects, properties, and relations. There is, as they point out, a surplus of candidate sequences available that are each equally serviceable. If so, any choice among these candidates will be arbitrary. In this paper, I show that, unless a controversial assumption is made regarding the nature of nonsymmetrical relations, none of the proffered candidate sequences are in fact adequate to the play the role. Moreover, as I argue, the most promising alternative theory of relations—one that avoids the problematic assumption and, in addition, fits most naturally into the sequentialist’s framework—fails to meet a basic requirement: it cannot distinguish between the proposition that Brutus stabbed Caesar and the proposition that Caesar stabbed Brutus. The upshot is that the conspicuously structured entities that are widely assumed to be up to the task of “playing the proposition role” shed no light on the very structure they are invoked to represent.

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  1. 1.

    Although I will not argue for it here, a related conclusion holds for the recent proposals of Hanks (2011, 2015) and Soames (2010, 2014, 2015), according to which a proposition is a certain act type. To the extent that these proposals deploy the propositional building blocks utilized either by Russellian or Fregeans, the same pessimistic moral applies.

  2. 2.

    For expository purposes, I will take The Theorist to be a Russellian. Nothing turns on this.

  3. 3.

    More would have to be said to make this a properly sufficient condition: we must ensure, for example, that Caesar and Brutus are assigned their positions with respect to the same instance of stabs. Otherwise (2) will also be made true at an \(E^{\prime }\) at which Brutus stabs someone other than Caesar and Caesar is stabbed by someone other than Brutus. Formulating this condition proves remarkably difficult. See Sect. 3.1, below.

  4. 4.

    I am indebted to Peter Hanks for raising this objection at the Barcelona conference. Thanks also to an anonymous referee for comments that led to further clarification.

  5. 5.

    The question about representation gives rise to two ways of reading my argument in this paper: as attacking a metaphysical thesis about propositional structure, which claims that propositions are structured in the way that The Theorist maintains, and as attacking an expressive thesis about propositional structure, which claims that The Theorist’s representations (themselves explicitly structured) adequately capture the structure intrinsic to propositions. My claim against Schiffer is, in effect, that the expressive thesis is false: sequences fail to capture a crucial aspect of the propositions they purport to represent, namely, their structure. But this follows from my argument against the metaphysical thesis. If to be structured is to be composed out of individuals, properties, and relations in the manner of (1) and (2), then propositions are not structured. But the very reason that such a sequence cannot be the proposition that (say) Brutus stabbed Caesar also precludes it from representing this proposition—if, by representing it, one means, as Schiffer does, to model its defining features. Thanks to an anonymous referee for urging clarification of this point.

          My arguments are in the same vein as those of Keller (2013), who argues that sequences fail to illuminate an equally crucial aspect of propositions: the relation (constituency) they bear to their building blocks. (For more on constituency, see note 35, below.) Critics of the sequential analysis include Bealer (1998) and Merricks (2015).

  6. 6.

    Proponents of directionalism include Russell (1903: §218) and, more recently, Hochberg (1999). (See MacBride (2012) for a rich and illuminating comparison of Hochberg and Russell on relations.)

  7. 7.

    Most of the literature on relations takes the plural noun ‘Caesar and Brutus’ to signify something we might think of as Caesar-and-Brutus: a plurality whose first member comes first in order of predication and whose second member comes last. I will follow suit (unless context indicates otherwise), noting, however, that the practice is problematic. Thus, in what follows I will often drop the phrase ‘in that order,’ taking it to be implicitly understood. For discussion, see van Inwagen (2006: pp. 457–461).

  8. 8.

    Note that ‘a and b’ is not functioning as a plural term here, and is thus an exception to the rule stated in note 7.

  9. 9.

    I follow Fine in speaking of “the application of [R] to its relata” and of R’s “applying... to its relata” (Fine 2000: pp. 1, 19). I take these phrases to be referring to the situation of R’s being true of its relata. The latter phrase, unlike the former, avoids any suggestion that what is being referred to is a mental or linguistic act. As Fine writes (slightly reworded): the completion of the relation R by a and b is the state of the objects a and b standing in the relation R (4).

  10. 10.

    I’m indebted here to MacBride (2007: p. 27).

  11. 11.

