Objectual aboutness

ABSTRACT

A de re objectual subject matter is a subject matter of the form how $a_1$, $a_2$ … are, whereas a de dicto objectual subject matter is a subject matter of the form how Fs are. An objectual subject matter is either a de re or de dicto objectual subject matter. This paper provides a systematic theory of aboutness towards objectual subject matters which, if correct, illuminates both the nature of intrinsicality and the general nature of aboutness. In addition to the familiar distinction between partial and complete aboutness, I argue that there is a further orthogonal distinction between pervasive and hereditary aboutness, resulting in four ways in which a sentence, proposition or state of affairs can be about an objectual subject matter. I state and defend a number of basic principles concerning these varieties of aboutness as they apply to objectual subject matters and propose non-reductive analyses of them in terms of each other. I also propose an analysis of aboutness towards de re objectual subject matters in terms of aboutness towards de dicto objectual subject matters and put forward an attractive solution to a puzzle regarding aboutness due to Nelson Goodman.

KEYWORDS:

• Aboutness
• intrinsicality
• metaphysics

Acknowledgments

Thanks to Ethan Brauer, Simon Goldstein, Peter Hawke, Jan Plate, Lavinia Piccolo, Daniel Waxman and an anonymous referee for Inquiry for their valuable comments on this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For simplicity, I am assuming in this paper that there are abstract objects, where abstract objects include properties, states of affairs, numbers and sets. If nominalism (which is the view that there aren't any abstract objects) is independently defensible, then the theory of objectual aboutness given in this paper should be able to be modified so that it avoids commitment to abstracta. Such a modified theory, for example, would describe what sentence tokens are about and would have consequences regarding nominalistically friendly counterparts of sentences like ‘The proposition that Trump Tower is made of steel is about the subject matter of how Trump Tower is’. Since subject matters are presumably abstract objects, such a theory (when combined with nominalism) would deny that there are any subject matters and would hold that sentence tokens of the form $⌜\varphi$ is about how … are$⌝$ can be true without there being any subject matter that ϕ is about and which ‘how … are’ refers to, or expresses.

2 Topics in philosophy that aboutness has been applied to include intrinsicality (Lewis, 1983; Marshall, 2016a, 2016b), the distinction between empirical and non-empirical statements (Lewis, 1988b), relevance logic (Read, 1988), belief and knowledge (Yablo, 2014, Ch. 7; Yalcin, 2011, 2018), partial truth (Yablo, 2014, Ch. 5), the logic of knowledge (Hawke, 2018), imagination (Berto, 2018), the identity conditions of propositions and states of affairs (J. Goodman, 2019, 61; Marshall, MSa), and the semantic paradoxes (Picollo, 2020). The topic of aboutness needs to be distinguised from semantic and metasemantic questions regarding what names refer to and what propositions and states of affairs sentences express.

3 I officially use ‘iff’ to express the material biconditional. For the sake of readability, I will often be loose with respect to use and mention conventions.

4 On this way of using ‘state of affairs’, a state of affairs can fail to obtain. States of affairs, on this usage, differ from facts, which are ways things are, or, in other words, obtaining states of affairs. I am neutral here on whether (as they are understood here) propositions are identical to states of affairs.

5 More carefully, two sentences that express states of affairs represent things as being the same way iff they express the same states of affairs.

6 If one thinks that states of affairs can concern subject matters but cannot be about them, then one can simply construe ‘about’ when I apply it to states of affairs in this paper as a technical expression that means what (on its relevant use) ‘concerns’ means.

7 In order to take into account context dependence, this assumption needs to be restated as: For any sentence ϕ, for any context c, ϕ in c is about what the proposition and state of affairs ϕ expresses in c is about. I will ignore this complication and other complications related to context dependence in the following.

8 Discussions of aboutness include Ryle (1933), Putman (1958), N. Goodman (1961), Parry (1968), Lewis (1988a, 1988b), Perry (1989), Yablo (2014), Fine (2016, 2017a, 2017b, 2020) and Hawke (2018). Some of this literature focusses on de re aboutness (such as Perry, 1989), while other work focusses on general aboutness (such as Lewis, 1988a, 1988b; Yablo, 2014; Fine, 2016, 2017a, 2017b, 2020; Hawke, 2018). Further literature that is relevant to the topic of de re aboutness is the literature on the distinction between qualitative (or purely general) states of affairs and properties on the one hand, and non-qualitative (or haecceitistic) states of affairs and properties on the other, where (intuitively) a state of affairs or property is non-qualitative iff it concerns some particular entity or entities. (See, for example, Hoffmann-Kolss, 2019).

