Abstract
Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the method is poorly behaved unless the background second-order logic is predicative.
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Notes
See [18] for an overview.
See [9].
The situation is analogous to that of the two theories ACA and ACA 0 of second-order arithmetic, from which our notation is inspired.
Note that the argument is independent of the particular choice of D 1. For any set D 1 of abstraction terms, the second-order parameters are chosen from the set ℘(T), which means there will be no natural way of eliminating such parameters from a given abstraction term t to yield a term in D 1.
We are here ignoring two options which strike us as desperate and unattractive. One is to exclude from our language all abstraction terms with free second-order variables. This would enable us to let the term set D 1 be the set of abstraction terms thus restricted, and let D 2 be the powerset of D 1. However, this would come at the cost of sacrificing some of the core applications of abstraction, which, as we observed in Section 2.2, require second-order quantification into the scope of abstraction operators.
A second desperate option is to operate with two (monadic) second-order domains: a narrower one, corresponding to the first of the above options, and a wider one, corresponding to the second option. Assume the language contains two sets of second-order variables. Then it is possible to let one set of second-order variables range only over the narrower domain and admit quantification with respect to such variables into the scope of abstraction operators, while not admitting this for the second set of second-order variables, which range over the wider domain. See [1, p. 119] for discussion of a closely related idea. However, this option strikes us as contrived and not much better than first constructing a term model and then adding a layer of second-order quantification ‘by hand’.
This limitation is not noted in earlier studies of grounded abstraction, such as [13].
[7] proves the theory to be exactly as strong as Q.
For instance, the ‘complex predicate’ has to be applied to the terms ε u(u = ε x.ϕ(x)) and ε u(u = ε x.ψ(x)).
For instance, this situation will arise if ϕ(x) is of the form ∃u∃v(u = v∧𝜃(u, v, x)).
Consider for instance the predicative version of Basic Law V formulated in a two-sorted language with one sort for ordinary objects and another sort reserved for extensions, where the abstraction terms on the left-hand side belong to the latter sort, while the right-hand side belongs entirely to the former. To construct a model, start with any domain D of ordinary objects, let the second-order quantifiers range over the powerset of D, let this powerset also be the domain of extensions, and finally interpret the operator ε as the identity function.
Note that the symbol ‘\(\sqsubseteq \)’ is being used ambiguously for the relation between one approximation and another that extends it, and for the relation between an approximation and an admissible refinement. In practice, this will cause no confusion.
See [1, p. 63].
Our notion of ε-rank is closely related to the notion of rank defined in [22, p. 364]. However, our definition avoids the problems afflicting the latter ([3, pp. 383–4] and [5, pp. 97–8, fn. 27]) because of the very limited nature of the term set D 1 on which our notion is defined: every member of our D 1 is a closed abstraction term. So although our definition and the theorems which it underpins can be regarded as a partial vindication of Wright’s project, it is important to realize the severity of the restrictions we impose.
[2, p. 219] sketches a closely related model, which he claims is a model for first-order Frege theory. We suspect Dummett intended the same model as ours but misdescribed it slightly by defining D n+1 as ‘the union of D n with the set of all its finite and cofinite subsets’: what we want are sets that are cofinite relative to the final model, not relative to D n . If so, then our natural term model, whose existence Dummett denies on pp. 220–22, turns out to be isomorphic to the very model that he himself invokes as a purely technical trick in order to prove a consistency result. See also [6] for a discussion of this sort of construction and its relation to Church-Oswald models of set theory. Finally, the corollary also shows that the model that arises naturally from our application of the groundedness procedure is very similar to the model employed to prove the consistency of Basic Law V with \({{\Delta }^{1}_{1}}\)-comprehension in [21].
Strictly speaking, [14] operates in a somewhat different setting, namely a first-order language with a two-place predicate ‘η’, where the intended reading of ‘x η y’ is ‘x is a member of the extension y’. However, it is straightforward to adapt his strategy to our setting. This is what we do in what follows.
See Section 2.6.
For a detailed description of the implementation of this strategy, see section 6 of [14].
We are grateful for valuable comments from Salvatore Florio, Jönne Kriener, Hannes Leitgeb, Jon Litland, Sam Roberts, Stewart Shapiro, Sean Walsh, Philip Welch, and two anonymous referees, as well as from the audiences of a Bristol workshop on ontological dependence and the mathematical logic seminar at the University of Oslo. Both authors were supported by AHRC-project Foundations of Structuralism and the second author also by an ERC Starting Grant (241098).
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Horsten, L., Linnebo, Ø. Term Models for Abstraction Principles. J Philos Logic 45, 1–23 (2016). https://doi.org/10.1007/s10992-015-9344-z
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DOI: https://doi.org/10.1007/s10992-015-9344-z