Abstract
Within the framework of Meinongianism, nothingness turns out to have contradictory features—it seems to be an object and not. In this paper, we explore two different kinds of Meinongian accounts of nothingness. The first one is the consistent account, which rejects the contradiction of nothingness, while the second one is the inconsistent account, which accepts the contradiction of nothingness. First of all, after showing that the consistent account of nothingness defended by Jacquette (Humana Mente 25:95–118, 2015) fails, we express some concerns on the general possibility of consistently characterizing nothingness. Secondly, starting from Priest’s inconsistent characterization of nothingness (Priest, One, Oxford University Press, Oxford, 2014a; Australas J Log 11(2):146–158, 2014b), we will introduce our own inconsistent account. The key idea of our account is to take nothingness as the complement of the totality. Finally, we will make formal sense of it by constructing an inconsistent mereological system, which is the development of the paraconsistent mereology proposed by Weber and Cotnoir (Synthese 192:1267–1294, 2015).
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Notes
The situation faced here is similar to the liar paradox. Indeed, paraconsistentists have argued that to take the liar sentence as an exception for the T-scheme is ad hoc. See Priest (2006).
See Givone (2003).
We see what this distinction is in Sect. 3.1.
More precisely, “every thought intends a first-person (internal to the thoughts of the thinking subject) transparently ostensible object, directly transcribable from the grammatical structure of the thought’s linguistic expression as what the thought is about” (Jacquette 2013, p. 99).
In the passage where he proposes this, Jacquette claims that the property N is characterized as “being intendable (positive) and having no constitutive properties and consequently nonexistent (negative)” (Jacquette 2015, p. 212). This is quite misleading, because, according to Jacquette himself, the property N is a constitutive property of N-nothing. What he should say is that N is characterized as being intendable (positive) and having no constitutive properties other than N and consequently nonexistent (negative).
Then, what is N at all? The best way to understand the property N might be to take it as a theoretical posit to ensure the intendability of N-nothing and the fact that it is indeed an object of thought.
If we assume the one-to-one correspondence between sets of nuclear/constitutive properties and objects (as Parsons 1980), (2) is incompatible with CP too. To see this, consider the singleton of the nuclear property of being red. The alleged one-to-one relation ensures that some object has the property of being red, which is the only nuclear property it has. Thus, it doesn’t have N.
This is so, if one accepts Parson’s definition of the property being possible, according to which x is possible iff it is possible for some existent object has all nuclear properties which x actually has. (cf. Parsons 1980, p. 101)
Discussing whether N-nothing has any nuclear property, Jacquette suggests the idea that every object must have at least one nuclear property (cf. Jacquette 2015, pp. 204, 212). If this is true, then the null object is not an object in this sense, since it has no nuclear property. However, the argument he gives for this idea is not promising. He seems to draw this idea from the fact that every object must be distinguished from any other object. He thinks that an object is distinguished from any other in virtue of the nuclear properties it has. But this is wrong. The null object is distinguished from any other object in virtue of the fact that, on the one hand, it lacks any nuclear property, and, on the other hand, any other object has at least one nuclear property.
As we will see in footnote 12, Richard Sylvan suggests the idea that nothingness is something which is characterized as not being an object.
Another inconsistent account of nothingness is proposed by Richard Routley/Sylvan. To begin with, he directly characterizes nothingness as what is not an object. Consistently with this idea, in his unpublished work, Routley/Sylvan claims that “nothingness is not an item” (Sylvan, Box23). In Routley/Sylvan’s terminology, ‘item’ is a synonym of ‘object’. He is also explicit in claiming that nothingness (also called, in his terminology, Bugboo) is contradictory and that such a contradiction must be accepted. He writes: “Here is a new object, Bugboo. Suppose that Bugboo is not an item. Then, Bugboo is also an item [albeit an entirely nondescript one]” (Sylvan, Box219). As we have seen in the previous section, Routley/Sylvan takes being an object as being describable. Since nothingness is characterized as not being an object and this is a description of nothingness, nothingness is describable, and therefore, it is an object. Unfortunately, Routley/Sylvan’s position is only sketched and a more detailed account of nothingness is not available due to his premature death.