    See also Fine (2000: pp. 2–3) and MacBride (2013: p. 9). MacBride suggests that this sort of consideration is inconclusive. As he writes: “It may be how reality is itself jointed that favours selecting ‘R’ rather than ‘\(\mathbf{R}^{*}\)’ as the expression that corresponds to a genuine universal. In other words, there may be a natural rather than a logical foundation for admitting a nonsymmetric relation but not its converse” (Ibid; emphasis in text). This strikes me as deeply implausible, for precisely the reasons that Russell provides. Moreover, as MacBride himself notes, even if “reality actually selects” only one among a relation-converse pair—say before from before, after—it surely remains possible for reality to have selected after. So we can have possible worlds differing only in the fact that in one the Big Bang is before the Big Crunch whereas in the other the Big Crunch is after the Big Bang. But these two possibilities are obviously the same. See further, MacBride, pp. 9–10.

  12. 12.

    I am oversimplifying here. Fine does not claim that all nonsymmetric relations are neutral. His concern is simply to show that we can make sense of differential application without assuming directionalism. For him, a neutral relation is by definition one in which there is no meaningful notion of converse. But he does not deny the possibility of directed, or biased, relations. (See Fine 2000: p. 1, unnumbered footnote).

  13. 13.

    Fine here follows Williamson (1985).

  14. 14.

    The reference here is not to argument positions in EXEMP, but to the positions in R.

  15. 15.

    I have in mind here the implicit coordination of objects with positions.

  16. 16.

    As I understand it here, which means assuming both Identity and Uniqueness.

  17. 17.

    This is perhaps easiest to see when we consider directionalism. The directionalist can indicate the positions in loves in either of two ways: by order (first, second), or by role (lover, beloved). In loves, the first position is also the lover position while the second position is also the beloved position. The converse of loves is a relation in which the first position is also the beloved position while the second position is also the lover position. The application to positionalism requires merely that one drop the idea that locations are ordered, not that there are locations. (The positionalist is as much invested in the spatial metaphor as the directionalist.) This may make it harder to conceive of location switching, but the idea is not in itself unintelligible.

  18. 18.

    Armstrong (1997: p. 91) reports that Lewis suggested to him a similar treatment of symmetrical relations.

  19. 19.

    But see Donnelly (2016) for a response.

  20. 20.

    This isn’t to say what precise form the proposition takes on the positionalist approach. That question is addressed at the end of the current section.

  21. 21.

    Recall that the plural noun ‘Brutus and Caesar’ is being used, in the relevant contexts, as referring to a plurality whose elements are ordered.

  22. 22.

    In general, for an n-ary predicate, the exemplification relation will require (\(n \cdot \) 2) + 1 arguments.

  23. 23.

    In fact, although Fine does not consider this, one reason to specify stabs in (PRED 2) is to differentiate the exemplification of stabs from the exemplification of, say, its negation. Since stabs and its negation presumably share the same argument positions, we need a way of distinguishing their respective completions.

  24. 24.

    I owe this formulation to Phil Bricker.

  25. 25.

    R is a functional relation just in case Rxy&\({\textit{Rxz }}\rightarrow y=z\).

  26. 26.

    Again, it might strike one as odd to say that (2) needs an “interpretation,” since many (including The Theorist) would take (2) to possess truth conditions intrinsically. In fact, as I have been arguing, (2) leaves open (among other things) precisely how Brutus and Caesar are to be correlated with X and Y. If so, (2) cannot on its own possess a truth condition.

  27. 27.

    According to Braun (1993: p. 461), “[the] propositions expressed by ‘Bush is taller than Reagan’ and ‘Reagan is taller than Bush’ differ because Bush and Reagan occupy different positions within the structures of the propositions expressed by those sentences.” The approach in this paragraph can be viewed as an attempt to give this suggestion formal expression.

  28. 28.

    Where that phenomenon is understood in neutral, non-propositional terms.

  29. 29.

    Thanks to Bryan Pickel for discussion of the concerns treated in this and the next paragraph and for pointing me in the direction of Russell’s solution to the problem. (circa 1913)

  30. 30.

    Russell calls such relations “non-permutative”. Permutative relations are ones in which the positions are “homogeneous”—admitting arguments of the same logical type. Non-permutative relations contain distinct positions of distinct types. Hence, in the binary case, differential application is unproblematic: if non-permutative R applies to a and b it cannot apply to b and a. (See further Russell 1913/1992: pp. 122–123; Landini 2007: p. 57 and passim.)

  31. 31.

    For example, the event theorist will link a name’s occupying a certain grammatical role with a given aspect of events—an aspect described in terms of thematic roles. If, say, ‘Pat’ is the subject of the verb ‘to sing,’ then the referent of ‘Pat’ is the agent of some singing event or other (Pietroski 2005: p. 47).

  32. 32.

