Abstract
In this paper we offer a new solution to the old paradox of nothingness. This new solution develops in two steps. The first step consists in showing how to resolve the contradiction generated by the notion of nothingness by claiming that the contradiction shows the indefinite extensibility of the concept of object. The second step consists in showing that, having accepted the idea of indefinite extensibility, we can have absolute generality without the emergence of the contradiction connected to the absolute notion of nothingness. The idea of indefinite extensibility allows us to have our cake (absolute generality) and to eat it too (avoid commitment to a contradictory notion of nothingness).
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Notes
The first work where this paradox has emerged is Parmenides’s poem On Nature (see Coxon 2009). After Parmenides the problem has been constantly studied in the history of philosophy: here I limit myself to refer to Plato’s Sophiest (see Cooper 1997) and to the opening of Hegel’s Science of Logic (see Hegel 1969).
I will give an example in footnote 14 and another in footnote 15.
The best presentation and defense of dynamic abstraction is Linnebo 2018.
It is tempting to speak of an indefinitely extensible totality, but this is just loose talk to say that a concept corresponds to an indefinitely extensible sequence of definite totalities. In particular, if a totality is considered to be a set or a plurality, then – properly speaking – no totality is extensible, because adding an element to a set or to a plurality results in a new set and a new plurality. What is properly indefinitely extensible is a concept, and as a consequence, the sequence of its extensions.
Here the word ‘plurality’ must be taken as in plural logic, i.e. as a plural term denoting several things at once. A plurality is not a further object with regard to its elements; rather, it is simply these objects. The same term ‘plurality’ is just loose talk to be substituted with plural terms. Instead of saying ‘a plurality of dogs’ one should rather say ‘the dogs’. For an overview of plural logic, see Linnebo 2017; for a more in-depth study, see Oliver and Smiley 2013a, 2013b. Linnebo (2018, pp. 57–58) also argues in favor of the identification of definite totalities with pluralities of objects.
Of course, I am not presenting such conceptions of indefinite extensibility as interpretations of what Dummett had in mind. I am not interested here in Dummett’s exegesis, so I shall not concern myself with the question of whether the conception of indefinite extensibility presented here faithfully represents his views.
This was George Boolos’s position concerning the concept of set: there is no universal set, but there is the plurality of all sets.
This principle is known as modal collapse and has been defended by Linnebo 2010.
More on this can be found in Linnebo 2013, 2018; see also Fine 2006. I explained my preferred way of interpreting such a form of generality in Costantini 2018, to which I refer the reader for a more complete presentation of it. In the literature there is a great deal of agreement with the idea that the most appropriate modal logical system for such a modality is S4.2.
See also Casati and Fujikawa 2017, §1.
Voltolini 2015 explicitly interprets it as a non-existent object. Moreover, he argues that it is an inconsistent object, and so one can accept it only in the case one accepts an ontology of impossibilia.
Perhaps the most famous of Priest’s examples is the following: consider the sentence ‘God created the universe out of nothing’, and translate it by means of a quantifier: there is nothing from which God created the universe. But this is true also in the case in which the universe is eternal and God never created it. Therefore, the quantificational translation gives us a different sentence that can have a different truth-value with regard to the original one. This should show that in the original sentence ‘nothing’ is not quantifier, but a true noun phrase. However, this particular example has been challenged by Sgaravatti and Spolaore 2018, who provide an interpretation of the sentence without assuming ‘nothing’ as a singular term. As such, the example cannot be considered conclusive.
Consider the following pun: ‘Nothing is bigger than the universe; my hand is bigger than nothing, so my hand is bigger than the universe’. The pun is built on the different logical functions of the term nothing, respectively as a quantifier phrase and as a noun phrase. Of course, the derivation (my hand is bigger than the universe) is fallacious, precisely because of this equivocation, but the fact that the pun works is a clear clue that both uses of ‘nothing’ (as a quantifier and a noun phrase) are legitimate, at least in natural language.
Russell even defined as proper names the demonstrative exactly because they always have a referent.
I have in mind something like the subtraction argument formulated, for instance, by Baldwin 1996. However, Baldwin’s formulation presupposed that reality is composed of a finite number of objects in order to perform the subtraction. I will relax such a condition, and I will assume that, ideally, the subtraction can happen even if the number of objects that composes reality is infinite. In any case, nothing of what I say strongly depends on such an argument; the subtraction argument can just be used to have an intuitive idea of what ‘nothingness’ should denote.
Of course, whether or not the quantifiers used in these definitions are ontologically neutral will depend on the meta-ontology one prefers.
That there is such an object is guaranteed by premise 1.
This is clearly the reason why we dropped Priest’s characterization of nothingness as the mereological fusion of the empty set. Since there is only one empty set, then there will be only one empty fusion, and so only one nothingness. On this point one might complain that we are actually cheating. Since our characterization of nothingness differs from Priest’s, it is not a solution to a paradox to show that the paradoxical argument fails for something else. However, as can easily be appreciated by looking at the derivation of the paradox above, Priest’s identification of nothingness with the fusion of the empty set does not play any role in that derivation. As such, our proposal is a direct solution to the paradox above, which is essentially the same paradox considered by Priest. Moreover, if one wants to block our solution by assuming that nothingness is the fusion of the empty set, one should argue for such a claim, otherwise the assumption would beg the question against the present proposal.
Abstractionist approaches are well-known from the literature in philosophy of mathematics. They go back to Frege’s attempt to ground arithmetic on second-order logic and an impredicative version of Basic Law V. Principles like Basic Law V are said ‘abstraction principles’ and they have the form of an equivalence between an identity statement and an equivalence relation over a certain domain. Schematically, #α = # β ↔ α ≈ β. For a general introduction to the abstractionist’s approach, I suggest Cook 2009. Linnebo 2018 provides an account of what he calls ‘dynamic abstraction’, i.e. the idea that the passage from the equivalence relation to the identity statement in an abstraction principle (i.e. the passage from right to left) may lead to an expansion of the domain over which the equivalence relation has been defined.
Of course, ID-N and V should also be understood in modal terms, because they are applicable however you can expand the universe of discourse.
As explained in footnote 11, the modality is captured by S4.2.
I have to thank the first reviewer for the first objection, and the second reviewer for the second objection.
For the ‘something from nothing’-transformation see Shiffer 1996.
A reviewer pointed out that there are more examples of occurrences of nothingness as a noun-phrase of those discussed in the present paper. For instance, ‘Sartre and Heidegger both wrote on nothing’, and ‘Nothing is something that puzzled many philosophers since Parmenides’ are both sentences where ‘nothing’ is a noun-phrase.
See Aquinas 1962, I, q. 65, a. 3. For Aquinas, the creation is not ex nihilo causae efficientis et finalis (the efficient and final cause is God himself).
Such a formula means that the creation is from the nothingness of the creature (ex nihilo sui) and the nothingness of creature’s matter (et subiecti). Again, see Aquinas 1962, I, q. 65.
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I would like to thanks Matteo Plebani and two anonymous referees for this journal for helpful comments on previous versions of this paper.
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Costantini, F. Extending Everything with Nothing. Philosophia 48, 1413–1436 (2020). https://doi.org/10.1007/s11406-019-00144-x
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DOI: https://doi.org/10.1007/s11406-019-00144-x