Abstract
The question of the analyticity of Hume’s Principle (HP) is central to the neo-logicist project. We take on this question with respect to Frege’s definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege’s definition of number, it isn’t analytic, and if HP is taken to be primitive there is only a very narrow range of circumstances where it might be taken to be analytic. The latter discussion also sheds some light on the connections between the Bad Company and Caesar objections.
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Notes
See footnote 4 below.
Frege’s formulation of HP is developed in Frege (1980, §§63–73).
Some authors prefer ‘NxFx = NxGx...’ or something similar, treating the cardinality operator as a variable-binding term-forming operator. In the presence of full second-order comprehension, which is part of the background logic, the two formulations are equivalent (Burgess 2005, §2.6).
Wright introduces this question as part of his response to the objection that HP is not analytic on the grounds that not every concept has a number (e.g. is self-identical). In short, his point is that a restriction is needed such that substitutions for ‘F’ are restricted to those concepts such that the question, “How many Fs are there? makes sense—or at least has a determinate answer...” (Wright 1999, p. 12), e.g. count nouns and expressions for sortal concepts.
For the history of the use of these different conceptions among mathematicians dating back to the medieval period see especially (Mancosu 2016, chapter 3).
We realize that more work will have to be done to compare the cardinalities of disjoint collections, but as we will see, that is possible. See also the references in footnote 5.
How (or whether) one is to account for the analyticity of general logical laws is left unspecified.
It is important here that “primitive truths” are understood as definite propositions that are among the general logical laws.
Frege makes this point explicitly stating, “To study such conceptions is not useless by any means; but it is to leave the ground of intuition entirely behind. If we do make use of intuition even here, as an aid, it is still the same old intuition of Euclidean space, the only one whose structures we can intuit.” (Frege 1980, §14).
It is worth mentioning that one may view what Frege says in (1980, §14) to be in tension with the statements he makes in Frege (1980, §3). In particular, if within a theory T, there is a proof of a statement \(\varphi \) from the axioms (i.e. primitive truths) of T and each step of the proof appeals (only) to general logical laws and (admissible) definitions, but there is a consistent theory \(T^*\) in which \(\lnot \varphi \) does not entail a contradiction, should \(\varphi \) be considered analytic? If one privileges the narrower criterion in Frege (1980, §3), one might be inclined to answer: yes. Yet, if one privileges the wider criterion in Frege (1980, §14), one might be inclined to answer: no. We will understand the relationship between the narrower and the wider criteria as follows. The wider §14 criterion is to be used when assessing statements that do not admit of proof (in a theory): the primitive truths of a theory (to ensure that the primitive truths of the theory are logical) and definitions (to ensure that the definiendum clearly has the same Sinn as the definiens in all domains—see, footnote 16 below). Now, if there is a proof in a theory \(T^\prime \), of a statement \(\varphi \), that begins with primitive truths of logic (i.e. axioms which satisfy the §14 criterion) and each step of that proof appeals only to general logical laws and definitions (i.e. definitions which satisfy Frege’s §14 criterion), then \(\varphi \) satisfies (3\(_F\)) and is analytic according to Frege’s criteria in §3 and §14 (since \(\varphi \) is provable by general logical laws from primitive truths and definitions that hold with respect to every subject matter, \(\varphi \) holds with respect to every subject matter). See Schirn (2017, §3) for an in depth discussion of the relevant passages.
Henceforth by “analytic” we mean “analytic in Frege’s sense” unless it is expressly noted.
Strictly speaking, Frege took HP to depend on the amended version of (N) given in the Grundgesetze; however, the differences between the two versions do not have a significant bearing on our argument.
As anonymous reviewer rightly pointed out, the epistemic status of (N) is irrelevant from a neo-logicist perspective, as neo-logicists take HP (or similar principles) as primitive. However, as we will get into nearer the end of the paper, this is not only an interesting application of Mancosu’s (2016) use of the part-whole principle in looking at topics related to neo-logicism, but also may provide insight both into Frege’s program and neo-logicist conceptions of analyticity.
In other words, if, with respect to a special science \(\varSigma \), (N) satisfies (A), then the sense (Sinn) of ‘N(F)’ is not the same as the sense (Sinn) of ‘Eq(F)’ within \(\varSigma \). Hence, (N) does not apply to any subject matter whatsoever (i.e. the sense of ‘N(F)’ is not always the same as the sense of ‘Eq(F)’). We understand Frege to be explicitly rejecting such definitions as admissible when he writes, “[T]he laws of logic presuppose concepts with sharp boundaries...Accordingly all conditional definitions, and any procedure of piecemeal definition, must be rejected. Every symbol must be completely defined at a stroke so that, as we say, it acquires a Bedeutung” (Frege 1997b, § 65).
