Mananas, flusses and jartles: belief ascriptions in light of peripheral concept variation

On a simple and neat view, sometimes called the Relational Analysis of Attitude Ascriptions, a belief ascription on the form ‘S believes that x is F’ is correct if, and only if, S stands in the belief-relation to the proposition designated by ‘that x is F’, i.e., the proposition that x is F. It follows from this view that, for a person to believe, say, that x is a boat, there is one unique proposition that she has to believe. This paper argues against this view. It fails, I contend, to make sense of peripheral concept variation. As we attribute and individuate concepts, two people’s concepts C1 and C2 count as e.g., concepts of boats even if their concepts have different extensions in peripheral, or borderline, cases of boats. Thus, A and B can believe that x is a boat through believing peripherally different propositions. It follows that there is no unique proposition that a person has to believe in order to believe e.g., that x is a boat.


Introduction
It seems natural to suppose that for a person to believe, say, that grass is green, she has to believe a specific proposition, namely the proposition that grass is green. If she believes some other proposition, she does not believe that grass is green. In this paper I will challenge this idea. Two persons who both believe that grass is green, may well do that through believing different propositions. This means that I will challenge the Relational Analysis of Attitude Ascriptions (my focus is on belief ascriptions, though the reasoning applies to other propositional attitudes as well). According to the Relational Analysis, applied to beliefs, a belief ascription of the form 'S believes that x is F' is correct if, and only if, S stands in the belief-relation to the proposition designated by 'that x is F', i.e., the proposition that x is F (cf. e.g., McGrath & Frank 2020;Moltmann, 2003). This view -often taken to be the standard view -gives us a straightforward and intuitively plausible view about what it takes to believe that something is the case. The Relational Analysis has been questioned in various ways (e.g., Bach 1997;Moltmann, 2003;Prior, 1971;Shier, 1996), but the challenge presented in this paper will be different from previous objections. It will challenge a specific implication of the relational analysis, namely the idea that there is some unique specific proposition that has to be believed in order to believe, say, that grass is green. Let us call this idea Unique Propositionalism. It might seem that this idea is trivially correct (at least as long as we accept that beliefs, and other propositional attitudes, are relations between agents and propositions): if you don't stand in the belief-relation to the proposition that grass is green, but to some other proposition, you don't believe that grass is green, but believe something else. The argument of this paper questions this idea.
I will argue that for most concepts, F, there will be many (slightly) different propositions such that, if you believe one of those propositions, then you count as believing that x is F. Most that-clauses, as we use them to attribute beliefs in everyday language, allow for slight variation in believed content. This further means that for most that-clauses, we cannot use that that-clause to pick out a specific proposition -there simply is not one unique proposition that can properly be described as the proposition that grass is green.
The problem with Unique Propositionalism, I will argue, is that it fails to make sense of a phenomenon that we can call peripheral concept variation (related to what some have called "the open texture" of concepts). My aim is to present a case to the effect that the possibility of peripheral concept variation presents a challenge for Unique Propositionalism. There are, as we shall see, ways of resisting the conclusion, but I hope to show that the rejection of Unique Propositionalism is at least a possibility that should be taken seriously.

The phenomenon of peripheral variation
Several philosophers have pointed to the phenomenon that, for many concepts (and terms), there are, in addition to clear cases of things that fall and do not fall within the extension of the concept (term), also borderline cases, or peripheral cases, that are less clear. Waismann (1945) gave some entertaining examples: "Suppose I come across a being that looks like a man, speaks like a man, behaves like a man, and is only one span tall -shall I say it is a man?" (122) Or consider an animal that shows every sign of being a cat, but: "What, for instance, should I say when that creature later on grew to a gigantic size? Or if it showed some queer behaviour usually not to be found with cats, say, if, under certain conditions, it could be revived from death […]? Shall I, in such a case say that a new species has come into being? Or that it was a cat with extraordinary properties?" (121-122) Based on such examples, Waismann argued that most empirical concepts have an "open texture": there are cases that we have not yet considered, and for which we therefore do not have established linguistic rules for whether they fall under the concept.
More recently, less fictional examples have been used to make basically the same point. "For example, the term 'jar' is open textured because whether a given container belongs to the extension may be something that has not yet been decided. There may be a kind of glass vessel that could equally well be called a 'jar' or a 'bottle'." (Gauker, 2017, p. 14) Let me refer to such vessels as "jartles". Ludlow (2014) discusses numerous other cases. For example; "there are open-faced sandwiches and wraps and burgers and I suppose croque-monsieurs that one might or might not put in the range of 'sandwich'." (p. 10) What counts as a book? An e-book? A bookish text only 30 pages long? Something not yet published? He concludes that "even after a millennium of shared usage the meaning is quite open-ended." (p. 1) The examples can be multiplied: A bus that can also drive on water (call it "fluss", for "floating bus"), is that (also) a boat? An object that is exactly like a banana except that it is manufactured in a laboratory -is such a man-made banana, or "manana" for short, a banana?
