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What propositional structure could not be

  • Article Type S.I. : Unity of Structured Propositions
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Abstract

The dominant account of propositions holds that they are structured entities that have, as constituents, the semantic values of the constituents of the sentences that express them. Since such theories hold that propositions are structured, in some sense, like the sentences that express them, they must provide an answer to what I will call Soames’ Question: “What level, or levels, of sentence structure does semantic information incorporate?” (Soames in Philos Mind Action Theory 3:575–596, 1989). As it turns out, answering Soames’ Question is no easy task. I argue in this paper that the two most promising ways of answering it, the Logical Form Account and the LF Account, are both unsatisfactory. This result casts doubt on the very idea that propositions are structured.

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Notes

  1. I use boldface in place of both mention quotes and corner quotes throughout. I’ll stick with the standard double quotes for direct quotation and scare quotes.

  2. In what follows, I suppress relativization to context and other parameters (e.g. value assignments) for readability.

  3. I’ll also drop the qualification ‘declarative, univocal’ in what follows.

  4. As stated here, the Primitive Entity Theory does not provide a criterion determining when two sentences express the same proposition. Because of this, it has sometimes been criticized as being a non-theory (see, inter alia, Richard 1990: p. 34). I think this is a legitimate concern, but since my aim here is not to defend a version of the Primitive Entity Theory, I cannot discuss it (but see Keller 2013 for discussion). However, there are versions of the theory (e.g. Alonzo Church’s Alternative (0)) that do provide such a criterion, and distinguish propositions very finely (see Church 1974). The fineness of grain objection discussed later in the paper would apply to such versions of the Primitive Entity Theory as well. So, it is worth noting that simply denying that propositions are structured is not sufficient to escape that objection. My point is only that the claim that propositions are structured like the sentences that express them (in the relevant sense)places pressure on one to accept one of the undesirable conclusions discussed herein (the Benacerraf dilemma for the Logical Form Account or the fineness of grain problem for the LF Account). Thanks to an anonymous referee for pressing this point.

  5. Note that I’ve characterized the three main theories of propositions in terms of their relationship to sentences, and the latter two presuppose that propositions are compositional sentential semantic values, either by being determined by or composed of the semantic values of the relevant sentential constituents. This way of setting things up, as well as the assumption that propositions play the role of sentential semantic values, is questionable and, indeed, is rejected by some friends of propositions. Nonetheless, I’ll adopt this assumption here. Propositionalists who reject this way of viewing propositions may just find it to be a further reason to reject the accounts of propositions that are under scrutiny here. It’s also worth noting that Bryan Pickel (forthcoming) has recently defended an alternative version of the Theory of Structured Propositions, according to which they are compositional sentential semantic values but are not (necessarily) composed of the semantic values of the constituents of the sentences that compose them. Because Pickel’s view of the relation between the semantic values of sentential constituents and the propositions that sentences express differs from standard accounts of structured propositions, it’s not quite clear whether the objections raised in this paper succeed against a view like his.

  6. It has a distinguished pedigree in the history of analytic philosophy—versions of the theory are defended by Bolzano, Frege and, at some points in their careers, Moore, and Russell. More recently, variants of the Structured Proposition Theory are defended by Joseph Almog, Kent Bach, David Braun, Peter Hanks, Jeffrey King, Nathan Salmon, Scott Soames, and Jason Stanley, among others.

  7. The locus classicus of the argument against the Possible Worlds Theory is Soames (1987). For an example of the sort of argument outlined here against the Primitive Entity Theory, see King (2007): p. 6; Richard (1990) gives a related argument. For a response, see Keller op cit.