It is perhaps more accurate to describe NSA as a branch of mathematical logic as a certain amount of mathematical logic is integral to its presentation.
We are assuming that the sphere of NSA constitutes the sphere of some special science. This is plausible as it is a coherent theory about a particular domain, whose reliance on sets precludes it from belonging to pure logic.
Strictly, using sets here (as opposed to concepts) is a departure from Fregean terminology, however, doing so will make things simpler.
Those familiar with the construction of the hypernatural numbers, or who wish to take or take our word on the matter can feel free to skip this section, and similarly for the following section where we define numerosities.
If there is no finite set in an ultrafilter, it is non-principal.
For a more general discussion of ultrafilters, see Komjáth and Totik (2008).
To illustrate with a toy example, if \(\langle s_n\rangle =\langle 0,3,4\rangle \) and \(\langle r_n\rangle =\langle 1,3,4\rangle \), then \(\langle s_n\rangle \approx _\mathcal {U}\langle r_n\rangle \) iff \(\{2,3\}\in \mathcal {U}\). Keep in mind this example is meant merely as an illustration. \(\mathcal {U}\) does not contain any finite sets. For an actual example, see Sect. 4.3 below.
It follows from the manner in which Frege defines cardinal numbers (Frege 1980, §§77–86) that, for any set of natural numbers F, the value of Eq(F) is equal to the standard cardinality of F.
We are following the lead of Benci and Di Nasso (2003, p. 52) and Wenmackers and Horsten (2013, p. 48) in calling this number, \(\alpha \). Although, strictly speaking, Wenmackers and Horsten do not call \(num(\mathbb {N}_0)=\alpha \). Rather they stipulate that \(num(\mathbb {N}_1)=\alpha \). Following their stipulation, \(num(\mathbb {N}_0)\) should be \(\alpha +1\). We’ve chosen to overlook this detail to keep things simple.
See Wenmackers and Horsten (2013, p. 50) for a brief explanation of how addition on \(\mathbb {^*N}_0\) is defined.
Frege uses \(\infty _1\) rather than \(\aleph _0\).
The result that (N) can be consistently denied within NSA means that (N) is either synthetic or false. However, the result is not, in itself, sufficient to decide between the syntheticity or falsity of (N). For this reason we take no stand on this issue (likewise, for HP).
BLV: \(\forall F \forall G (\epsilon F = \epsilon G \leftrightarrow \forall x (Fx \equiv Gx))\).
This is not to say that it would be impossible to find a consistent theory of extensions, the objects of which could be used in the formulation of (N), and HP derived therefrom. However, our current best theory of extensions is Zermelo–Fraenkel set theory. If we then take extensions to be governed by such a system, we would have to show that the axioms of ZF are analytic. And if we can do that we can declare victory for logicism without having to worry about (N) or HP, other than to perhaps pick out which sets to call the natural numbers.
It strikes us that Fregean analyticity as we have represented it here differs enough from the standard Kantian or Quinean accounts of analyticity that it may provide such a status even if we may not consider it to be a species of analyticity proper. Discussion of this possibility would take us too far afield, but the second author hopes to address it in the near future.
Again, the abstraction operator is sometimes presented as a variable-binding, term-forming operator. See footnote 3 above. Additionally, the variables bound by the abstraction operators can be of any order or arity, though in general it’s APs involving first level concepts that are of particular interest.
It’s straightforward to construct a model of HP (see e.g. Boolos 1998, Chapter 9). Additionally, HP plus full axiomatic second-order logic, known as Frege arithmetic (FA) is equiconsistent with PA\(^2\).
In fact results reported by Cook (2017) and Walsh and Ebels-Duggan (2015) might give us reason to think that HP is special from certain mathematical perspectives. As those results have little to do with definitions of number however, we don’t think it likely that HP’s admissibility as a definition would follow.
See Mancosu (2015, §9) for an overview as well as a discussion of some issues related to NSA and Caesar.
An AP, \(\varGamma \), is inflationary if it entails there be more \(\varGamma \)-abstracts than there were objects in the original domain. BLV is inflationary; HP is inflationary on finite but not infinite domains.
See Cook and Ebert (2005), who calls this the ‘C-R problem, for more discussion in the context of neo-logicism.
Again see Mancosu (2015, §7), where he also uses the word ‘insidious’. We liked it enough to use it and add this footnote.
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Acknowledgements
Thanks to Richard Zach, Nicole Wyatt, and Dave Liebesman for helpful comments on (many) earlier drafts. Thanks as well to anonymous reviewers whose comments improved this paper. Finally, thanks to audiences in Calgary for their attention and questions.
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Darnell, E., Thomas-Bolduc, A. Is Hume’s Principle analytic?. Synthese 198, 169–185 (2021). https://doi.org/10.1007/s11229-018-01988-8
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DOI: https://doi.org/10.1007/s11229-018-01988-8