9 See N. Goodman (1961, 2).

10 Formally, ‘ϕ, ‘x’ and ‘y’ in (Part⇒Whole) are singular variables. (Part⇒Whole) should be understood as being tacitly prefixed with the quantifier expression ‘For any sentence ϕ, for any x, for any y’. Other formulas in this paper should be similarly understood as being implicitly prefixed with suitable universal quantifier expressions that bind their variables.

11 See, for example, Marshall and Weatherson (2018) and Marshall (2021).

12 In this paper, I will assume that these properties (and other similar properties, such as being red and being spherical) are intrinsic. If these examples of intrinsic properties are rejected, we can replace them with other properties.

13 Given the distinction between pervasive and hereditary aboutness, ‘x is completely about how x is’ can be understood in terms of either pervasive complete aboutness or hereditary complete aboutness. I will argue in Section 3.6, however, that these readings are equivalent in the sense that a property is intrinsic according to (1) under the first of these readings iff it is intrinsic according to (1) under the second of these readings. Given this is the case, we can arguably use (1) with either of these readings to analyse intrinsicality. Being hereditarily completely about x is what I called in Marshall (2016b) being intrinsically about x. As I argued in Marshall (2016b), there are arguably notions of intrinsicality other than that characterised by (1), and these other notions can arguably all be analysed in terms of intrinsic aboutness. If this is correct, then these other notions of intrinsicality can also be analysed in terms of objectual aboutness. For other (arguably less standard) notions of intrinsicality, see Figdor (2008, 2014) and Plate (2018).

14 For example, using the theory of objectual aboutness developed in this paper, I argue in Marshall (MSc) that intrinsicality can be given a broadly logical analysis, the provision of which has been a long standing aim of philosophers working on intrinsicality. For surveys on intrinsicality, see Marshall and Weatherson (2018) and Cameron (2009).

15 For necessitism, see, for example, Williamson (2013). For S5 quantified modal logic, see, for example, Menzel (2018). I am also assuming de re modal absolutism, according to which objects have modal properties absolutely, as opposed to merely relative to sortals or counterpart relations. For a non-absolutist account of de re modality, see Lewis (1971).

16 Examples of contentful predicates in English are ‘is red’ and ‘is a property that is not instantiated by itself’ (provided ‘property’ and ‘instantiates’ are contentful), while examples of predicates that aren't contentful in English are the nonsense word ‘is a ugibooghtw’ and ‘is as tall as the present king of France’ (when there is no present king of France and given the Fregean account of ‘the’). While some restriction on what states of affairs and properties there are arguably needs to be put in place in order to avoid certain versions of the Myhill-Russell paradox and Russell's paradox for properties, I will not assume any specific restriction on these entities here.

17 Options for understanding second order quantifiers include understanding them in terms of first order quantification over either sets, properties or linguistic entities, understanding them in terms of plural quantification, understanding them in terms of endless disjunctions and conjunctions, and understanding them primitively along the lines of Prior (1971) and Williamson (2003). For discussion of some of these options, see Williamson (2003).

18 See Lewis (1986, 59–61). For further discussion of Lewis's notion of fundamentality, see Sider (1993), Marshall (2012) and Dorr and Hawthorne (2013).

19 These assumptions regarding fundamentality aren't required by the final overall theory of objectual aboutness given in Section 3, since this final theory does not rely on the distinction between quantificational and non-quantificational states of affairs.

20 Since in this section I only discuss aboutness ascriptions to sentences expressing non-quantificational states of affairs, in this section formulas containing ‘ϕ should be taken to be tacitly prefixed with ‘For any sentence ϕ expressing a non-quantificational state of affairs’.

21 For a defence of this picture of parthood, see Sider (2007). As Sider points out, this picture does not require the highly controversial view that a whole is identical to its parts defended by Baxter (1988a, 1988b).

22 The translation is from Burnyeat (1990).

23 x is a (mereological) fusion of the Ys iff: (i) for any y that is among the Y s, y is part of x, and (ii) for any part z of x, for some y among the Y s, z and y share a part.

