Abstract
Say that an audience understands a given utterance perfectly only if she correctly identifies which proposition (or propositions) that utterance expresses. In ideal circumstances, the participants in a conversation will understand each other’s utterances perfectly; however, even if they do not, they may still understand each other’s utterances at least in part. Although it is plausible to think that the phenomenon of partial understanding is very common, there is currently no philosophical account of it. This paper offers such an account. Along the way, I argue against two seemingly plausible accounts which use Stalnaker’s notion of common ground and Lewisian subject matters, respectively.
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Notes
Because the focus here is on the relation between propositions required for partial understanding, it does not matter for present purposes how exactly Carla came to interpret Anna’s utterance as she did. For present purposes, we may imagine that Carla thought that Anna was speaking in code, that she has idiosyncratic beliefs about the meanings of certain English sentences, etc.
The need to capture these judgments is especially pressing given arguments from abundance (see above, p. 3). Given those arguments, typical conversations will be roughly like Color and Meeting in that the proposition expressed by an utterance and the proposition the audience takes the utterance to express will be slightly different; thus, unless we are ready to accept that we typically misunderstand other’s utterances completely, we need an account of partial understanding which predicts that cases like Color and Meeting are cases of partial understanding. At the same time, adequate accounts of partial understanding should predict that not just any conversation is a case of partial understanding. Given how different the propositions involved in Fuel and Squirrels are, those two cases can be safely taken as paradigms of cases in which the audience completely misunderstands the speaker’s utterance; hence the requirement that theories of partial understanding predict that Fuel and Squirrels are not cases of partial understanding.
For a related view of communication involving vague terms, see Fara (2000).
As I warned in the introduction, Contextual Equivalence and all the other views I will consider concern exclusively the relationship between the proposition expressed by an utterance and the proposition the audience takes that utterance to express required for partial understanding. Thus, for the purposes of the discussion, I set aside other potential requirements for partial understanding, such as the use of reliable interpretation mechanisms, adequate conditions for hearing an utterance, identification of assertoric force, and so on. Strictly speaking, all the views I will consider should start as follows: “Insofar as the proposition expressed by a given utterance and the proposition that the audience takes the utterance to express are concerned, the audience partially understands the utterance just in case ...” Here and whenever I state a potential account of partial understanding, I elide this clause for the sake of perspicuity.
This could be the case if, for example, it is public information that Paula is Anna’s friend, and that Paula isn’t friends with anyone whose house isn’t color g.
I am assuming that the proposition expressed by Anna’s utterance is either (1)’s compositionally determined semantic content in the context of Anna’s utterance, or a proposition which Anna meant in the Gricean sense (see above, p. 1); i.e., a proposition that Anna intends her audience to believe and such that Anna intends that her audience recognizes that intention. However, an anonymous reviewer proposes a different notion of expression, according to which an utterance expresses all the propositions that exclude from the context set exactly the worlds that the speaker intended to exclude. According to this notion of expression, if it is part of the common ground that Anna’s house is color g if and only if Paula doesn’t support the use of fossil fuels, then it is impossible for Anna’s utterance of (1) to express the proposition that Anna’s house is color g without also expressing the proposition that Paula doesn’t support the use of fossil fuels. If that’s the case, then, because Carla thinks that Anna’s utterance expressed the proposition that Paula doesn’t support the use of fossil fuels, and that proposition is in fact expressed by Anna’s utterance on the present notion of expression, Carla understood Anna’s utterance at least in part. I am skeptical of the resulting notion of expression (after all, it entails that Anna’s utterance of (1) expresses the proposition that Anna’s house is color g and 2 + 2 = 4; see Harris 2020 for independent reasons for skepticism) but, granting for the sake of the argument that the present notion of expression helps address the worry of overgeneration, it still doesn’t address the concern of undergeneration I present in the next paragraph in the main text.
In linguistics, partitions of logical space are often taken to be the semantic contents of interrogative sentences; see e.g. (Groenendijk and Stokhof, 1982, 1989; Roberts, 2012) Yablo (2014) takes subject matters to be divisions (as opposed to partitions) of logical space, but this choice does not make a difference for our purposes.
If A and B are partitions, A refines B iff every member of A is a subset of a member of B.
