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I am immensely grateful to the commentators for all of their thoughts and probing questions about TO. My replies will focus on one or two central points from each article. Inevitably, I need to be selective. First, for (§<) to hold, the facts that figure on the two sides of ‘ < ’ need to be distinct. I agree. Indeed, this is what distinguishes my asymmetric approach to abstraction from the more traditional symmetric approach espoused by Frege, the neo-Fregeans and Rayo, where the two sides of an abstraction principle are regarded as merely different ways to ‘carve up’ one and the same fact.2 Second, (§<) raises tricky questions about the individuation of facts. Suppose two distinct specifications are equivalent: α ∼ β . Are § α = § β and § α = § α one and the same fact? The answer will depend on how abstraction terms contribute to the representation of facts. One option is that the abstraction terms make a ‘purely referential’ contribution (in the sense of Quine (1960, 177)): an abstraction term first picks out an object, which is then slotted into the fact. On this analysis, the fact that § α = § β is identical with the fact that § α = § α . Another option is that an abstraction term contributes both a specification and the object thus specified. Then, ‘ § α = § β ’ represents a fact a concerning the two specifications α and β , namely, that they determine one and the same abstract. Since the language for which I try to secure an interpretation is an artificial one, I get to choose. I choose the former option, on the grounds of its greater simplicity (cf. TO, p. 19). Third, suppose my choice is granted, such that ‘ § α = § β ’ represents the fact that a certain object is self-identical. This raises the question of what it would be to ground a fact of self-identity. Understandably, CDP find this question problematic. There is neither room nor need for any grounding here, they think, because self-identity is ‘a kind of “logical fact”, a universal feature of reality’ (p. 12). They conclude that ‘although metaphysical grounding might sound as a natural and interesting connection it is not a viable road’ (ibid.). A better option, they propose, would be to invoke conceptual grounding, understood as ‘a relation amongst truths, which is objective, non-causal and explanatory in nature (as is the case with metaphysical grounding), but which holds in virtue of the concepts these truths contains’. (Sereni and Zanetti make a similar proposal.) While I am agnostic about the prospects of appeals to conceptual grounding, I believe CDP are too quick to give up on metaphysical grounding. (Henceforth, I omit the qualification ‘metaphysical’.) I agree with CDP that it would be obscure to ask of an existing object what more might be required to ground its self-identity. However, following much of the abstractionist tradition, I use a negative free logic where § α = § α is tantamount to the statement that § α exists.3 And I do not find it an obscure idea that α ∼ α , if true, should ground the existence of § α . More importantly, this idea figures at the heart of an account of abstraction and grounding that Louis deRosset and I have recently worked out. We start with the thought that a sufficiency statement φ ⇒ ψ records an explanatory argument (or a ‘grounding potential’, as I put it in TO). We proceed to use such arguments to derive information about grounding.4 This yields an attractive theory of abstraction and grounding, we argue, which solves CDP's problems – as well as some additional problems identified by Donaldson (2017). No sooner have deRosset and I solved some problems due to Donaldson than he points out another. The new problem concerns specifications that have only contingently the properties relevant to some form of abstraction. A clay sphere might instead have been shaped as a cube. A pen pointing north might have pointed east. How, then, might shape or direction abstraction on these specifications work? This is not an idle question. For as Donaldson observes, ‘abstractionism is attractive in large part because it offers us a general theory of types and tokens’ (p. 11). In Section 6.3.3 of TO, I discuss whether it is permissible to abstract on a relation ∼ that is only contingently a partial equivalence. I defend a negative answer on the grounds that the language in which we talk about the resulting abstracts would be too modally fragile: while α ∼ β is in fact a basis for asserting § α = § β , it might easily not have been so. Donaldson's problem is related but different. My problem is that what is a partial equivalence might fail to be so at other possible worlds. His problem is, loosely speaking, that what is a partial equivalence might fail to be so across possible worlds. In our clay example: a as it is fails to be congruent with a as it might have been. Two possible responses suggest themselves. Extending the strict policy of TO, I could prohibit the offending forms of abstraction on the grounds of excessive modal fragility. A promising alternative, though, would be to permit the abstraction but acknowledge its modal fragility. One way to do so would be to let the abstraction work, not on ordinary objects such as our piece of clay, but on objects qua shaped in a particular way. It is entirely plausible that a -qua-thus-shaped should necessitate the existence of s . On this proposal, the specifications would be precisely as modally