Two remarks for clarification. First of all, the reader may wonder why we move from Meinongianism to mereology. However, this switch is, by no means, to change the subject. The mereology defended in what follows makes sense only if we metaphysically work in the framework of Meinongianism: Indeed, Meinongianism, in particular, CP and IT, motivates us to take nothingness as a contradictory object. Secondly, in our mereology, nothingness has many contradictory features. Since we work in the framework of modal Meinongianism, there are two available options: Either the inconsistent features of nothingness hold in the actual world or only in some impossible worlds other than the actual one. Note also that only the first option commits us to dialetheism.
In addition to (3) and (5), Priest (2014c, p. 439) also seems to hold that self-identity is the same property as objecthood. We do not commit ourselves to this strong metaphysical claim. What we use is the equivalence specified in (3) and (5).
For a preliminary attempt to add C to PM, see Casati and Fujikawa (2015).
We owe this point to an anonymous referee.
It is worthwhile emphasizing that this is not the only purpose of presenting a formal model. Indeed, our mereological system helps us to investigate some features of nothingness that would have been more difficult to be detected without any formal tools. For instance, through our formal system of mereology, we can precisely characterize the emptiness of nothingness (see Sect. 5.3).
Even though we have a good reason to assume C, someone may still think that the non-totalness of the totality is too weird. However, the non-totalness of the totality is not as weird as it appears at the first sight. See discussion concerning the emptiness of the complement of the totality in Sect. 5.3.
We owe this point to an anonymous referee.
For a related discussion, see footnote 33.
Since the main aim of the present paper is a metaphysical one, our use of the technical tools is instrumental: This means that the formal system introduced in the present work is meant to clarify, support and give a better understanding of the philosophical account of nothingness. With this attitude we have decided to engage with Weber and Cotnoir’s mereology, but we do not commit ourselves to the view that this is the only possible way of doing so.
This is because \(x \le y\) iff \(x \sqcup y = y\), where \(\sqcup \) is the binary sum operation defined as follows: \(x \sqcup y := lub[z | z=x \vee z =y]\) (see Weber and Cotnoir 2015, p. 1281).
One may wonder whether every true sentence about \(lub[x|x=x]\) has a true negation. Fortunately, this is not the case. For example, in the model \(M_{{\mathrm{PM}}^{\mathbf{CB }}}\) in Appendix 3, \(lub[x | x = x] = lub[x | x = x]\) is true only and \(lub[x | x = x] \ne lub[x | x = x]\) is simply false.
The readers may expect that (14a) reflects the non-objecthood of nothingness. However, this is not the case, in the sense that the truth of (14a) needs not depend on the fact that nothingness is not an object. Indeed, even though it is a theorem of \(\hbox {PM}^{\mathbf{C }}\) that \(\overline{lub[x | x = x]} \ne \overline{lub[x | x = x]}\), it is not its theorem that \(\overline{lub[x | x = x]} \not \le lub[x | x = x]\) (this is shown by the first model in Appendix 3). (14a) is just the immediate corollary of C.
We owe this to an anonymous referee.
Take any x. Suppose that \(x \le \overline{lub[x|x=x]}\). From (11b), \(\overline{lub[x|x=x]} \le x\). From this and PM2, it follows that \(x = \overline{lub[x|x=x]}\).
As suggested in footnote 18, the argument presented here helps us to understand what C exactly means. C says that something is not a part of the totality. Thus, something is outside of the totality. However, as shown by the present argument, the alleged thing which is outside of the totality should be understood as the shape of the totality, not as an additional bump attached to it.