24 See Sider (2007, 54).

25 Sider (2007), for example, endorses the intimacy picture, but rejects both (IN) and (MN) and endorses counterpart theory. While the theory of objectual aboutness can presumably be modified so that it is compatible with such a theory of modality (if such a theory is independently acceptable), the required modification might be non-trivial to give. On the other hand, it is easy to avoid relying on the assumption of mereological necessitarianism if the theory of objectual aboutness developed in this paper is modified so that it accords with the non-intimacy picture of parthood in the way indicated above.

26 There are two other important varieties of aboutness, which might be called exact complete aboutness and exact partial aboutness, which I will not discuss in detail in this paper. In the case of singular de re aboutness ascriptions to sentences expressing non-quantificational states of affairs, these varieties of aboutness may be intuitively characterised by (EP1) and (EC1).

• EP1. ϕ is exactly partially about x iff: ϕ (either accurately or inaccurately) completely describes how x is (that is, it leaves nothing about how x is unspecified) and might also be about more than how x is.

• EC1. ϕ is exactly completely about x iff: ϕ (either accurately or inaccurately) completely describes how x is (that is, it leaves nothing about how x is unspecified) and is not about anything more than how x is.

‘Ellie is unit negatively charged’ (where Ellie has no proper parts), for example, is pervasively completely about Ellie, but it isn't exactly completely about Ellie. This sentence isn't exactly completely about Ellie since, for example, the sentence does not describe how Ellie is mass-wise, and hence it leaves something about how Ellie is unspecified. Fine (2017b, 680) arguably uses ‘exactly about’ for ‘exactly completely about’ and ‘entirely about’ for ‘exactly partially about’. He also arguably uses ‘partly about’ for ‘hereditarily partially about’ and ‘about … in its entirety’ for ‘hereditarily completely about’. As far as I am aware, Fine does not discuss either pervasive partial aboutness or pervasive complete aboutness, nor, as far as I know, have these two varieties of aboutness been discussed elsewhere.

27 x is a mereological atom iff x has no proper parts.

28 The theses endorsed by the theory of aboutness I will present in this paper are held to hold necessarily by this theory. As a result, this theory endorses, for example, not just (Hereditary-P1) and (Pervasive-P1), but also their necessitations. $⌜\left[\mathrm{\exists x}\right|\varphi ]\psi ⌝$ symbolises $⌜$For some x such that ϕ, $\psi ⌝$ and $⌜\left[\mathrm{\forall }x\right|\varphi ]\psi ⌝$ symbolises $⌜$For any x such that ϕ, $\psi ⌝$.

29 This follows from the definition of ‘fusion’ in footnote 23.

30 Goodman's response to his puzzle is to distinguish between what he calls absolute aboutness from relative aboutness, where absolute aboutness is a two-place relation between a sentence and an object, whereas relative aboutness holds between a sentence and an object relative to another sentence. According to Goodman, ‘Maine experiences cold winters’ is neither absolutely about New England, nor absolutely about Aroostook county. Instead, the sentence is only about these things relative to certain other sentences. In this respect, ‘Maine experiences cold winters’ is no different from any other sentence, since, for example, on Goodman's account, in this relative sense of aboutness, every sentence is about New England: that is, for each sentence ϕ, there is a sentence ψ such that ϕ is about New England relative to ψ. Other arguably problematic consequences of Goodman's account of aboutness are that logically equivalent sentences are absolutely about the same things and that ‘T(Biden)’ fails to be absolutely about Trump even in the case where ‘T’ expresses the property of being next to Trump. Another solution to Goodman's puzzle is given by Hawke (2018, 7) who responds to the puzzle by endorsing (Part⇒Whole) and rejecting (Whole⇒Part).

31 The predicates and names quantified over in (Atomic-1), and throughout this paper, need to be logical predicates and logical names, rather than merely expressions with the same syntactic features as predicates and names. One important feature of logical names and predicates is the following: For any logical name a referring to x, for any logical name b referring to y, for any logical predicate F, ⌜$\mathrm{Fa}$⌝ describes (either accurately or inaccurately) how x is and ⌜$\mathrm{Fa}$⌝ describes x as being the same way (or same-wise) as ⌜$\mathrm{Fb}$⌝ describes y as being.