This is in line with theories on which the content of an interrogative sentence is a partition of logical space. According to those theories, a conjunction of interrogatives expresses the coarsest partition to refine each of the conjuncts (see e.g. Groenendijk & Stokhof, 1989).
More recent accounts of sentential subject matter due to Hawke (2018) and Plebani and Spolaore (2021) also face the problems I raised in this section. Like the standard Lewisian account, both of these accounts predict: (i) that the proposition that Anna’s house is color g and the proposition that it’s not the case that Paula supports the use of fossil fuels have an informative entailment in common about the subject matter of the color of Anna’s house and Paula’s position with respect to fossil fuels (namely, their disjunction); and (ii) that the proposition that Anna’s house is color \(g'\) has no informative entailments about the subject matter of whether Anna’s house is color g. Thus, together with Subject Matter Dependence, these two accounts incorrectly predict that Carla partly understood Anna’s utterance in Color/Fuel and that David doesn’t understand Anna’s utterance at all even though he knows exactly which proposition she expressed. See (Abreu Zavaleta, 2018, Chap. 2) for further discussion of a view like Plebani and Spolaore’s in relation to arguments from abundance.
In using the notion of content parthood to study partial understanding, I follow a recent trend of applying the notion to problems that have so far resisted philosophical treatment. For example, Shumener (2017, 2022) applies the notion of content parthood to the debate about laws of nature; Elgin (2018) uses it to analyze what it is for an argument to beg the question using the related notion of analytic containment; Weiss (2019) uses it in the interpretation of Sextus Empiricus’ theory of conditionals; Fine (2022); Gemes (2007); Yablo (2014) discuss content parthood in relation with verisimilitude, and Davies (2021b) uses content parthood to develop a semantics for speech reports.
Fine does not use possible worlds. In Fine’s framework, where I say that a truthmaker is a superset of another, he would say that the former is part of the latter. This difference does not matter for present purposes.
The third clause is weaker than Fine’s (2017a), who imposes the stronger requirement that every falsitymaker for P be itself a falsitymaker for Q. Yablo (2014) replaces clause (iii) with the requirement that every falsitymaker for P be a superset of some falsitymaker for Q. Neither alternative suits our purposes. Given plausible assumptions, Fine and Yablo’s replacements for clause (iii) entail that the content of the proposition that Anna’s house is red is not part of the content of the proposition that Anna’s house is scarlet. Other than that, the differences between the present definition and Fine’s and Yablo’s do not matter for our purposes; within the present implementation of truthmaker semantics, all three definitions entail that conjunctions have their conjuncts as parts, that disjunctions are not generally part of their disjuncts, and that content parthood is a preorder.
That content parthood is reflexive follows immediately from the reflexivity of \(\supseteq \). That content parthood is transitive follows from the transitivity of \(\supseteq \). Fine (2016, 2017a) imposes further requirements on propositions which make content parthood antisymmetric. For the sake of simplicity, I do not impose such further requirements.
Proof. Take arbitrary propositions P and Q. We show that \(\text {P}\le \text {P}\wedge \text {Q}\). (i) Take an arbitrary \(x\in t(\text {P})\). It follows from (6) that \(x\cap x'\in t(\text {P}\wedge \text {Q})\) for some \(x'\in t(\text {Q})\). Since \(x\supseteq x\cap x'\), it follows that there is a truthmaker for \(\text {P}\wedge \text {Q}\) of which x is a superset. And, since x was chosen arbitrarily, every truthmaker for P is a superset of a truthmaker for \(\text {P}\wedge \text {Q}\). (ii) Take an arbitrary \(s\in t(\text {P}\wedge \text {Q})\). It follows from (6) that there are \(x\in t(\text {P})\) and \(x'\in t(\text {Q})\) such that \(s=x\cap x'\). Since \(x\cap x'\subseteq x\), it follows that there is a truthmaker for P of which s is a subset. And since s was chosen arbitrarily, every truthmaker for \(\text {P}\wedge \text {Q}\) is a subset of a truthmaker for P. (iii) Take an arbitrary \(x\in f(\text {P})\). It follows from (6) that \(x\in f(\text {P}\wedge \text {Q})\). Thus, by the reflexivity of \(\subseteq \), there is a falsitymaker for \(\text {P}\wedge \text {Q}\) of which x is a subset. And, since x was chosen arbitrarily, every falsitymaker for P is a subset of a falsitymaker for \(\text {P}\wedge \text {Q}\). From (i)–(iii), it follows that \(\text {P}\le \text {P}\wedge \text {Q}\).