The authors above have mainly discussed the phenomenon of borderline (or peripheral) cases with an eye at the implications for the meaning of terms. Weismann uses it to critique verificationism about meaning, and Ludlow argues that word meanings are underdetermined and dynamic. My concern is the implications for the extensions of concepts, and the further consequences for belief (and concept) ascriptions.
The phenomenon in question can be described in terms of classifications, or classificatory dispositions. People classify things (and properties, relations etc.), i.e., sort them into distinct categories. Many classifications are tied to words of a public language -we classify things as jars, as bananas, cats, chairs etc. This allows us to communicate about these things. There is a general pressure to classify things in the same way as other people, and tie the classifications to the same words. Otherwise communication fails. It is thus part of learning a language to learn to classify, say, certain things as being jars and other things as not being jars. But there are also actual and potential cases that (i) there has not yet been a sufficient communicative pressure to classify in the same way -as being e.g., jars or non-jars, bananas or non-bananas -because they have not been around or have not been salient enough to us, and (ii) the classification of which cannot be unambiguously extrapolated from previous classificatory conventions. For example, there has been no pressure to establish a convention according to which mananas are classified as bananas or not, or jartles as jars or not. These are peripheral cases, rather than central or clear cases of either belonging or not belong to the class.
When it comes to such peripheral cases, we can expect two things. First: many people have less clear dispositions to classify in either way, they may waver or hesitate. Second: different people are disposed to classify differently (since there has been no pressure to converge). Some people may classify mananas as bananas, some may not. Indeed, when you talk to people about such cases, both tendencies are clearly present.
The relevant kind of variation is variation in what we may call fundamental (or ideal) classifications. Of course, people's classifications of central cases vary as well. You may e.g., mistakenly identify a central case of a non-banana (e.g., a plastic banana) as a banana (or reversely) because it looks like a real (fake) banana in the dark, or due to wishful thinking (e.g., being really hungry). But regarding peripheral cases we can expect variation in people's fundamental dispositions to classify -e.g., variation regarding whether a manana is classified as a banana, or a fluss as a boateven between people who agree about all the properties of the manana and the fluss (all properties, that is, except the property of being a banana or the property of being a boat), and even between people unaffected by distorting factors such as wishful thinking. For there has been no pressure to converge on one specific classification. 1

Variation in the extensions of concepts
One straightforward way of understanding the abovementioned phenomenon is in terms of concepts: when two people differ in fundamental classificatory dispositions regarding flusses -one includes them in the class to which she connects the word 'boats', one doesn't -it plausibly indicates that they utilize, and have connected to their use of 'boat', concepts with different contents, where these contents are such that the concepts have different extensions in at least some possible worlds (worlds with flusses). One concept's extension includes core cases of boats and flusses, the other includes only the former. This makes sense because, first, both concepts are plausibly concepts it is possible to have and utilize in ones thinking, and second, being fundamentally disposed to include or exclude flusses, seems to be good prima facie evidence that one utilizes one rather than the other concept. Plausibly, then, when A and B use the term 'boat', concepts with different extensions underlie their use. Further, given that the concepts we utilize make up the contents of our thoughts, when both sincerely assert, about the same object, "x is a boat", what underlies their 1 How does open texture differ from (ordinary) vagueness? Any view about this will be controversial, partly since it is controversial what vagueness is. For example, if we think that having borderline cases suffices to make a concept vague, then open texture is a sort of vagueness; but if giving rise to instances of sorites paradox is characteristic of vague terms and concepts, then open texture is something else. I see no need to take a definite stand on this issue here, since my argument does not depend on it. But there may be one relevant difference between typical examples of vagueness, and the examples of open texture that are the focus in this paper. Take "heap", as an example. It is at least fairly plausible to think that being a competent speaker involves having a sense that, somewhere on the path from one grain of sand to a thousand grains, it is unclear whether it is a heap of sand or not. Someone who uses "heap" to draw a precise line between heaps of sand and non-heaps, at, say, one hundred grains of sand, either displays a lack of competence with regard to the vague nature of the meaning of the term, or intentionally uses it with a stipulated non-standard meaning. For typical cases of open texture, on the other hand, people are allowed to go either way -treating e.g., mananas as unclear cases, as clear members, or as clear non-members of the extension of "banana" -without displaying conceptual incompetence. As we shall see, the latter claim about these cases is important to the argument in this paper (though the contrast against typical cases of vagueness is not). Thanks to an anonymous referee for suggesting that I bring up the relation to vagueness. respective assertions are beliefs in different propositions (propositions that have different truth-values in at least some possible cases, cases where x is a fluss). 2 There are of course potential objections. Depending on one's view about what determines conceptual content, one could reject the claim that variation in people's fundamental classificatory dispositions shows that they utilize concepts with different extensions. First, some might object on the basis of social externalism (e.g., Burge 1979) that the classifications of individuals don't matter to conceptual content. On this view, conceptual content depends on the use of terms or concepts in one's community. In response, it can first be noted that, if we accept social externalism, the argument in this paper can be restated: concepts can still differ between linguistic sub-communities where people classify peripheral cases differently. Second, a less defensive response is available. Arguably, considerations about peripheral cases show that social externalism has limited scope. As noted above, for central cases there is a pressure to converge in classification (within a linguistic community). We thus try to adjust our ways of classifying so that they sync with how other people classify. It makes sense that for our classifications of these cases, we defer to the use of other people -if our own classificatory intuitions are not in sync with those of the majority in our community, we are prepared to think that their classifications are the correct ones. But for peripheral cases there will be no established consensus in the community to defer to, and the idea of deference to the use of the community requires such a consensus.
