Abstract
Two uses of modal logic to explicate mathematics—due primarily to Hilary Putnam and Charles Parsons—are compared and contrasted. The approaches differ both technically and concerning ontology. Some reasons to push the former approach in the direction of the latter are articulated and discussed.
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- 1.
- 2.
Though as Stewart Shapiro reminds me, the descendants of Aristotle get along much better than those of Abraham. I believe this says something about the broader value of the form of reasoned debate that philosophy often illustrates.
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See Putnam (1967, p. 22). Cf. also p. 21, where Putnam locates ‘any philosophical significance’ of his modal approach in its ability to avoid the need for maximal models of Zermelo set theory.
- 4.
Strictly speaking, the empty set is not a model. But this causes no problem in practice.
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This approach differs from Parsons’s in making more explicit use of plural logic and in offering a much simpler translation from the language of ordinary set theory into the language of modal set theory. (Parsons’ translation is a combination of the double negation translation of an intuitionistic language, composed with Gödel’s translation of this language into the modal language.)
- 6.
See Linnebo (2013, Sect. 6.1).
- 7.
Parsons’ own translation is substantially more complicated, as explained in footnote 5.
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Although the modal logic of my approach is strictly weaker than the one used on the Putnam approach, the opposite holds for their theories of quantification: Roberts’ is free, while mine is not. Consequently, my logic entails the Converse Barcan Formula, which requires that the domains be non-decreasing along the accessibility relation. This means that necessity on my approach corresponds, on the Putnam approach, to necessity assuming the continued existence of all the objects there are. (More precisely, ‘\(\Box \ldots \)’ on my approach corresponds to ‘\(\forall xx \big (\forall y (y \prec xx) \rightarrow \Box (Exx \rightarrow \ldots )\big )\)’ on Putnam’s, where E is a plural existence predicate.) In light of this, (G), in my setting, makes a claim that goes beyond anything ensured by Roberts’ logic in his setting, namely that two possible extensions of the ontology can be ‘merged’ into a single common extension. This observation will be important as we go along.
- 10.
For instance, the deducibility relation can be given by the logic PFO of Linnebo (2012a) minus the plural comprehension scheme.
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See Linnebo (2013) for details. As we will see below, it is problematic to add an analogous reflection principle to Putnam’s approach.
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Thus, (EP) is closely related to the claim that for every inaccessible there is a larger inaccessible.
- 13.
A possible example would be an ontology of (first-order) facts, according to which there is the fact that there are so-and-so many things just in case there are that many things. (Thanks here to Peter Fritz.).
- 14.
See e.g. Hellman (1989, pp. 68ff).
- 15.
As Hellman (1989, pp. 42–3) is well aware.
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Of course, it is a consequence of Williamson (2013)’s necessitism —the view that necessarily everything necessarily exists—that there can be no incompossibles. Williamson’s preferred analysis of the example is that the two coexisting possible knives cannot simultaneously be ‘chunky’, which in this case comes to being realized in spacetime. However, necessitism provides no solace in our present context, as its fixed maximal domain of objects would clash with our emphasis on set theoretic open-endedness.
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I have in mind Roberts’ ‘stability’ axiom, (S), which he shows goes significantly beyond (the adaption to our present setting of) Hellman’s accumulation principle. The axiom is the Putnam translation of the logical truth \(\forall \bar{x} (\exists \bar{y} \phi \rightarrow \forall z \exists \bar{y} (z=z \wedge \phi ))\).
- 18.
As Hellman (2015) observes, this principle needs to be restricted so as to prevent reflection on Extendability, which (as I had pointed out) would result in an inconsistency. While this problem concerns the interaction of reflection and higher-order quantification, the problem described in what follows is qualitatively different, as it arises for a purely first-order sentence. It should be noted that Hellman has now given up this attempt to do reflection in a modal-structural setting (prompted in large part by Roberts (forthcoming)); see Hellman (2015, fn. 22) and Hellman (forthcoming).
- 19.
We are here relying on the plausible assumption that the knives in question aren’t ‘metaphysically shy’.
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The latter states that, if xx are no more numerous than some things yy which form a set, then xx too form a set.
- 21.
An obvious adaptation of Roberts’ Stability axiom would work from a technical point of view. So again, modal structuralists’ entitlement to this axiom needs to be assessed.
- 22.
Cf. Fine (1994).
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Might this conception nevertheless be compatible with holding fixed the non-mathematical circumstances? I cannot rule this out. If modal resources could be added that enable such carefully controlled changes, this would yield a version of the ‘two separate modal dimensions’, to be discussed below, but where both dimensions concern the circumstances, not the interpretation of the language.
- 24.
The same impression is given by Parsons (1974) , where it is clear that what shifts are interpretations, not the circumstances.
- 25.
Similar ideas are found in Kitcher (1983, ch. 6).
- 26.
Cf. Hellman (1989, Ch. 2), which also suggests semantic or linguistic means of selecting to ‘form sets’ . Allowing this kind of ‘set formation’ represents no danger of paradox provided we allow the domain to expand. Consider for instance the objects rr that are all and only the non-self-membered objects. When we introduce their set \(r = \{rr\}\), the domain expands. Paradox would follow only if we insisted—misguidedly—that r be in the original domain. See Linnebo (2010) for a detailed analysis.
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In fact, there will be some connections between the ‘dimensions’, as an interpretation will depend on the objects in terms of which it is defined. But this complication does not matter for present purposes.
- 29.
I am grateful to Peter Fritz, Jon Litland, Stewart Shapiro, and the participants in a workshop on Modality in Mathematics and the Oslo Mathematical Logic Seminar for helpful discussion of the material presented in this article. Special thanks to Geoffrey Hellman and my former doctoral student Sam Roberts, whose astute comments on earlier versions of the article have prompted to substantial improvements. This article was completed during a period of research leave enabled by the Oslo Center for Advanced Study, whose support I gratefully acknowledge.
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Linnebo, Ø. (2018). Putnam on Mathematics as Modal Logic. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_14
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