32 x overlaps y iff x and y share a part.

33 In (HP2), (PP2), (HC2), (PC2) and other formulas it appears in, ‘the Xs’ is a plural variable that is tacitly bound by the universal quantifier expression ‘for any Xs’. As a result of this and the similar treatment of ‘ϕ’ described in footnote 20, (HP2), (PP2) and (HC2) and (PC2) should be understood in this section as being tacitly prefixed with ‘For any sentence ϕ expressing a non-quantificational state of affairs, for any Xs’. In addition to the above four varieties of aboutness, there are four further varieties of aboutness whose characterisations can be obtained from the right-hand-sides of (HP2), (PP2), (HC2) and (PC2) by replacing ‘ϕ is about how some of some of the Xs are’ and ‘ϕ is about how all of all of the Xs are’ with ‘ϕ is about how some of all of the Xs are’ (which can be understood as ‘For any x among the Xs, ϕ is about how some of x is’) and ‘ϕ is about how all of some of the Xs are’ (which can be understood as ‘For some x among the Xs, ϕ is about how all of x is’). While these further varieties of aboutness arguably have some utility, I will not discuss them here due to lack of space and because they are arguably less important than the varieties of aboutness discussed in the main text.

34 x overlaps one of the Ys iff, for some y that is among the Y s, x overlaps y. ⌜$\left[\mathrm{\exists }X\right|\varphi ]$ (ψ⌝ symbolises ⌜For some Xs such that ϕ, ψ⌝ and ⌜$\left[\mathrm{\forall }X\right|\varphi ]$(ψ)⌝ symbolises ⌜For any Xs such that ϕ, ψ⌝.

35 What I am calling an analysis is what is sometimes called a metaphysical analysis or an identification. (See, for example, Dorr 2016.) Examples of analyses are true instances of ⌜For it to be the case that ϕ is for it to be the case that $\psi$⌝ and ⌜Necessarily, for any x, for it to be the case that x is F is for it to be the case that x is $G$⌝. I assume that, if ϕ and ψ express states of affairs, then: ⌜For it to be the case that ϕ is for it to be the case that $\psi$⌝ is true iff ϕ and ψ express the same state of affairs.

36 A sentence of the form ⌜$\varphi$ iff $\psi$⌝ provides an analysis iff ⌜For it to be the case that ϕ is for it to be the case the $\psi$⌝ is true. A sentence of the form ⌜Necessarily, for any x, x is F iff x is $G$⌝ provides an analysis iff ⌜Necessarily, for any x, for it to be the case that x is F is for it to be the case that x is $G$⌝ is true.

37 If propositions are states of affairs, then a non-quantificational proposition is just a non-quantificational state of affairs. On the other hand, if a proposition is a state of affairs under a mode of presentation (or a way of grasping a state of affairs), which I take to be most important rival to the former view, then a non-quantificational proposition is a non-quantificational state of affairs under a mode of presentation.

38 Recall from footnote 12 that I am assuming that the property of being green is intrinsic. I will also assume that the properties of being an emerald and (perhaps implausibly) being a flying pig are intrinsic. As in the case of being green, we can replace these properties with other properties if one rejects this assumption.

39 Recall from footnote 3 that ‘iff’ is being used to express the material biconditional.

40 We can change the example if this is denied.

41 I am assuming that the property of being made of promethium is intrinsic. Like in other cases, if this intrinsicality assumption is rejected, then the example can be changed.

42 Let p be the property of being such that no part of one that is made of promethium is green, and let q be the property of being such that no thing wholly other than one is a thing. Another response to this third argument against (Extensionality) is to claim that, while, on the notion of intrinsicality characterised by (1), (Extensionality) is incompatible with p being intrinsic and q being non-intrinsic, this doesn't give us a strong reason to reject (Extensionality) since there are other notions of intrinsicality on which (Extensionality) is compatible with these classifications. A quick reply to this response is that: (i) p is intrinsic and q is non-intrinsic on the most fundamental notion of intrinsicality on which the property of being identical to Paris is intrinsic, and (ii) this notion of intrinsicality is the notion characterised by (1). See Marshall (2016b) and Marshall (2021) for discussion of the different varieties of intrinsicality.