It may also be objected, from the stance of causal externalism, that since the concept of bananas is a natural kind concept, even if some people are disposed to include things not belonging to that natural kind, this provides no evidence about the content of the concept: it refers to the natural kind that actually causally regulates our use of the concept irrespectively of what people think about the reference. Here we can also give a defensive reply: if these considerations are correct, then we can restrict the argument to other than natural kind concepts, e.g., to the concepts of boats and jars. But there is also a non-defensive line of reply. We don't want to exclude the possibility of talking and thinking about (disjunctive) classes of the following sort: things that are of a certain natural kind or are (in certain ways) sufficiently similar to things belonging to that natural kind. Consequently, we do not want to exclude the possibility of having a concept with these (and only these) things in its extension. For example, we want to be able to say that one can have the natural-kind concept of bananas, but also that one can have the concept of natural-kind bananas + mananas. But what would then be a sign of the latter concept rather than the former governing the use of some person's (or community's) use of a term t? Plausibly that the fundamental classifications tied to t include also mananas.
A similar objection starts in the idea of reference magnetism, that is, the idea that the content of concepts (and terms) might depend partly on dispositions to classify, but also on what is the most natural property in the vicinity (Lewis, 1983(Lewis, , 1984Sider, 2011). For example, even if some people are disposed to include mananas in the same class as bananas, the concept that they use refers to natural-kind-bananas and not bananas + mananas, since this is a more natural referent (or, differently put, that classification better carves nature at its joints). Again, however, we arguably want to say that one can have a concept that refers to natural-kind-bananas + mananas, and that certain fundamental classifications is evidence of having that concept. Further, even main defenders of reference magnetism acknowledge that for most nontechnical terms/concepts, a view that implies that the term refers to the most natural referent in the neighborhood even though speakers use it differently, is uncharitable and therefore implausible (see Sider 2011 Chap. 3).
Indeed, the main point of this section can be made in terms of charitability. S's fundamental classificatory dispositions inform us about what S is prepared to include in the extension of a concept. So if we want to understand what it is S is thinking (and what S commits to and not in thinking something), it is, unless some further reason is given, uncharitable to suppose that the extensions of the concepts that make up S's thoughts differ from what S is fundamentally disposed to include in those extensions. Consequently, peripheral classificatory variation gives us reason to hold that the concepts that govern different people's use of the same term have peripherally different contents, i.e., contents such that their extensions differ with regard to some possible peripheral cases. 3 Let us call this the Actual Peripheral Concept Variation claim: peripheral classificatory variation is evidence of different people having concepts with peripherally different contents. In the next section it will be useful to distinguish this from the Possible Peripheral Concept Variation claim, according to which it is possible for different people to possess concepts with peripherally different contents. This weaker claim is, as we will see, sufficient in the argument against Unique Propositionalism to be presented, but the stronger claim allows further conclusions.
Of course, the short considerations in this section have not conclusively established either of these two claims. But for the purposes of this paper it suffices that there is a reasonable case in favour of the them, since the main aim is to show that peripheral variation presents a challenge for Unique Propositionalism that should be taken seriously.
Before we move on to present that challenge, one further clarification is in place. The variation claim defended here is not about the semantics of open textured terms. The shared conventional meaning of terms cannot vary in the way I have claimed that conceptual content does; it is shared, after all. Perhaps the best way to account for open texture in semantics is to say that the conventional meaning of e.g., 'banana', is such that the extension of the term is indeterminate regarding mananas. We can also express this as the view that the (unique) concept that is semantically expressed by (or semantically associated with) the term, is indeterminate. This view is consistent with (and does not challenge the case for) the variation claim I have made above: that people, as a matter of psychological fact, associate peripherally different concepts with the term; and use these different concepts when they form their thoughts and beliefs (the beliefs that underly their assertive use of the term). This is because the concept that is semantically expressed by a term in virtue of its conventional meaning need not be the concept that as a matter of psychological fact governs an individual speaker's use of the term.