fragile as the abstraction that they underwrite. By contrast, it would be a bad idea to respond by denying (1). For I wish to use principles of this form to explain why cardinal numbers are necessary objects, thus meeting a request by Donaldson. Letting e e be the empty plurality, we obtain □ ( e e ≈ e e → E ( 0 ) ) .5 Since the antecedent holds of (logical) necessity, it follows that □ E ( 0 ) . This, in turn, ensures that the singleton-plurality consisting of 0 only is self-equinumerous – indeed necessarily so. Since this self-equinumerosity necessitates the existence of 1, we easily derive □ E ( 1 ) . Continuing in this way, we can prove that every natural number exists necessarily. An analogous argument shows that every pure set exists necessarily. A second manifestation of Donaldson's new problem remains to be addressed. While a (necessitated) abstraction principle enables comparisons of the shapes of objects within any one possible world, its ability to underwrite comparisons across possible worlds is unclear. But such comparisons seem possible. To borrow an example from Donaldson, we can meaningfully say that a projected church would have had exactly the same shape regardless of which of two competing proposals, differing only with respect to their choice of materials, had prevailed. Moreover, I agree with him it would be unwise for abstractionists to secure such cross-world comparisons by embracing Lewisian modal realism, according to which actual and merely possible specifications exist on a par and so are straightforwardly comparable. To illustrate my alternative proposal, consider cardinal numbers. Suppose we followed Frege and the neo-Fregeans in abstracting cardinal numbers from (Fregean) concepts. This would result in modally fragile abstraction, with the associated problems discussed above. Let F be true of all and only my children. Although cardinality abstraction on F in fact yields the number 2, it might easily have given a different number. It is better, therefore, to proceed as in TO and do cardinality abstraction on pluralities, which have their numbers essentially. For example, let c c be my children. Then necessarily, if c c exist, they are two. To enable cross-world comparisons of cardinal numbers, all that remains is to find pluralities that exist across all possible worlds. Pluralities of pure sets or of cardinal numbers are natural options. These pluralities can serve as rigid ‘measuring rods’ that enable the desired comparisons. For example, while c c are equinumerous with the plurality of 0 and 1, it might easily have been the case that there are some objects who are all and only my children and who are not equinumerous with said measuring rod. In the case of shapes, our rigid measuring rods might be regions of space, as opposed to physical objects located at these regions. For a region plausibly has its shape essentially and can – modulo familiar (and serious) Leibnizian worries – be compared across possible worlds. Sereni and Zanetti (henceforth, S explain how two of – namely, and my account of objects – are as forms of On both abstract objects exist but are from facts. I for this They proceed to a it is natural to this in terms of grounding which I to metaphysical this to grounding with the of that both and I of S to of of as (p. this an abstraction principles are as of grounding, then if the relevant principle is the might fail to respond to our (ibid.). I am not in a to S about of Let instead explain why I believe the offers all the metaphysical that we both in general and in to make sense of grounding. The as I is first and a to the that there is way to the of logic to We choose which objects to out. I the objects that we out are and exist of Let first make some on I about some has to do with an answer to every question about the The not ability to to it is a to the that more than what one might have is required to a which we can then put to the question objects are there in my it this is not a we need to what kind of objects the has in On the it more than to a question. we have a an In we not to answer questions that are not with some form of there is the that on the the objects we out will be on thus out. is, had we (or not out the objects, they would not have I do not why any such should my account of to numbers or physical In both we obtain an account of what it to ground the existence of the relevant the existence of 2, for example, is in the existence of any while the existence of a is in the existence of of In neither then, do the objects depend on us for their and physical are different from objects existence is on us and our and By or could have a or a from into to be a way of the of conceptual in the of To a certain it be that principles of are by and thus that objects are in terms of the concepts that to (p. first my approach to he with some about the of numbers. There are two different abstractionist of the natural numbers. The has regarded the numbers as in the sense that they are by abstraction on as discussed above. In of TO, I an alternative abstractionist which the natural numbers as by abstraction on the equivalence of in their While I both as I that the better our actual thought and this is in large part This that should the of of In some our of this is in with account of the of (p. I this and work, such as that can us the and of the I agree with much of what he for example he the of the for our and its conceptual to any of or its In my