Let P and Q be propositions such that \(\text {P}=\langle \{x\}, \{x'\}\rangle \) and \(\text {Q}=\langle \{y\}, \{y'\}\rangle \), where \(x\not \supseteq y\) and \(y\not \supseteq x\). Then \(\text {P}\vee \text {Q}=\langle \{x, y\},\{x'\cap y'\}\rangle \). Since \(y\not \supseteq x\), not every truthmaker for \(\text {P}\vee \text {Q}\) is a superset of a truthmaker for P. So \(\text {P}\vee \text {Q}\not \le \text {P}\).
Proof. Let P and Q be propositions, and suppose that there is a substantive proposition which is part of both P and Q. Let R be that proposition. Since R is substantive, it follows that \(\bigcup t(\text {R})\ne \emptyset \) and \(\bigcup t(\text {R})\ne \bigcup S\). Thus, there is a state \(r\in t(\text {R})\) such that \(r\ne \emptyset \) and \(r\ne \bigcup S\); i.e., there is a non-trivial state r in \(t(\text {R})\). Since R\(\le \)P, it follows that \(r\supseteq p\) for some \(p\in t(\text {P})\), so \(r\supseteq \bigcap t(\text {P})\). And, given that R\(\le Q\), similar reasoning shows that \(r\supseteq \bigcap t(\text {Q})\). So there is a non-trivial state s such that \(s\supseteq \bigcap t(\text {P})\) and \(s\supseteq \bigcap t(\text {Q})\), namely, \(s=r\). Thus, if there is a substantive proposition which is part of both P and Q, there is a non-trivial state s such that \(s\supseteq \bigcap t(\text {P})\) and \(s\supseteq \bigcap t(\text {Q})\). By contraposition, if there is no non-trivial state s such that \(s\supseteq \bigcap t(\text {P})\) and \(s\supseteq \bigcap t(\text {Q})\), then there is no substantive proposition which is part of both P and Q.
Note that this proposition is not the same as the disjunction G\(\vee \lnot \)F; by definition, G\(\vee \lnot \)F has two truthmakers, whereas the proposition I define in the main text has only one.
Note that these simple propositions essentially correspond to standard Kripke valuations. Where P is a proposition that normal language users could easily take to be expressed by an utterance of an atomic sentence, standard Kripke valuations would identify P with the set p of worlds in which P is true, whereas the present approach would identify P with the pair \(\langle \{p\},\{W\setminus p\}\rangle \). In this respect, the present approach is (once again) conservative from the perspective of standard possible-worlds semantics.
There are many other propositions that G and G\('\) both have as parts. For example, let \(g^*\) be the region of the color spectrum comprised of g and \(g'\). Then the proposition that Anna’s house is color \(g^*\) is part of both G and G\('\)—since every world in which Anna’s house is color g or \(g'\) is a world in which Anna’s house is color \(g^*\), and any world in which Anna’s house is not color \(g^*\) is a world in which Anna’s house is neither color g nor color \(g'\).
Although Common Content captures the minimum of similarity between propositions that results in partial understanding, it does not capture differences in the degree to which different audiences understand a given utterance—or, to put it another way, it only captures understanding to a non-zero degree. For example, suppose that Paula takes Anna’s utterance of (2) to express the proposition that Anna is in Boston. Common Content predicts that Paula partially understood Anna’s utterance, since the proposition AB and the proposition that Anna is in Boston have may parts in common; for example, the proposition that Anna is in the US East Coast, the proposition that Anna is not in Milan, etc. This seems the right prediction; although proposition AB and the proposition that Anna is in Boston are not intuitively as similar to each other as AB and BC, they still seem similar enough to consider this a case of partial understanding. After all, both AB and the proposition that Anna is in Boston are about Anna’s location, and the present case still seems radically different from cases of complete misunderstandings like Fuel and Squirrels. Nevertheless, a complete theory of partial understanding should be able to capture the intuition that, although Bob and Paula both understand Anna’s utterance of (2) to some extent, the former understands better than the latter. Using Common Content as the basis, we can build a graded notion of understanding by imposing a measure on the state space, with similarity between propositions P and Q depending on how big a proportion of P and Q their greatest common part covers. But, as I mentioned in the introduction, I leave the matter of degrees of understanding for future work. Thanks to an anonymous reviewer for bringing up the case discussed in this footnote.