Peripheral variation and belief ascriptions
What is required for S to have a concept of F, e.g., of boats? (Or equivalently: the boat concept? I use "boat concept" and "concept of boats" interchangeably: both refer to the concept that needs to be involved when one forms e.g., the thought that x is a boat.) In general, we require at least that she has a concept with a specific extension. Suppose you say "Beautiful boat!", pointing to a regular blue sailboat, and S responds, "that's not a boat, it's blue". It turns out that S is (fundamentally) disposed to classify exactly like you (and most people), except for not including blue objects as boats. We can conclude either of two things: Either she utilizes in her thinking, and has connected to the term 'boat', a concept the extension of which includes all central cases of boats, except blue ones. In that case, this plausibly is not the boat concept, but some concept with a narrower extension. (This is consistent with the claim that the term "boat", when she uses it, conventionally expresses the boat concept. This can be the case due to the public meaning of the term, even if she herself does not possess the boat concept.) Or we may attempt to resist this conclusion, and say that she still possesses the boat concept -this is reasonable only to the extent that we can make it plausible that she has a concept with the right extension (one that includes blue boats), but due to poor grasp of the concept misclassifies blue boats as not being boats. Either way, having the boat concept involves having a concept with the right extension (one that includes blue boats).
Similarly, consider a person who fundamentally classifies e.g., motorcycles and cars as boats, calling them 'boats' despite knowing that they are vehicles for travelling on land rather than water. Suppose we take this to show that the concept that she uses, and has attached to the term 'boat', is a concept whose extension includes all vehicles. Again, it is unreasonable to say that her concept is the concept of boats. Rather, it is a wider concept. Alternatively, suppose that we want to say that it could still be that the concept she has is the concept of boats -plausibly, this is reasonable only if it can be argued that (despite her over-inclusive classification) the extension of her concept does not include all vehicles. This, again, speaks for the conclusion that having the concept of boats requires having a concept with the right extension.
Corresponding considerations hold for our ascriptions of beliefs to such divergent classifiers. Suppose the "over-inclusive classifier" points to a motorcycle and (sincerely) says, 'that is a really fast boat'. Given that we know that her utterance is due to her using 'boat' about all vehicles (rather than e.g., somehow misperceiving the object as looking like a typical racer boat), we might find it implausible (and uncharitable) to say that her belief (the one that triggers her utterance) is a belief the truth of which requires that x -the referent of 'that' -is a (fast) vessel for travelling on water. Rather, her belief is correct if x is a fast vehicle of any kind. If so, then plausibly her belief is not a belief that x is a fast boat. It only makes sense to ascribe to her the belief that x is a fast boat if it can somehow be argued that she has a belief which is true only if x is a fast vessel for travelling on water.
The examples and reasoning above suggest that having a concept of F -e.g., of boats -requires having a concept with a specific extension (and that having a belief that x is F is having a belief that is true only if x is in that extension). However, when we turn to variation in extension regarding peripheral cases, this no longer seems right. These are cases where no common classificatory standard has been established. It thus wouldn't make sense to require a specific classification of, or extension in, these cases when we attribute concepts of e.g., boats and bananas. And indeed, if we look at what we go on when we actually attribute concepts, it seems that we don't require that. For example: Suppose that A is disposed to fundamentally classify flusses as being boats, and that B fundamentally classifies them as not being boats. If having the concept of boats required having a concept with a certain specific extension in fluss-cases -say, having a concept that includes flusses, rather than one that excludes flusses -then we would have to say either that one of A and B is conceptually incompetent, or that one of them is disqualified from having a concept of boats. But both seem implausible. It is intuitively implausible to conclude from variation in such peripheral cases that one of the parties is conceptually incompetent, or that one of them lacks the concept in question (and thereby lacks the ability to form, say, a belief that something is a boat). Rather, it seems plausible to say they have different ideas about whether flusses are boats (thus both having thoughts about boats, and utilising a boat concept), without either one displaying a lacking grasp of the concept.
Likewise, suppose that A says, "mananas are bananas", while B says "they are not, though they are very banana-like, they are fake-bananas since they don't come from banana plants". Suppose this reflects their fundamental classificatory dispositions. Intuitively, I'm inclined to think that neither is disqualified from being a competent "banana-thinker". Rather they have different views about whether mananas are bananas. A thinks that they are bananas, B thinks that they are not. If this is correct, since both can think that something is/isn't a banana, both of them utilize a banana concept. 4 It seems, then, that when we attribute concepts to people -e.g., concepts of bananas, jars, and boats -central and peripheral cases play different roles. For example, we think of two people as having thoughts about, and therefore concepts of, bananas, even if their concepts have peripherally different extensions. More generally, when we attribute a concept of F to people (on the basis of how they classify things): (i) we require a specific extension in central cases of Fs and not-Fs, so that central cases of Fs fall within the extension and central cases of non-Fs fall outside of it; but (ii) we tolerate differences in the extension with regard to peripheral cases of Fs. That is, peripheral cases can fall either within or outside the extension, and it still counts as a F concept. Thus, two concepts with peripherally different contents -different in such a way that the concepts have different extensions regarding some peripheral cases of bananas -can both be banana concepts, as long as their contents are such that they have the same (and the right) extension in central cases of bananas and non-bananas. 5 If this is correct, it is misleading to talk of the banana concept (or the concept of bananas), since there are several banana concepts with slightly different extensions.