though, I some from for example I that is to by and of some while that the might out to be too to be by any pure (p. In I I have a in my for the and of cardinal of our But I up with the cardinal within And I it as to an alternative, – this is to be we a better, with our actual we to the of numbers. In with my that numbers are of His argument with an there been the in would have (p. 12). But then a is there to believe in the existence of numbers of the existence of number (p. I the existence of numbers to make on the world, account one such namely, that there are that number (p. make this to be with It is true that of representation in our thought and language are at in by facts about But this not that the objects are thus In I to objects an where they depend on the existence of to a where they depend on by (cf. the Section and the of abstraction, the that of objects might make it to prove that objects a and exist but to whether they are the same object or Let explain this and it in discussed in the Section the ground for asserting an ‘ § α = § β ’ is the equivalence of the associated α ∼ β . In of the discussed in TO, this equivalence is within our the of two and the of two all are that In some the specifications α and β be pluralities of objects, which are not in fact within our but figure only in a account of how can be In other it some to determine whether the specifications are A case in are physical which are by of (cf. the Section two of in the relation is not to these two but on the physical to which they This the that physical from abstract objects (cf. TO, and To whether the by two of are it is not to these two By contrast, to whether the cardinal numbers by two pluralities are it is to these two for whether or not they are equinumerous is to the two pluralities in a new and different of namely, where the specifications in question are and the objects that they are where there is to whether two specifications are is a the case where is the set of the as of on a and the equivalence relation is the one that holds two in case for any of they the same (p. the resulting relation is not which it can be to whether two objects thus are identical or are we to make of The example shows that abstraction not on its an This need not be a p. objects an abstraction principle any additional (p. The equivalence α ∼ β can be more or But the § α = § β additional Moreover, we can the of the The example a relation where the equivalence of two specifications is not to the specifications but a of This the example an interesting in the case of physical the relation is But it is in with the by two arguments in TO for the of by the Section one and both arguments he instead the that there are abstract The former argument a from the of to I there is any viable such in the Section the only in to my in a The of and (p. not to the but to the of the can a first of a be to a statement of The argument a from my example of the of to physical to the of of why the principle holds in the case of physical is that holds (p. this not my My to with an easily example, to the general the interesting question of whether of a can my account of to physical but its to the of the abstract. To answer the let us the of my of in the Section it is permissible to start as if there are F This is because our language has in terms of This to physical and if a interpretation that the to F s at would be the I out in similar ways for and such an interpretation is My of this on the form of in the in which the interpretation is there is relevant the two My is while the example of physical not much to the general of the of the former to a of the is an Suppose we ‘ ∼ a ’ as is part of one of in case it is part of one of The resulting of is attractive but if there are objects that do not into For example, any two of would be I agree with not only on this option, but on the he My only is that he up the for a too be a partial on a that the principle of → ( ) . Then there is a partial on a , on which a is such ) holds for all pluralities of of of then ( ) = for every object of there are of such that ( ) = and the of Moreover, and are up to I this question of for abstraction (p. One is that we obtain a and of the existence of a is the (p. numbers and other objects have the has in the in particular given that neo-Fregeans a with into (ibid.). The then, is that my of objects on a to existence of some the (ibid.). An answer would be that TO is with what we abstraction, that is, abstraction on objects or pluralities Frege and the for example, I cardinality abstraction on pluralities of objects, not on concepts (cf. Section of these the that logic by its my account any to and instead to I to give a more interesting answer as Although TO on abstraction, I am to what we abstraction, understood as abstraction on concepts or the TO offers only some on abstraction let a more answer to I concepts as am not to a of on which such any new for the By a as I that all it to such a is to a from objects to is much the of in the by This is then by any specifications are concepts. I do not why this would the of the resulting abstract to that (or are a than (or I I find the two of on a I am to make my is a kind of that the existence of and that the existence of objects that are not (p. I to though, that my not the of a a all that it is to a from objects to But so can be The problem from the of my where as in the Section These an Suppose we try to use a φ ( ) to a Suppose