  8. See e.g. Salmon (1986) and Soames (2002).

  9. I discuss the former question, regarding the nature of propositional constituents, in Keller op cit.

  10. Soames has since abandoned the traditional structured proposition theory for a theory of cognitive propositions (see Soames (2015); King et al. 2014). However, opting for a cognitive theory does not obviate the need to answer Soames’ Question. For example, Peter Hanks, who also holds a cognitive theory, writes: “Now, in the case of declaratives, philosophers have drawn the conclusion that the propositional content of a declarative has to be more finely grained than a set of possible worlds. It has to be something with constituents andstructure that mirrors the structure of the sentence that expresses it.... That is the kind of view I am proposing here” (Hanks 2015: p. 192, emphasis added). Soames’ own updated view of propositional structure is more complicated (see e.g. his discussion of Russell’s Gray’s Elegy example in King et al. 2014: chapter 6)—too complicated to discuss here. It is less clear on Soames’ new view how propositional structure is related to sentence structure.

  11. There’s a way of reading Soames’ question so that it looks like a question every propositionalist (not just proponents of structured propositions) has to answer, i.e., as a question about what structure two sentences need to share in order to express the same proposition. The broader context of the above quotation, however, suggests that Soames thinks that this kind of sentence-structure will be paralleled by the structure of the propositions they express. It is this question—what sort of sentence-structure does propositional structure parallel?—that is my focus here. Thanks to an anonymous referee for pressing me to clarify this point.

  12. My objection does not target accounts of propositional structure in terms of the structure of cognitive acts (as in Soames’ most recent work in, e.g., King et al. 2014—see footnote 10 above). Also, there may be other ways of developing the Logical Form and LF Accounts, or formulating an account in terms of sentence structure that does not rely on traditional logical form or LF. I think the objections I raise here could be generalized to some of these other accounts, but it would be beyond the scope of this paper to carry out that task.

  13. See Gilmore (2014), Keller op cit., King (2007): chapter 3, and Armstrong and Stanley (2011) for discussion of whether or not propositional constituents should be construed as parts and of the claim that propositions violate Uniqueness.

  14. In what follows, double brackets are used to indicate the denotations of the enclosed items. So a boldfaced declarative sentence enclosed in double brackets indicates the proposition expressed by the sentence. This notational convenience is taken from Ripley (2012).

  15. Unless propositions may change their truth-values over times, in which case this claim should be relativized to times.

  16. An anonymous referee points out that on a view that uses ordered ‘tuples to represent propositional structure, it looks like P1 won’t suffer from this problem, since this view will distinguish the constituents of the relevant propositions. For example, on such a view, the example propositions could be represented with <Loves, <Desdemona, Othello\(\gg \) and <Loves, <Othello, Desdemona\(\gg \). Though, on this analysis, the two propositions have the same “ultimate” constituents, they have different “non-ultimate” constituents (whatever is represented by the ordered pairs). But what exactly are these non-ultimate constituents supposed to be? What parts of the propositions corresponds to the relevant ordered pairs? It’s not clear to me that they (i.e. the ordered pairs) should be taken to represent constituents, but to encode the order of the constituents. Because of this, I’m not sure that this should be taken as a view on which an appeal to bare structure works, since it seems like the ordered ‘tuples are encoding constituent-arrangement (i.e. what the ordered pair members of the ‘tuples represent is the order of the constituents related by loves). If that’s the case, it’s a version of P2.

  17. And even the appeal to positions or slots might not be enough. See Ostertag (2013) for discussion.

  18. In what follows, consider the discussion to be confined to so-called “eternal” sentences. For simplicity and readability, I want to leave aside knotty issues about context-sensitivity. They will not affect the main arguments or conclusions of this paper.

  19. And perhaps more obviously, and more to the point, the propositions they express have different constituents.

  20. This objection is revisited below in the discussion of King’s view in Sect. 4.1.

  21. For example, see inter alia (Bernard Bolzano 1837; Frege 1918; Moore 1899; Russell 1903; Russel 1905; Soames, 1989; Braun 1993; Salmon 1986).

  22. May (1998) and Pietroski (2015) seem to be optimists, while Szabó (2012) is a pessimist.

  23. For a fuller discussion of the model-theoretic analysis of logical consequence and the place of propositions in that framework, see Blanchette (2000).

  24. The idea is not that the logical form of a natural language sentence s is to be identified with a particular well-formed formula in a particular language L—there will be infinitely many formulae (in infinitely many languages) that adequately formalize s.