We can note here that some philosophers have thought that concepts with different extensions can be F concepts for reasons independent of considerations about open texture. In discussing Hirsch's "Quantifier Variance Thesis" -i.e., the view in metametaphysics that there are different existence concepts none of which is privileged (see. e.g., Hirsch 2011) -Matti Eklund writes: The quantifier variance thesis speaks of different existence concepts. This invites questions about how concepts are individuated. But the thesis gets its bite from the associated claim that some purported entities may 'exist' in one sense of 'exist' but not another: that the different concepts can differ in extension. So as far as the talk of concepts in a variance thesis are concerned, we can think of concepts as being different exactly when they have different extensions. (Eklund, 2020, p. 189) The idea here, then, is structurally identical to the point I have made above: even though two concepts have different extensions, they may both count as e.g., existence concepts, or more generally, F concepts.
Turning to beliefs, consider two people who believe e.g., that bananas are yellow. If they have banana concepts with peripherally different contents (so that they have different extensions in at least some possible worlds), concepts they employ when forming their beliefs, then their beliefs are beliefs in different propositions. (Given the assumption that two propositions are different if they have different truth-values some situations, a speaker is free to assert Pa and free to assert ¬Pa, without offending against the meanings of the terms, or against any other rule of language use. Unsettled entails open. The rules of language use, as they are fixed by what we say and do, allow someone to go either way. Let us call this the opentexture thesis." (p. 10) On my view, the sense in which we are free to go either way in a peripheral case of F, is this: irrespectively of which way you go (i.e., how you are disposed to classify) this will not make us refrain from attributing a concept of F to you, or think of you as conceptually incompetent. 5 This is not to say anything about which conceptual changes might occur in the future. If mananas become prevalent, perhaps there will be a pressure towards a conceptual choice and for some reason mananas come to be treated as clear members in the extension of 'banana'; and those who do not classify mananas as members will be treated as conceptually incompetent. That is, open textures can become closed. This is consistent with the present claim, however: the claim that, as we presently classify concepts, both concepts that include and concepts that don't include mananas in their extension can count as banana concepts. in at least some possible worlds. I'll discuss the suggestion that we drop this assumption in the very end of this paper.) So they have beliefs that we report with the same that-clause (they both believe that bananas are yellow), even though they believe different propositions. This challenges Unique Propositionalism about that-clauses in attitude ascriptions, since it questions that we can correctly claim "A believes that bananas are yellow" only if A stands in the belief-relation to some specific proposition designated by the that-clause. Rather, what we should say is that such a belief ascription is correct only if A stands in the belief-relation to some proposition in a certain set of propositions (a set of propositions that share truth-values in all possible central cases of the involved concepts).

Clarifications
A few clarifications will help explain the view just arrived at. First, to be clear, the proposed view does not deny that, when we believe that x is F, we always do that through believing some specific proposition. But which specific proposition that is can vary between different instances of belief that x is F. In this sense, belief that x is F is multiply realizable by beliefs with different propositional contents. 6 Second, we should note that the view proposed is not a contextualist view: it is not the view that, for a particular utterance of "S believes that x is a banana", the context will decide which member of the set of merely peripherally different propositions S needs to believe for the utterance to be true. The view is better labelled a "leewayview": Believers may have any of the peripherally differing concepts as their concept 6 There are affinities between the view I propose and the view proposed by Buchanan (2010). Buchanan's main thesis here (which he argues for based on considerations orthogonal to the considerations in play in this paper) is that the semantic value of the that-clause in attributions like, "Chet says/means that George W. Bush lives in Washington" is typically not a proposition but a proposition type (Ψ). Roughly, speakers don't mean propositions, they mean proposition types. But Buchanan also suggests (but without developing in detail) the idea that a belief report with the same that clause ("S beliefs that George W. Bush lives in Washington") is true if, and only if, S believes some proposition of that type (362 − 63). This part of Buchanan's view is similar to my suggestion that belief reports in general are true only if S believes some proposition in a certain set (rather than: only if S believes some specific proposition). Further similarities and differences between the two views will have to be explored elsewhere.The view proposed is also in certain respects similar to Shier's (1996) and Bach's (1997) views. They both argue that in order for a belief report to be correct, there is not one specific proposition that the believer needs to believe. A belief report does not specify a specific proposition that the believer believes, but only describes or characterizes what she believes. There are various different things (according to Shier these "things" are propositions, Bach hesitates to call them that) the believer can believe that makes this characterization -the belief report -correct. This is similar to my view. But their ideas are in crucial respects different. They hold that the that-clause of e.g., "Mary believes that grass is green" denotes a specific proposition, the proposition that grass is green, and that the belief report is true only if what Mary believes is some more specific/fine-grained proposition (or "thing") which requires the truth of that proposition. So for the belief report to be correct, Mary either needs to believe the proposition denoted by the that-clause, or believe something that entails the truth of that proposition. This is different from the idea that I suggest, and their views do not help accommodate peripheral concept variation. On the idea that I suggest, there is no specific proposition denoted by the that-clause. Rather, there are various peripherally different propositions (that need not be more specific versions of each other, or some other proposition) such that, if the believer believes any of these propositions, the belief report is correct. of bananas, and then they may use that concept to form the belief that x is a banana. Accordingly, we can call it: Peripheral Leeway Propositionalism. 7 Third, this view implies that it is not the case that, when A reports "B believes that x is a banana", the that-clause of the report designates the same proposition that A would express if using the embedded sentence ("x is a banana") to make a standalone assertion. Nor does it designate the proposition that A would believe if she believed that x is a banana. For both of these propositions might (peripherally) differ from the proposition that B happens to believe when she believes that x is a banana. Indeed: the that-clause does not designate any specific proposition at all. Instead, the thatclause serves to specify which set of propositions, one of the members of which B must believe for the report to be correct.