43 While (Extensionality) is false for the sense of aboutness on which (25) is about how flying pigs are without there being any flying pig it is about, there might be another sense of aboutness on which (Extensionality) does hold. On this other arguably more derivative sense of aboutness, a sentence ϕ might be said to be about how x is if ϕ is about how Fs are (in the sense we are interested in) and x is an F. For example, if Em is some particular emerald, then, since ‘Every emerald is green’ is about how emeralds are (in the non-derivative sense of aboutness) and Em is an emerald, this sentence is about how Em is on this derivative sense of aboutness. In contrast with the notion of aboutness we are concerned with in this paper, on this derivative sense of aboutness, what a sentence is about is plausibly a contingent matter. It is useful to note that, assuming Russell's theory of definite descriptions, if x is the only thing that satisfies F, then, although $⌜$The F is $G⌝$ will not typically be about how x is on the notion of aboutness we are concerned with in this paper on which what a sentence is about is a necessary matter, it is about x on this alternative derivative sense of aboutness.

44 In (HP3), (PP3), (HC3) and (PC3), ‘ϕ’ is a singular variable and ‘F’ is a one-place predicate variable. These and other formulas containing these variables should be understood in this section as being tacitly prefixed with ‘For any (contentful) sentence ϕ, $\mathrm{\forall F}$’, where ‘$\mathrm{\forall F}$’ is a second order quantifier expression quantifying into one-place predicate position. As in the case of de re aboutness discussed in footnote 33, we can also characterise four further varieties of aboutness that can appear in de dicto aboutness ascriptions by employing the phrases ‘some of all of’ and ‘all of some of’ in place of ‘some of some of’ and ‘all of all of’. Consider, for example, sentence (A).

• A. All hydrogen atoms are green.

On the intended readings of the respective characterisations: (i) (A) is about how all of all hydrogen atoms are, (ii) (A) is about how some of all of H2O molecules are (but not about how all of all H2O molecules are), (iii) (A) is about how all of some atoms are (but not about how some of all atoms are), and (iv) (A) is about how some of some molecules containing oxygen are (but is neither about how some of all such molecules are nor about how all of some such molecules are). Due to lack of space I cannot discuss these further varieties of aboutness in this paper.

45 z overlaps a G iff, for some y that is a G, z overlaps y.

46 If we wish, we can make the cases more similar by replacing ‘thing that is self-identical’ with ‘thing that is wholly distinct from every non-self-identical thing’ in this argument.

47 Say that F is impossible iff it is impossible for something to be an F. One response to the above argument against the combination of (HP3$\leftarrow$PP3) and definition (${\le }^3$-Attempt2) is to adopt a view on which a sentence cannot be about how Fs are (with respect to any of the varieties of aboutness) if F is impossible, and thereby deny that (36) is pervasively partially about non-self-identical things. (For discussion of such a view, see Marshall, MSb.) At best, all that this response achieves is to shift the problem with (HP3$\leftarrow$PP3) to (PP3$\leftarrow$HP3). The reason for this is that, if no sentence is about Fs if F is impossible, then, given definition (${\le }^3$-Attempt2), since (thing that is non-self-identical ${\le }^3$ thing that is self-identical), (PP3$\leftarrow$HP3) has the false consequence that ‘Every emerald is green’ is not pervasively partially about emeralds. Hence, whether or not sentences can be about how Fs are when F is impossible, at least one of (HP3$\leftarrow$PP3) or (PP3$\leftarrow$HP3) is false if ‘${\le }^3$’ has definition (${\le }^3$-Attempt2).

48 To see that (${\ne }_W^3$) is the natural definition of the de dicto notion of being wholly distinct from (and hence is the natural de dicto analogue of (${\ne }_W^2$)), it is helpful to note that (${\ne }_W^3$) entails (A).

• A. $F{\ne }_W^{3}G$ iff: either (i) F is possible and G is impossible, (ii) F is impossible and G is possible, or (iii) F is possible, G is possible, and, necessarily, for any x that is F, for any y that is G, x is wholly distinct from y.