So far is a metaphysical justification and explanation of non-substantiality of the overlap due to the complement of the totality. It would be good to reflect it in our formal system as well: In particular, it would be good to formally define the notions of substantial overlap and non-substantial overlap. Unfortunately, it is not easy to do this. Let’s see why. For the sake of simplicity, let us assume that only the complement of the totality is empty. Then, a natural idea is that x substantially overlaps with y iff they have a common part other than the complement of the totality. For example, following Priest (2014a, p. 98, n. 38), one may define it as (18).
This definition doesn’t work in our theory, since the complement of the totality is not self-identical. The fact that any two objects have it as a common part, together with this, establishes the truth of the right-hand side of the definition. One may hold that the notion of substantial overlap will be appropriately defined by appealing to a sentential operator which expresses the ‘consistency’ of a formula. Echoing da Costa (1974), let use \(\circ \) as such a consistency operator. \(\circ A\) means that A is consistent, where the consistency of A is understood as ‘ruling out’ the contradiction A and \(\lnot A\) in an appropriate way. Then, one may define substantial overlap as follows:
Two objects overlap with each other iff they have a common part and it is consistently the case that the common part is not the complement of the totality. Since the complement of the totality is not only non-self-identical but also self-identical, two objects which have the complement of the totality as their unique common part do not satisfy the right-hand side of (19). The problem of this definition is that incorporating the consistency operator will make the whole theory trivial: By using the consistency operator, we can define the absurdity constant \(\bot \) as \(\circ A \wedge A \wedge \lnot A\), since for any A and B, \(\circ A \wedge A \wedge \lnot A \vdash B\) holds; and adding an absurdity constant to \(\hbox {PM}^{\mathbf{C }}\) will make it trivial (see Sect. 5.2).
We owe this point to an anonymous referee.
A proof goes as follows: Since (13a) holds, \(\overline{lub[x | x = x]} \sqcup lub[x | x = x] \le lub[x | x = x]\) holds. Given that for any x and y\(x \le x \sqcup y\) (cf. Weber and Cotnoir 2015, p. 1281), \(lub[x | x = x] \le \overline{lub[x | x = x]} \sqcup lub[x | x = x] \). Therefore, \(\overline{lub[x | x = x]} \sqcup lub[x | x = x] = lub[x | x = x]\).
\(lub[ x | x \ne x ]\) is defined in \(\hbox {PM}^{\mathbf{C }}\), since \(\overline{lub[x | x = x]} \ne \overline{lub[x | x = x]}\).
PM itself doesn’t ensure that something is not self-identical. Thus, to incorporate the sum of non-self-identicals in PM, we need to add some assumption. C entails that something is non-self-identical, but of course, this is not the one and only choice. For example, to obtain the sum of non-self-identicals, one may add the following assumption to PM.
One may wonder whether the sum of non-self-identicals is the complement of the totality in PM+(23). In fact, the model in Appendix 3 shows that, even in PM+(23), the sum of non-self-identicals needs not be the complement of the totality, since it satisfies all axioms of PM and (23), but doesn’t validate \(lub[x | x \ne x] = \overline{lub[x | x =x]}\).
On this topic, some initial work has been done by Priest (2014a).
Some initial work on inconsistent grounding theories has been already done by Priest (Draft), Casati (2017; Forthcoming).
Here we appeal to argument by cases, which is valid in DKQ. See Weber and Cotnoir (2015, p. 1289).
For simplicity, we use ‘0’ and ‘1’ as terms in the theory for 0 and 1 respectively.
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Acknowledgements
We would like to thank the following people for their helpful feedback and comments to some of the early drafts of the present paper. Francesco Berto, Ricki Bliss, Aaron Cotnoir, Yasuo Deguchi, Jay Garfield, Hitoshi Omori, Keiichi Oyamada, Graham Priest, Steward Shapiro, Zach Weber.
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Casati, F., Fujikawa, N. Nothingness, Meinongianism and inconsistent mereology. Synthese 196, 3739–3772 (2019). https://doi.org/10.1007/s11229-017-1619-1
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DOI: https://doi.org/10.1007/s11229-017-1619-1