We can also put this point in terms of concepts. When A reports "B believes that x is a banana", A does not use her own banana concept to pick out what it is B believes; nor does she use the conventional meaning of 'banana' (which might, perhaps, be indeterminate). For according to the view I propose, the report can be correct even if Bs belief involves neither of these concepts, but instead involves a concept peripherally different from them. In the belief-report, 'banana' instead serves to specify that the concept involved in the belief is a member of one specific set of (merely peripherally different) concepts; namely the set of concepts that one may associate with the term 'banana', without being disqualified as an incompetent banana-thinker. (Or, in other words: the set of banana concepts.) To report that "B believes that x is a banana" is to report that B believes that x falls under one such concept. In effect, 'banana', as used in the that-clause, therefore also serves to specify the set of propositions which is such that B has to stand in the belief relation to one of its members, for the report to be true.
Fourth, it should be emphasized that this is not a complete view of belief ascriptions. This holds for the view I have argued against as well: Unique Propositionalism tells us that for each belief report of the form "A believes that x is F" there is a unique proposition such that the report is true only if A believes that proposition. But it does not specify sufficient conditions for the truth of the report: i.e., it does not say that believing the proposition always makes the report true. One reason one might want to resist saying that, has to do with Frege/Kripke puzzles. Even if "Lois Lane believes that Clark Kent can fly" and "Lois Lane believes that superman can fly" require that Lois Lane believes the same proposition, some have thought that in certain situations (e.g., situations where Lois Lane has seen superman fly, but doesn't know that Clark Kent is superman) the former report can be false even if the latter is true. If so, believing that Clark Kent can fly requires something more than believing that specific proposition, e.g., believing it under a certain mode of presentation. Peripheral Leeway Propositionalism rejects Unique Propositionalism, since it tells us that, for each belief report of the form "A believes that x is F" there is a set of propositions such that 7 Note that this view is different from the "multiple proposition" view suggested by Hodgson (2018). Hodgson argues that since many sentences of the form "x is F" express multiple propositions when used by a speaker (since meaning underdetermines what is said), when such a sentence serves as an embedded clause in a belief report, the report expresses that the believer stands in the belief relation to each of these propositions. Peripheral Leeqway Propositionalism instead holds that the believer stands in the belief relation to one member of a set of propositions.
(i) the report can be true in virtue of A believing any member of this set, and (ii) it is true only if A believes one of these members. But this is not to say that A's believing a proposition from the set is sufficient for the report to be correct. For all Peripheral Leeway Propositionalism states, for some reports and some situations, it might be the case that the proposition also has to be believed under a certain mode of presentation for the report to be correct.

Possible and actual variation
The case above against Unique Propositionalism is independent of the actual existence of peripheral variation. What matters is, first, that it is possible for different people to have concepts and beliefs with peripherally different contents -this is the Possible Peripheral Concept Variation claim from Sect. 3 -and second, that, to the extent that people do have beliefs with such peripherally different contents, they are attributed with the same that-clause.
However, if we also accept -as I suggested in Sect. 3 that we should -the Actual Peripheral Concept Variation claim, we can draw a further more substantial conclusion. According to this claim, peripheral variation in fundamental classification is evidence of peripheral variation in the contents of the concepts in question. Thus, since we can expect there to be peripheral classificatory variation (Sect. 2), we can expect there to actually be peripheral variation in the contents of concepts. This means that we can expect different people to have peripherally different F concepts -e.g., peripherally different concepts of bananas -and also that people's beliefs attributed with the same that-clause -e.g., their beliefs that bananas are yellow -have peripherally different contents.