49 Arguments that are analogous to the first and third argument given against (Extensionality) in Section 3.1 can be given against (Intensionality). One of these arguments, for example, is the following:

Suppose, for reductio, that (Intensionality) holds. Since it is impossible for there to be a non-cubical cube and it is impossible for there to be a non-spherical sphere, non-cubical cubes are necessarily coextensive with non-spherical spheres. Since ‘No non-cubical cube is green’ is pervasively partially about non-cubical cubes, it therefore follows from (Intensionality) that this sentence is also pervasively partially about non-spherical spheres. Since ‘No non-cubical cube is green’ is not at all about non-spherical spheres, it follows that the reductio assumption is false and hence (Intensionality) does not hold.

A very quick response to this argument is that, since being a non-cubical cube and being a non-spherical sphere are both impossible, the claim that ‘No non-cubical cube is green’ is about how non-cubical cubes are and not at all about how non-spherical spheres are isn't compelling and can be reasonably rejected given the overall theoretical virtues possessed by the theory of aboutness presented in this paper. For a more developed response to this and other arguments against (Intensionality), and for further reasons to endorse the thesis, see Marshall (MSb).

50 An attractive explanation for (Nothingness) and (Separation) is provided by the theses (A) and (B): (A) if N is impossible, then, on any of the varieties of aboutness, ϕ is about how Ns are iff ϕ is intuitively about how nothing is; and, (B) if F is possible, then, on any of the varieties of aboutness, if ϕ is about how Fs are, then ϕ is intuitively about how something is. (A) entails (Nothingness), since it entails that, if $N_1$ and $N_2$ are both impossible, and ϕ is about how $N_1$s are (on at least one variety of aboutness), then ϕ is intuitively about how nothing is, which is the case iff ϕ is about how $N_2$s are (on each of the varieties of aboutness). (A) and (B) jointly entail (Separation), since, if F is possible, N is impossible and ϕ is about how Fs are (on at least one variety of aboutness), then by (B), ϕ is intuitively about how something is, and hence it is not intuitively about how nothing is, and hence, by (A), it is not about how Ns are (on any of the varieties of aboutness). For further articulation and defence of this explanation for (Nothingness) and (Separation), see Marshall (MSb).

51 If it is claimed that, if x is an emerald, then x would still be an emerald had it not existed, then we can change the example.

52 Recall from footnote 12 that the property of being spherical, like the property of being cubical, is being assumed in this paper to be intrinsic.

53 An alternative response is to suitably modify the intuitive characterisations (HC1), (PC1), (HC2) and (PC2).

54 As noted in footnote 31, the predicates and names quantified over in (Atomic-1) are logical predicates and names. (Atomic-2) is to be understood similarly.

55 As stated in footnote 12, I am assuming that these properties are intrinsic. As also stated in footnote 12, these properties can be replaced with others if this assumption is rejected.

56 This is a consequence of (A).

• A. [$\mathrm{\forall F}$][$\mathrm{\forall G}$] (if G≰³F, then [$\mathrm{\exists H}|H{\le }^{3}G$ and $H{\ne }_W^{3}F$).

To verify (A), suppose G≰³F. Then either: a) F is possible and G is impossible, b) F is impossible and G is possible, or c) F and G are both possible, and, possibly, for some x, x is G and x is wholly distinct from every F. If either (a) or (b) obtain, then $G{\ne }_W^{3}F$, and hence, since $G{\le }^{3}G$, the consequent of (A) obtains. Suppose instead that (c) obtains. Let ‘K’ abbreviate ‘is G and wholly distinct from every F’. Then K is possible, $K{\le }^{3}G$ and $K{\ne }_W^{3}F$, and hence the consequent of (A) obtains. The consequent of (A) therefore obtains in all three cases (a), (b) and (c), which establishes (A).

57 A further consideration in favour of this theory of aboutness (which I cannot discuss here) is that it is supported by what are arguably the most attractive solutions to the puzzle concerning aboutness discussed in Marshall (MSb).

58 Strictly speaking, this definition needs to be modified in order to relativise ‘it is a priori that ϕ’ to a way of being linguistically competent with ϕ. This is because two people can both be linguistically competent with a public language sentence ϕ, but have different linguistic competencies with the sentence, so that ϕ is a priori knowably true for one of them but not the other. I will ignore this complication here and assume that, for any sentence, ϕ, there is a single way of being linguistically competent with ϕ such that, necessarily, everyone linguistically competent with ϕ is linguistically competent in that way. I will also assume that, given the single type of linguistic competence required to be competent with ‘Phosphorus’ and ‘Hesperus’, it is not possible for someone to know that ‘Phosphorus is not wholly distinct from Hesperus’ is true purely on the basis of reason and their linguistic competence.