an object a the to one of the But a fail to the to a My is that the of concepts be to of a Since is a I will on some of the questions it on what I to be its Donaldson, a better of the necessary existence of the objects of pure I such a in Section of these that to abstraction him to familiar abstraction such as (p. I that my use of abstraction is and only a to an I first use a of the familiar abstraction the to to the ordinary – with a that have p. In TO, I the resulting account of by of modal logic the Section I each familiar abstraction principle into of existence and which are then to the modal The resulting principles are a but there is if to the familiar abstraction principles than the of TO, I can to that namely, by an alternative of abstraction on a logic than (cf. the 11). While I that each of abstraction is that this of of such I agree. TO, my is that for that result from abstraction is and (p. (p. This to There is more for than for we the need for by abstract objects to exist on the basis of the possible existence of I to such abstracts some in TO and While I that TO offers account of such abstraction, I believe an account can be – by abstracting on existing concepts that the relevant merely possible For I with into the the concerns the of general My is and traditional I a if it can be given as a plurality and merely if On the one I argue, with the that every can be by a such On the other I argue, with the that there is an But I with traditional by that this is merely that is, a this For one the argument that I use to any given use of the of and thus on a merely the Section For there are such as that of every object, that we can make sense of any plurality, we can make sense of so as to find objects and that but given the of we can make sense of its For we might this will the form of a new and different which not only but some merely Although the are the central are in the Section the and as as logic is We can use this to a language has all and only the possible of the so The to the of logic to the a universal plurality all of its The of this would be to an e , all of the at which we the that merely (p. for an that is merely with respect to the would then be actual is, given as a with respect to the e . It is only a to use the universal plurality to a which would the with which we Two types of are possible. One is to ground and any argument that the but merely This is the in TO for that in is that some can be of natural numbers an But this is the the of which enables us to make sense of the – thus The by contrast, is not a any if this to the is not For one one can a more account of how a merely is to work, this be as a For more arguments have been – not only by but by and in of my the first the that has of a number with the can give a for the answer and – I a all numbers – only the that it in the of number to have the , can give a for the answer for other ground for is p. in that for other ground for is that is not only our but metaphysical what might ground or explain a An is true by the of a A universal by contrast, be true by its – these not all be We with across a merely to what we the and But a universal need not be true by its it can be true by its in the of the relevant concepts that the It for example, in the of number that any number have the , which would explain and . This interpretation of the and universal can be for at a particular of the of our on the basis of that is at that as we might put both types of can be true by at the This ensures that both types of are as the of not An is or true, by the of a at the relevant And this will not as the A universal is or true, by by of at the relevant as in the case of the these will not as the types of then, of at a of the and are the might is a more way to make the is well on traditional 1 are while 1 are For a to an will be as the new to a true universal be on the interpretation of the by Since a statement is true by the existence of a the of any of 1 and is as a 1 statement about This ensures that any true will true how the might There is a to be On the natural this interpretation of the for a while every the of the of only the of To there are arguments that are of a of merely on its By contrast, a way to the idea of an but merely This of an arguments by that every has a The of a universal will be what objects we proceed to and will thus that the universal remains I wish to by what I as the concerning the out in some some new and and one The relation my of sufficiency and grounding has been in Section of the I sufficiency as a form of grounding of these This the way for an alternative on the of TO, namely, in terms of By thus grounding into the statement of the its which there all to the Section of the and and of these In TO, I to the idea of abstraction the of modal An alternative has We can all use of and instead use a which is more than traditional This alternative to be more the Section and Section of these TO entirely on abstraction is, abstraction on there has recently been on abstraction is, abstraction on concepts or the and of these a better of abstraction and merely a to of To this I wish to of what only a possible namely, the idea of Section of these to Louis Rayo, and the of the at the of for and This by the of number

Political science · Logic, Reasoning, and Knowledge · Philosophy · Philosophy and History of Science · Philosophy and Theoretical Science

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