We can add one element to the argument above. The argument presented thus far has emphasized that, as we actually attribute beliefs and concepts, we discount differences in peripheral cases. To this we can add, that doing so also makes sense in relation to our communicative interests. Thinking (and talking) of people as believing that x and not believing that y, etc., lets us keep track of (and communicate about) people's cognitive commitments. We said above that peripheral cases in relation to concept C are cases such that whether they are included or not in the extension of C has been of little or no significance to people (e.g., because they have not been sufficiently prevalent or interesting to sort out or communicate about). Consequently, there has been no pressure towards consensus with regard to inclusion or inclusion of such cases, and we therefore can expect variance. But it also follows that there has been no reason to be bothered by facts about whether a person (or the extension of her concept) includes or excludes such cases, when we classify her as having or not having the concept in question: her cognitive commitments with regard to these cases will be irrelevant to us. Not only that: there is good reason not to care about extension in peripheral cases when we attribute F concepts and beliefs that x is F. Compare to colour individuation and colour concepts: for everyday purposes, having extremely fine-grained colour concepts -one for each nuance -would, given our limited psychologies, be more harmful than beneficial for efficient communication about which things have what colour. Similarly, for everyday purposes, individuating concepts and beliefs in a way that is sensitive to fine-grained differences in extension in periph-eral cases, would be disadvantageous for communication about who believes what. Thus, there is good reason for why we treat people as, e.g., (competent) bananathinkers whether or not they (and the extension of their concepts) include mananas.

Belief-individuation: Believing the same thing
Usually, we speak of two people who have beliefs that can be correctly reported with the same that-clause -e.g., both believe that x is a banana (or more generally, that x is F) -as believing the same thing. But according to Peripheral Leeway Propositionalism, two beliefs may be attributable with the same that-clause without being beliefs in the same proposition. This means that, in one clear sense they don't believe the same thing -they believe different propositions -even though their beliefs are attributable with the same that-clause. Should we take this to imply that our main everyday way of individuating beliefs is wrong? Not necessarily. As long as we are clear that we are using two distinct senses of "believing the same thing", we can speak of people believing the same thing in one sense (they both believe, say, that x is a banana), while believing different things in another sense (they believe different propositions).
It may be objected that the first sense is not really to believe the same thing (in any sense), but very similar things (very similar propositions). 8 We can return to the colour analogy to see that this objection rests on a too rigid view on talk of "sameness". We have no problem saying that two green objects are of the same colour -green -even though one is slightly more light green. They have the same colour -green -in virtue of having one of the many more precise colours the having of which counts as having the colour green. Likewise, two people who believe that x is a banana, can be said to believe the same thing -that x is a banana -in virtue of believing something (a proposition) the believing of which counts as believing that x is a banana. (I am not saying here that people in general, or philosophers, think that their talk of "believing the same thing" should be understood in this way, and think that believing different propositions can count as believing the same thing. Rather, I propose it as the best construal of such talk, given that Peripheral Leeway Propositionalism is correct.) To continue the analogy, someone could ask whether the two objects have the exact same colour, and one may respond: no, they are slightly different shades of green. And if we ask whether the two persons believe the exact same thing, one may respond (if one has come to accept Peripheral Leeway Propositionalism): no, they believe slightly different propositions. Asking for such precisifications is to initiate a move between two senses of sameness, none of which is more correct. (Having said this, it should also be noted that the main point of this paper -i.e., that beliefs in peripherally different propositions can be attributed with the same thatclause -does not rest on these views on what "believing the same thing" should be taken to mean. One could instead conclude that, if Peripheral Leeway Propositionalism is correct, then people who we ordinarily take to believe the same thing, often do not really believe the same thing in any sense.)

Objections and replies
I end the paper by considering four objections to argument presented:

The proposed view entails that, if A believes that x is a banana, and B believes
that x is not a banana, they don't necessarily disagree (both can believe something true). This actually strikes me as the right result. When A and B have different views about, say, whether mananas are bananas, they plausibly don't have a real factual disagreement, but merely a verbal one. How about disagreements that plausibly are real? Suppose A believes there are bananas in the kitchen, and B believes there are no bananas in the kitchen. Given the different extensions of their banana concepts, it is not the case that they disagree in the sense that at most one of their beliefs can be true. This is admittedly counterintuitive. However, in most ordinary contexts we are warranted to suppose that their difference doesn't depend on the inclusion/exclusion of certain peripheral cases, and thus that there is an underlying disagreement: A believes that there are core-bananas in the kitchen and B believes that there isn't. At most one of those beliefs is true.