    Although the directionalist doesn’t claim that relations have positions, he is as much committed to their existence as the positionalist. The views differ only on whether the assignment of objects to positions respects an intrinsic ordering. The directionalist believes it does, the positionalist does not.

  33. 33.

    That said, I am sympathetic to the worry, how S and \(S^{*}\) can differ with respect to their similarity to T. Intuitively, they seem equally similar to T.

  34. 34.

    Fine points out that it is possible to nail down, via description, the manner in which the exemplar is completed, and then to fix the reference of a manner term using this description Fine 2000: p. 23; see also MacBride 2007: p. 46). For example, we can introduce a name to refer to the manner in which stabs is completed at T by Brutus and Caesar. This would allow us to specify how S and \(S^{*}\) differ without invoking the exemplar itself: we now indicate directly the specific manner in which one, but not the other, is the completion of its relation. If so, differential opposites such as S and \(S^{*}\) can be said to differ intrinsically after all—we need not invoke a relation to a contingently existing exemplar. Still, this maneuver does not establish that the two differ structurally. As Dodd (2007) shows, two apparently structured entities (e.g., musical works) may differ in virtue in virtue of their intrinsic properties and yet not differ structurally.

  35. 35.

    One way of addressing the problem of propositional structure, not considered here, is through the parthood relation. This might initially strike one as hopeless. Doesn’t the problem of differential constitution show quite clearly that a proposition cannot be the mereological sum of its parts? After all, a stabs b and b stabs a, while composed of the same parts, are nonetheless distinct. Gilmore (2014) develops an account of the parthood relation that has the resources to respond to this worry; his theory of parthood is designed to explain the difference between these propositions without appealing to a notion of structure. It should go without saying that a theory of parthood that renders a theory of structure otiose would be an important achievement, solving at one stroke two puzzles. In my view, the approach doesn’t succeed. Let me briefly say why.

          Gilmore’s central idea is to relativize parthood to position: to say that x is a part of y is to say that, for some positioni, xat i is a part of y. (In fact, he appeals to a further parameter, but I will set that aside to simplify the discussion.) It follows that nothing is a part of a propositionpsimpliciter: to be a part of p is to be a part of pat a given position. Once we are given a complete set of parameterized parthood claims–claims of the form: xat iis a part of p (for every i in p)—we have all the relevant structural information regarding p. With respect to the proposition stabs(a,b), such a set will contain the information that a is a part of stabs(a,b) at position 1 and that b is a part of stabs(a,b) at position 2 (and, in addition, that stabs is part of that same proposition at, say, position 0).

          As Gilmore acknowledges, this approach assumes directionalism (191, note 50), since positions are ordered numerically. If so, it doesn’t provide a satisfactory account of structure, for reasons given above (Sect. 2.1). While it is possible, as he shows, to reformulate the approach without assuming an implicit ordering of positions, the reformulated approach is, on examination, merely an excessively cumbersome variant of positionalism and thus equally unsatisfactory (see Sect. 3.1).

  36. 36.

    This paper was presented at the Ninth Barcelona Workshop on Reference, June 22–24, 2015. I’m grateful to Bjørn Jespersen and Manuel García-Carpintero for organizing the conference and to the audience at my talk for stimulating comments and discussion. Phil Bricker, Amanda Bryant, Ray Buchanan, Peter Hanks, Antonella Mallozzi, Oliver Marshall, Bryan Pickel, Frank Pupa, Rosemary Twomey, and two anonymous referees provided helpful comments on an earlier draft. Special thanks to Phil for conversations and comments that helped to clarify my understanding of Fine’s view and to Bryan for helping to bring me up to speed on the pre-Finean literature on positionalism.


  1. Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press.

    Google Scholar 

  2. Bealer, G. (1998). Propositions. Mind, 105(1), 1–32.

    Article  Google Scholar 

  3. Braun, D. (1993). Empty names. Noûs, 27(4), 449–469.

    Article  Google Scholar 

  4. Dodd, J. (2007). Works of music. Oxford: Oxford University Press.

    Google Scholar 

  5. Donnelly, M. (2016). Positionalism revisited. In A. Marmodoro & D. Yates (Eds.), The metaphysics of relations. Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780198735878.001.0001.

  6. Dorr, C. (2004). Non-symmetric relations. In D. Zimmerman (Ed.), Oxford studies in metaphysics (Vol. 1). Oxford: Oxford University Press.

    Google Scholar 

  7. Fine, K. (2000). Neutral relations. Philosophical Review, 109(1), 1–33. doi:10.2307/2693553.

    Article  Google Scholar 

  8. Gaskin, R., & Hill, D. J. (2012). On neutral relations. Dialectica, 66(1), 167–186. doi:10.1111/j.1746-8361.2012.01294.x.