  25. But see King (2007) and (2011) for defense of the view that propositions are language-dependent. I discuss King’s view in Sect. 4 below. See also Schiffer (1996) and (2003) for an account on which propositions are language-created.

  26. Here, and in what follows, I use numerals as abbreviations of the sentences that follow them. A boldface numeral is an abbreviation of the quotation-name of the sentence; hence, following the convention described above, a boldface numeral in double brackets indicates the proposition expressed by the relevant sentence.

  27. Russell acknowledges a formal treatment of denoting phrases in terms of quantifiers and variables in the language of logic, but thinks the propositions expressed by these formal translations are distinct from their natural language counterparts.

  28. As Soames (2014) points out, the incomplete symbol thesis is unnecessary: Russell could’ve treated descriptions as complex quantifiers.

  29. See e.g. Salmon, (1989) and Braun (1993).

  30. Of course, the translation of a given sentence will take place in the context of translating the language as a whole, and adequacy should be understood against that background.

  31. One might initially think (perhaps being misled, in part, by the similar names of the theories) that structuralism in the philosophy of mathematics is analogous to the Structured Proposition Theory, and wonder why the friend of structured propositions isn’t able to employ the same strategy as the structuralist in avoiding both horns of Benacerraf’s dilemma. (Structuralists deny that to be a number is to be an object of a certain sort. Rather, to be a number is just to be a place in the right sort of structure, and there are infinitely many such structures that are suitable role-players for the natural numbers (see Resnick 1997; Shapiro 1997).) But this view of the analogy between the two kinds of Benacerraf problem is mistaken. Structuralism is analogous to deflationism about propositions, according to which there are various different abstract structures that “play the role” of propositions, but no “real” propositions. However, structured proposition theorists are realists about propositions, similar to non-reductionist realists about the numbers.

  32. See King (2007) and Soames (2010). For further discussion of this problem, see, inter alia, Jespersen (2012) and Keller op cit.

  33. For a simple quantified sentence like Some boy smokes, a Venn diagram would also serve to represent its logical form, but the problem is that Venn diagrams do not do so well when it comes to sentences involving relations. Their limited applicability rules them out of the running for possible representations of propositional logical form. Thanks to an anonymous referee for discussion.

  34. Thanks to an anonymous referee for pressing the point that, even within FOL, there are a variety of ways to represent a proposition’s logical form. However, I want to forestall an objection by the structured proposition theorist by noting that, in some cases of differently structured but equally adequate FOL-sentences, the structured proposition theorist will be able to make a principled decision in favor of one over the other. For example, I think it’s plausible e.g. for a structured proposition theorist to distinguish the propositions expressed by a sentence and its double-negation, and, likewise, to prefer a non-negated FOL-sentence \(s^{*}\) as the translation of a natural language sentence s that contains no (natural language) negating expressions over \(s^{*}\)’s double-negated counterpart. For example, she has a principled reason to prefer Rj as the translation of John runs to \({\varvec{\sim }}\)\({\varvec{\sim }}\)Rj. The reason for this is that propositions are individuated not just by their structure, but by their constituents, and [[It’s not the case that it’s not the case that John runs]] plausibly has constituents that [[John runs]] lacks (viz., the semantic contributions of each occurrence of it’s not the case that). However, this sort of response will not generalize to all cases of differently structured but equally adequate FOL-translations—in particular, it does not work for natural language sentences that do not wear their logical structure on their sleeve (e.g. cases of logically equivalent translations of definite description-containing sentences). There are difficult questions about the natural language-formal language interface that I do not have sufficient space to discuss here, but I hope it suffices to note that there are fairly clear cases where there is a principled distinction among adequate translations for the structured proposition theorist to make and clear cases where there does not seem to be a basis for any such distinction, even if some unclear cases remain.

  35. See e.g. King (2007): p. 8 on brackets and commas.

  36. See Bealer (1982, 1998).

  37. See Szabó op cit.

  38. See discussion in Stanley 2000.

  39. Since mirroring is a reflexive relation, a proposition could satisfy this definition in virtue of having the very structure of the relevant sentence at LF (as on the version of King’s theory in King 1996).