Davies (2021a) does not explicitly endorse Expression Closure or Existential Identification, but I attribute them to him because they explain his endorsement of (i) and (ii). According to Davies, although audiences won’t typically be able to know exactly which propositions an utterance expresses, they can still know that certain propositions are parts of the propositions expressed. As Davies puts it, there are “parts of the unidentifiable proposition [the strongest proposition expressed] that a context-sensitive sentence expresses in context which can quite plausibly be identified on the basis of the available evidence in the context” (Davies, 2021a, p. 12386). Thanks to an anonymous reviewer for suggesting this interpretation of Davies’ view, which they base partly on Davies’ earlier (2019).
Alternatively, Davies could claim that audiences use contextually salient information to make inferences about what propositions are parts of propositions expressed by an utterance. However, given the description of Color and Meeting, there is no contextually salient information that the audience can use to make such inferences (indeed, as I argued in Sect. 3.1, the absence of such contextual information in Color and Meeting is the reason why Contextual Equivalence undergenerates). It is thus difficult to see how this alternative would fare better than Interpretation Closure.
There is also an important difference between Davies’ implementation of truthmaker semantics and the present one. For all Davies says, it is possible that, for any compatible propositions P and Q with truthmakers p and q, respectively, there is a state of affairs r that is part of p and part of q. In that case, P and Q will have a substantive content part in common, namely, the proposition R whose only truthmaker is r and whose only falsitymaker is the fusion of the falsitymakers for P and Q—note that, given the semantics for disjunction, this proposition is different from P\(\vee \)Q, whose truthmakers will be p and q. The existence of propositions such as R is desirable in certain cases, such as the case of color: if the only shades of red are crimson and scarlet, then the state whereby x is red is essentially a state r in which x is crimson or scarlet, but which does not specify which. Thus, we can treat the proposition that x is red as the proposition whose truthmaker is r and whose falsitymaker is the fusion of falsitymakers for the proposition x is crimson and the proposition that x is scarlet. This proposition is different from the proposition that either x is scarlet or x is crimson: the former is part of both the proposition that x is crimson and of the proposition that x is scarlet, but the latter is not. As desirable as this particular prediction is in the case of redness, without further constraints, Davies’ implementation of truthmaker semantics overgenerates; it entails that any two substantive compatible propositions have a substantive content part in common, which defeats the purpose of introducing content parthood to explain partial understanding in the first place (this is also a feature of Fine’s implementation, but it doesn’t matter for his purposes because Fine is only interested in the logical properties of content parthood). The model I introduce in Sect. 4 (see especially Sects. 4.1 and 4.2) does not have this problem. On that model, there are compatible propositions with no substantive parts in common—a result that is partly captured by (8), and illustrated by Common Content’s predictions in cases like Fuel and Squirrels.
Strictly speaking, discourses are better understood as sequences of utterances, but I take them to be sets of utterances for the sake of simplicity.
The proof that if \(\text {P}\in \Gamma \) then \(\text {P}\le \bigwedge \Gamma \) is as in fn. 15.
See Abreu Zavaleta (2022) for development of this argument.
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Acknowledgements
The core proposal in this paper was first presented at the Institut d’Etudes Avancées de Paris in the Spring of 2017, but it only found its present form in the Fall of 2018. Thanks to the audience at IEA Paris and to audiences at Hamburg University (Summer 2019), Universidad Nacional Autónoma de México (Fall 2019), Syracuse University (Spring 2020) and University of Colorado at Boulder (Fall 2020) for valuable comments and discussion. Thanks to anonymous reviewers for constructive criticism, and thanks especially to Erica Shumener for her insightful comments and suggestions on innumerable drafts of this paper.
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Abreu Zavaleta, M. Partial understanding. Synthese 202, 41 (2023). https://doi.org/10.1007/s11229-023-04268-2
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DOI: https://doi.org/10.1007/s11229-023-04268-2