    Article  Google Scholar 

  9. Gilmore, C. (2014). Parts of propositions. In S. Kleinschmidt (Ed.), Mereology and location (pp. 156–208). Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199593828.003.0009.

    Google Scholar 

  10. Hanks, P. (2011). Structured propositions as types. Mind, 120(1), 11–52. doi:10.1093/mind/fzr011.

    Article  Google Scholar 

  11. Hanks, P. (2015). Propositional content. Oxford: Oxford University Press.

    Google Scholar 

  12. Hochberg, H. (1999). Complexes and consciousness. Stockholm: Thales.

    Google Scholar 

  13. Jubien, M. (2001). Propositions and the objects of thought. Philosophical Studies, 104(1), 47–62. doi:10.1007/0-306-48134-0_10.

    Article  Google Scholar 

  14. Keller, L. (2013). The metaphysics of propositional constituency. Canadian Journal of Philosophy, 43, 655–678. doi:10.1080/00455091.2013.870735.

    Article  Google Scholar 

  15. King, J. (2007). The nature and structure of content. Oxford: Oxford University Press.

    Google Scholar 

  16. Landini, G. (2007). Wittgenstein’s apprenticeship with Russell. New York: Cambridge University Press.

    Google Scholar 

  17. MacBride, F. (2007). Neutral relations revisited. Dialectica, 61, 25–56. doi:10.1111/j.1746-8361.2007.01092.x.

    Article  Google Scholar 

  18. MacBride, F. (2012). Hochberg’s micro-metaphysical relations: Order all the way down. In E. Tegtmeier (Ed.), Studies in the philosophy of Herbert Hochberg. Ontos: Heusenstamm. doi:10.1515/9783110330557.87.

    Google Scholar 

  19. MacBride, F. (2013). How involved do you want to be in a non-symmetric relationship? Australasian Journal of Philosophy, 92, 1–16. doi:10.1080/00048402.2013.788046.

    Article  Google Scholar 

  20. Merricks, T. (2015). Propositions. Oxford: Oxford University Press.

    Google Scholar 

  21. Ostertag, G. (2013). Two aspects of propositional unity. Canadian Journal of Philosophy, 43(5–6), 518–533. doi:10.1080/00455091.2013.870725.

    Article  Google Scholar 

  22. Pietroski, P. (2005). Events and semantic architecture. Oxford: Oxford University Press.

    Google Scholar 

  23. Russell, B. (1903). The principles of mathematics. London: George Allen & Unwin Ltd.

    Google Scholar 

  24. Russell, B. (1913/1992). The theory of knowledge: The 1913 manuscript, ed. by E. R. Eames in collaboration with K. Blackwell. London: Routledge.

  25. Salmon, N. (1986). Frege’s puzzle. Cambridge: MIT Press.

    Google Scholar 

  26. Schiffer, S. (2016). Cognitive propositions. Philosophical Studies, 173, 2551–2563. doi:10.1007/s11098-016-0634-2.

    Article  Google Scholar 

  27. Soames, Scott. (1987). Direct reference, propositional attitudes, and semantic content. Philosophical Topics 15: 47–87. Reprinted in Philosophical Essays, Volume II. Princeton: Princeton University Press, 2009. Page references are to the original.

  28. Soames, S. (1989). “Semantics and semantic competence.” Philosophical Perspectives 3: 575–596. Reprinted in Philosophical Essays, Volume I. Princeton: Princeton University Press, 2009. Page references are to the original.

  29. Soames, S. (2010). What is meaning?. Princeton: Princeton University Press.

    Google Scholar 

  30. Soames, S. (2014). Cognitive propositions. In J. King, S. Soames, & J. Speaks (Eds.), New thinking about propositions. Oxford: Oxford University Press.

    Google Scholar 

  31. Soames, S. (2015). Rethinking language, mind, and meaning. Princeton: Princeton University Press.

    Google Scholar 

  32. van Inwagen, P. (2006). Names for relations. Philosophical Perspectives, 20, 454–477. doi:10.1111/j.1520-8583.2006.00115.x.

    Google Scholar 

  33. Williamson, T. (1985). Converse relations. Philosophical Review, 94(2), 249–262. doi:10.2307/2185430.

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Ostertag, G. Structured propositions and the logical form of predication. Synthese 196, 1475–1499 (2019).

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  • Nonsymmetric relations
  • Neutral relations
  • Kit Fine
  • Benacerraf problem
  • Positionalism
  • Structured proposition
  • Russellian proposition