59 An objection to (79) is that the a priori theory should not be construed as endorsing (Atomic-1), since, while (Atomic-1) is a consequence of the theory of aboutness given in Sections 3.1–3.2 and 3.4, it is not a consequence of the theory obtained from those subsections by replacing ‘necessarily’ with ‘it is a priori that’ and the a priori theory is best understood as the theory so obtained. A response to this objection is that, even if this claim is correct (which might be disputed), we can modify the argument against the a priori theory by replacing ‘Phosphorus’ and ‘Hesperus’ with names a and b that refer to the same mereological atom and are such that it is not possible for someone to know $⌜a=b⌝$ is true purely on the basis of reason and their linguistic competence with this sentence. We can then replace ‘is spherical’ with a predicate F expressing a fundamental (and intrinsic) property, which would make it the case that $⌜\mathrm{Fa}⌝$ expresses a non-quantificational state of affairs. We can then argue as follows: Since b is a mereological atom and $⌜\mathrm{Fa}⌝$ is about how some of b is, $⌜\mathrm{Fa}⌝$ is about how all of b is, and hence $⌜\mathrm{Fa}⌝$ is pervasively partially about b. The argument against the a priori theory would then proceed as in the main text with ‘Phosphorus’ and ‘Hesperus’ being replaced with a and b.

60 Classical mereology is taken here to be the formal theory whose underlying logic is that of classical predicate logic (with two-place quantifier expressions and a lambda abstracta) and whose axioms consist of strong supplementation and three further axioms stating that parthood is reflexive, antisymmetric and transitive.

61 Two properties are incompatible with each other iff it is not possible for something to instantiate both of them, where the relevant notion of possibility is (as it is throughout this paper) absolute (or metaphysical) possibility.

62 This is because every logical truth (as it is defined here) is a necessary truth (given the background assumptions made in this paper, such as necessitism and the assumption that parthood is necessarily reflexive, antisymmetric and transitive).

63 I am assuming that, if an object is red* (as opposed to merely having a part that is red*), then it is fully red* (or, in other words, red* all over), and that being red* is incompatible with overlapping anything green* and vice versa.) We can modify the argument if this assumption is rejected.

64 I am assuming that ‘It is not the case that there is something that is both red* and green*’ expresses the same state of affairs as ‘¬ [$\mathrm{\exists x}|x$ is-a-thing] ([$\mathrm{\lambda v}$(v is-red* and v is-green*)](x))’, where the latter sentence can be regarded as a sentence in L where ‘is-a-thing’, ‘is-red*’ and ‘is-green*’ are simple predicates expressing the properties of being a thing, being red* and being green* respectively.

65 I am assuming that (relative to any assignment of values to ‘x’ and ‘G’) ‘x overlaps a G’ expresses the same state of affairs as ‘[$\mathrm{\exists u}|\mathrm{Gu}$][$\mathrm{\exists w}|w$ is-part-of x](w is-part-of u)’, where this sentence may be taken to be a fundamental sentence in L, and where ‘is-part-of’ is a simple predicate expressing parthood. Given this assumption, ‘[$\mathrm{\forall x}|$[$\mathrm{\lambda v}$(v is-red* and v is-green*)](x)][$\mathrm{\forall z}|z$ is-part-of x][$\mathrm{\exists u}|u$ is-red*][$\mathrm{\exists w}|w$ is-part-of z](w is-part-of u)’ can be taken to be the fundamental sentence in L expressing the state of affairs expressed by ‘For any x that is red* and green*, for any z that is part of x, z overlaps something that is red*’ that is a theorem in classical mereology.

66 Recall from footnote 28 that the theses endorsed by the theory of aboutness presented in this paper, such as (HC1$\leftarrow$HP1-${\mathrm{G}}^{\ast }$), (Atomic-1) and (Re$\leftarrow$Dicto-PP1), are held to hold necessarily by that theory.

67 This can be done using the same kind of argument that was used in Section 2.2 to show that (11) follows from (PP1⇒HP1) and (Foundation-P1)).

Additional information

Funding

Research in this paper was supported by an Early Career Scheme grant from the Research Grants Council of Hong Kong SAR, China (LU23607616).