Abstract
Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities.
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Notes
See also Physics 207a6. The Physics can be found in, e.g., Aristotle (1941).
Hamkins and Linnebo (2019) consider a closely related view which they call Grothendieck potentialism; more on Hamkins and Linnebo below.
The concept of potential infinity is thus similar to Dummett’s notion of indefinite extensibility, cf. Dummett (1978).
The analogous result for intuitionistic logic is proved in Linnebo and Shapiro (2017).
Of course, this will depend to some extent on which pieces of standard mathematics we accept. While the \(^\lozenge \) interpretation allows us to make perfectly good sense of arithmetic without accepting actual infinities, perhaps accepting even the potential existence of infinite sets requires the existence of actual infinities.
As usual, \(\langle x_1,\ldots ,x_n\rangle \) denotes the number which codes the sequence \(x_1,\ldots ,x_n\), and if n and m are codes of two sequences, then \(n^\frown m\) codes the result of concatenating n and m.
I should note that Linnebo and Shapiro (2017, note 12) acknowledge that S4.2 would not be appropriate for a theory of free choice sequences. So I am not objecting to their account, but supplementing it in a way they should be comfortable with.
See Sacks (1990).
More recently, Doder et al. (2010) have proved that the first-order logic of branching time with this operator is complete for a non-standard class of models. (They also invoke an infinitary proof system for this completeness result, though the infinitary rule of their system governs an operator which I will not consider here.) Roughly, their models stand to my standard models as Henkin models of second-order logic stand to standard (or ‘full’) models of second-order logic. Somewhat more precisely, the truth of ‘inevitably \(\phi \)’ in my models depends on what happens in all branches; the models of Doder, Ognjanović, and Marković’s include a set \(\Sigma \) of branches, and the truth of ‘inevitably \(\phi \)’ in their models depends only on what happens in the branches in \(\Sigma \).
In computer science, the logic of branching time is known as Computational Tree Logic (CTL); there is also slightly more expressive extension CTL\(^*\). My operator \(\mathcal {I}\) can be expressed in these systems by the operator combination \(\mathsf {AF}\). See Gabbay et al. (1994, Ch. 4, 2000, Ch. 3) for an overview of first-order temporal logic and logics of branching time as well as for further references.
On the other hand, the results of Hodkinson et al. (2002) are more informative in that they show even the two-variable fragments of their logics to be unaxiomatizable.
Cf. the discussion of substitutional interpretation and adding constants in Button and Walsh (2018, 13–18).
The material referred to here can be found in Sacks (1990, Ch. 1).
Cf. Niebergall (2014, 256–7): “Personally, I simply have no ordinary understanding of [the phrase ‘potentially infinite’], and I do not find much help in the existing literature. \(\ldots \) The reason is that those philosophers who are interested in the theme of the potentially infinite are usually drawn to it because they regard it as desirable to avoid assumptions of infinity (i.e., of the actual infinity), yet do not want to be restricted to a mere finitist position. An assumption of merely the potentially infinite seems to be a way out of this quandary: it seems to allow you to have your cake and eat it too.”
At least, this is so under the iterative conception of set. As an anonymous referee pointed out, this is not so clear under other conceptions of set, such as the logical conception.
At least in the cases that interest us, where there are infinitely many worlds.
This suggests an interesting task for the potentialist: using a potentialist metatheory, can one describe a class of potentially infinite models that provide an adequate semantics for potentialist discourse?
Thus Kripke models play a similar role in the potentialist’s argument as representation theorems do in the nominalist project of Field (1980). Thanks to Chris Pincock for suggesting the analogy.
Thanks to Chris Pincock, Stewart Shapiro, Neil Tennant, Wes Wrigley, and two anonymous referees for helpful comments on drafts of this paper.
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Brauer, E. The Modal Logic of Potential Infinity: Branching Versus Convergent Possibilities. Erkenn 87, 2161–2179 (2022). https://doi.org/10.1007/s10670-020-00296-3
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DOI: https://doi.org/10.1007/s10670-020-00296-3