Does the proposed view imply that one person can have two concepts of bananas
with peripherally different contents, and thus simultaneously truly believe that x is a banana (employing one concept), and truly believe that x is not a banana (employing the other concept)? No. The following view seems to capture our ways of attributing concepts: A person who is disposed to include mananas as bananas, may of course have the less inclusive concept as well, but this will not be her concept of bananas, since it is not the concept that she connects to the joint practice of classifying certain things as bananas (i.e., things to which the word 'bananas' applies). How about a person who has both concepts but connects none of them to the joint practice? 9 I'm actually not sure. Maybe the best we can do is to acknowledge that it is indeterminate whether she has a concept of bananas and which concept this is. If she applies one of the concepts but not the other to some object, then there might not be a determinate answer to whether it is correct to report that she believes that the object is a banana. (This may strike some as counterintuitive -of course there is a determinate answer to whether she believes it or not, they might say. But it strikes me as plausible that when we start to investigate the implications for belief attributions of underexplored phenomena like conceptual indeterminacy (open texture), unexpected upshots like this are likely to show up.) 3. The suggested view clashes with how we are used to talk of concepts and propositions. Just as it implies that there is no such thing as the concept of boats (or the boat concept), it also implies that there is no such thing as the proposition that x is a boat. For different propositions -propositions with different truth-values in possible peripheral cases -can count as propositions that x is a boat. But we can explain why it makes sense that we, in every day talk, lump together the different propositions that hide under the label "proposition that x is a boat" as "the proposition that x is a boat", and the different concepts that hide under the 9 Thanks to an anonymous referee for pointing to this possibility.
label "concept of boat" as "the concept of boats". For they don't differ in ways that are salient in relation to our communicative interests. So it is a useful way of speaking. But it is also something that good philosophy should correct -just like physics correct everyday thinking and talk in terms of absolute simultaneity and mass -while allowing that we continue to talk loosely about the concept of boats, and the proposition that x is a boat, without much communicative loss outside of philosophy (and in most philosophical discussions). 4. Can we not save Unique Propositionalism through finding a way of holding that there is one unique concept of e.g., bananas? A first alternative would be to say that there is e.g., one concept of bananas, but one without a determinate extension in peripheral cases. This, some might hold, is the lesson to draw from the fact that we tolerate different classifications of peripheral cases. It seems to me that this blurs the difference between two distinct questions, however: "which extension does S's concept C have?", and "Does C count as a F concept?" As noted in Sect. 3, we want to say that it is possible to have, and employ in ones' thinking, a concept that includes ordinary bananas and has a definite extension regarding mananas, e.g., includes them. If this is possible, and if we are disposed to think of someone who utilizes such a concept as having banana-thoughts (as I have argued above), then we should not conclude that there is only one indeterminate banana concept, but should rather conclude that there are several peripherally different banana concepts. (As discussed in Sect. 3, perhaps we should still conclude that there is peripheral indeterminacy in our language. If there is a shared conventional meaning of the relevant terms in question, that obviously cannot vary between individual speakers, and then the best conclusion from peripheral variance in use might sometimes be that the shared meaning involves peripheral indeterminacy.) A second alternative would be to stop individuating concepts (and propositions) by extensions, so that A's concept C and B's concept C' can be (instances of) the same concept, even though they have different extensions. Further, we could say that propositions are built up by these concepts that are not individuated by extensions, with the result that A and B could believe the same proposition when they believe that x is a banana, even if the propositions they believe diverge in truth-value relative to some possible worlds. If we think of concepts and propositions in this way, we can save Unique Propositionalism, thereby avoiding the conclusion that beliefs in different propositions can be attributable with the same that-clause. 10 There are problems with this suggestion, however. First, it is unclear which way of individuating concepts and propositions that would render this result. Second, much of the theoretical job that we want propositions to do, depends on two propositions being identical only if they share truth-values (in all possible worlds 11 ). For example, if we have two beliefs (or assertions), we want to say that one of them can be false and the other true (in the same possible 10 Thanks to an anonymous referee for this suggestion. 11 Or, if we want to express this more carefully: in all circumstances of evaluation. On some views, some propositions can vary in truth-value relative times, or standards of taste or other parameters within worlds (see e.g., MacFarlane 2014, ch. 4). We can ignore this complication here, however. world) only if they are beliefs in (or assertions of) different propositions. With this new proposal we cannot say that; for A and B believe the same proposition, but A's belief may be true while B's is false. So, either we have no explanation of the difference in truth-value of the beliefs, or we need to invent some new (or rather: old) technical notion: A and B may not believe different propositions, but they believe different old-school-propositions: i.e., what we thought of as "propositions" before we accepted the novel view that beliefs in the same proposition can differ in truth-value (in the same possible world). We now end up with the view that two beliefs can be attributed with same that-clause even if they are beliefs in different old-school-propositions. It seems that this is not really different from my proposal, but the same view in new terminological clothes, and therefore nothing has been gained. In any case, more would need to be said to develop this into a serious contender to Peripheral Leeway Propositionalism.

Conclusions
With that, I conclude the argument for the claim that, for many that-clauses, there is no unique proposition such that, the correctness of a belief ascription of the form "S believes that x is F" requires that S believes that proposition. Rather, there is a set of propositions such that (i) the report can be true in virtue of A believing any member of this set, and (ii) it is true only if A believes one of these members. I hope to have shown at least that this view -Peripheral Leeway Propositionalism -should be taken seriously, and that Unique Propositionalism -implied by the Relational Analysiscannot simply be assumed to be correct. 12 Funding Open access funding provided by University of Gothenburg.
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