  40. To ease readability, I’m leaving out generalization over contexts and variable assignments.

  41. Why does the proposition have to include the fact that PR encodes ascription (since it is already included in PR that R encodes ascription in English)? King claims that this is needed for the proposition itself (not just the interpreted sentence) to have truth-conditions (see King 2013: pp. 7–8). What exactly is it for PR to encode ascription? King explains: “Encoding ascription...is a relational property of the proposition itself: the property of being interpreted as ascribing what is at its right terminal node to what is at its left terminal node” (2013: p. 9). So, PR’s encoding ascription depends on thinkers interpreting it as such, via their cognitive access to the relevant sentence and, in particular, to the syntactic relation R that binds the lexical items of the sentence together at LF. There’s much to be said here about this aspect of King’s account, but it is not pertinent to the argument of this paper.

  42. King also countenances complex constituents: If there is a sub-tree rooted in a non-terminal node n of the propositional relation of a proposition p, then that sub-tree, along with the simple constituents at the terminal nodes dominated by n, is a complex constituent of p (2014: p. 210). There are no complex constituents in our example, but e.g. Two is the successor of one has as a complex constituent the PP sub-tree with the SVs of of and one at its terminal nodes.

  43. Burgess’ and Magidor’s objections are not in print, as far as I am aware, but King discusses them in 2013b.

  44. This example appears in Heim and Kratzer (1998) in a different discussion—not as an objection to King.

  45. Of course, this could also be represented by (2=1). Both logical forms are adequate—both encode the logical properties of the proposition expressed—so it makes no difference which one we choose.

  46. The difference could be encoded in set-theoretic notation by representing a relation like loving as operating on an ordered pair of items and a relation like identity as operating on a set of items.

  47. If nodes A and B are sisters, then A c-commands B and every node that B dominates and B c-commands A and every node that A dominates. C-command is not symmetric. Suppose A and B are sisters and B dominates C. Then A c-commands B and C, but C does not c-command A.

  48. King is aware that his modified account does not handle all of the problematic cases that have been raised against his original account (he gives an example of a sentence and its cleft-counterpart), but indicates that this is not a worry, since it handles the truly worrisome cases. See King (forthcoming). King has also argued that it would be ad hoc to identify [[4]]/[[5]], while distinguishing, e.g. [[1>2]] and [[2>1]]—which clearly ought to be distinguished. For according to King, the theorist who makes the aforementioned identification and distinction has no “principled explanation” of why syntactic differences make a propositional difference in the latter case but not the former. So, his theory accepts the cost of fine distinction, but reaps the benefit of a simpler semantics. The theorist who identifies [[4]]/[[5]] and the like incurs the cost of complicating her semantics by having to add separate clauses for symmetric and non-symmetric predicates. However, as Collins points out, this response holds no water against the theorist who is not antecedently committed to the Structured Proposition Theory; for the unstructured theorist, “such an ‘ad hoc and unprincipled’ situation vis-à-vis the role of combination might be perfectly OK, predicted even” (2013: p. 159). But my real point here is just that King’s modified account still results in very fine individuation of propositions—it’s not to offer a competitor view.

  49. King’s modified account does a better job with at least some translation cases, but it’s not clear that it will deliver the intuitive results for all such cases.

  50. See Searle (1992). For critical discussion of Searle’s “wall” argument, see James Blackmon (2013).

  51. I cannot discuss King’s propositional naturalism in any detail here, but see Keller (2017) and forthcoming for discussion.

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Acknowledgements

A version of this paper was presented at the Buffalo Logic Colloquium, March 2015, and the Unity and Individuation of Structured Propositions Conference: Ninth Barcelona Conference on Issues in the Theory of Reference, June 22-24 2015. I thank audiences at both places for their comments. Special thanks to Paddy Blanchette, David Braun, John Keller, Chris Menzel and two anonymous referees for this journal for helpful feedback.

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Keller, L.J. What propositional structure could not be. Synthese 196, 1529–1553 (2019). https://doi.org/10.1007/s11229-017-1585-7

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