For David Chalmers et al, eds., Metametaphysics, OUP forthcoming The Metaontology of Abstraction

ion as a legitimate form of implicit definition has been authenticated. And it is this, so it may be suggested, that requires the development of satisfactory answers to (M) and (E) and related questions. If it looks as if the truth of abstraction principles may turn on substantial metaphysical hostages, or as if there are special problems about knowing that they are true, or can be stipulated to be true, this appearance needs to be disarmed before the abstractionist can expect much sympathy for his proposals. We are here content to defer to this concern. Certain of the special features of abstraction principles—in particular their role in the introduction of a conceptually novel ontology—do suggest that some special considerations need to be marshalled, not to show that particular cases are in good standing, but to shore up their assimilation to the general run of implicit definitions for abstractionist purposes. Still, there is an important qualification to enter here concerning what exactly it is that we are agreeing to try to do—for very different conceptions are possible of what it is to give a satisfactory answer to question (E) in particular; that is, to justify the thought that a good abstraction is truth-preserving, right-toleft. One such conception which we reject is, we venture, implicit in maximalism. This conception has it, in effect, that it is, in some sense, possible—something we have initially no dialectical right to discount— for any abstraction to fail right-to-left unless some relevant kind of collateral assistance is forthcoming from the metaphysical nature of the world. There are, that is to say, possible situations—in some relevant sense of ‘possible’—in which an abstraction which actually succeeds would fail, even though conceptually, at the level of explanation and the understanding thereby imparted, everything is as it is in the successful scenario. Hence in order to make good that the right-to-left transition of an otherwise good abstraction is truth-preserving, argument is needed that some relevant form of metaphysical assistance is indeed provided. This is, seemingly, the way those who have advocated maximalism as neo-Fregeanism’s best course are thinking about the issue. The ‘possible’ scenario would be one in which not everything that could exist does exist —in particular, the denoted abstracts do not exist. And the requisite collateral consideration would be that this ‘possibility’ is not a genuine possibility—because maximalism is true (and is so, presumably, as a matter of metaphysical necessity.) Although the idea is by no means as clear as one would like, we reject this felt need for some kind of collateral metaphysical assistance. The kind of justification which we acknowledge is called for is precisely justification for the thought that no such collateral assistance is necessary. There is no hostage to redeem. A (good) abstraction itself has the resources to close off the alleged (epistemic metaphysical) possibility. The justification needed is to enable — clear the obstacles away from—the recognition that the truth of the right-hand side of an instance of a good abstraction is conceptually sufficient for the truth of the left. There is no gap for metaphysics to plug, and in that sense no ‘metaontology’ to supply. This view of the matter is of course implicit in the very metaphor of content recarving. It is of the essence of abstractionism, as we understand it — but, interestingly, if we have the proposal right, it is essential to the quantifier–variantist ‘rescue’ of abstractionism as well. 23 —perhaps this modality is: epistemically [metaphysically possible]! 24 —since on the quantifier variantist line here, or so we take it, the conservation in truth conditions, right-toleft, across a good abstraction is ensured purely by so understanding the quantification in the three possible existential generalisations of the left-hand side that the right-hand side suffices for their truth at a purely conceptual level, without collateral metaphysical assumption. It is a substantial thesis is that it is possible to do this. But it is a thesis about what meanings—concepts—there are, not about the World of the metaphysician.