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Fragmentation, metalinguistic ignorance, and logical omniscience

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To reconcile the standard possible worlds model of knowledge with the intuition that ordinary agents fall far short of logical omniscience, a Stalnakerian strategy appeals to two components. The first is the idea that mathematical and logical knowledge is at bottom metalinguistic knowledge. The second is the idea that non-ideal minds are often fragmented. In this paper, we investigate this Stalnakerian reconciliation strategy and argue, ultimately, that it fails. We are not the first to complain about the Stalnakerian strategy. But in contrast to existing complaints, we want to cause trouble for the strategy directly on its home turf. That is, we will advance our objection while granting both the plausibility of the fragmentation component—save for an extreme version of it—and that of the metalinguistic component. Once our central objection to the Stalnakerian strategy is in place, we will show how it negatively affects Adam Elga and Augustín Rayo’s recent attempt to apply the Stalnakerian strategy in the context of Bayesian decision theory.

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  1. The focus here is on the causal part of the causal-pragmatic picture. The pragmatic part has to do with how certain representational states get to count as beliefs—rather than, say, imaginings or hopes—in virtue of being closely connected with desires and actions.

  2. For critical discussions of the Stalnakerian strategy, see for example Borgoni et al. (2021), Jago (2014a), Field (2001), Forbes (1989), Robbins (2004), and Stanley (2010). For alternative proposals on how to deal with the problem of logical omniscience, see, for example, Berto and Jago (2019), Bjerring and Skipper (2019), Jago (2014a), and Dogramaci (2018).

  3. For motivation of this thought, see Stalnaker (1990), and for a critique, see Field (2001).

  4. To be sure, there are more questions that one could ask about the nature of fragments; for some of these questions, see Borgoni et al. (2021). But in line with most other people in the philosophical literature appealing to fragments, we will settle with the rough characterization above. Technically, what matters for our argument is that fragments correspond to sets of possible worlds that are complete and deductively closed, and everyone in the Stalnakerian camp would agree with this.

  5. Here are two examples that illustrate how existing critiques of the Stalnakerian strategy—in contrast to our approach—directly attack either the metalinguistic component or the fragmentation component. In Jago (2014a), the strategy is criticized for its characterization of mathematical and logical knowledge as, essentially, linguistic knowledge. For instance, failing to spot a particular winning strategy in chess seems hardly to reduce to a pure lack of linguistic knowledge. In Field (2001), it is argued that neither metalinguistic ignorance nor fragmentation can help account for the kind of behavior that is typically displayed when agents believe the impossible. For example, while the belief that a 60 degree angle can be trisected and the belief that a specific map requires more than four colors to color are both impossible, the types of behavior associated with these beliefs can be very different for mathematically untrained agents: one type involves the use of a compass while the other involves the use of color pencils. Yet, as Field argues, “[i]t does not seem […] that one can plausibly explain this difference in behavior in terms of different attitudes to sentences like ‘I will trisect a 60 degree angle’; and invoking ‘compartmentalized belief’ does not seem substantially more promising” (Field 2001, p. 103).

  6. By holding that an agent understands ‘S’, we intend to rule out cases such as one in which an agent does not grasp ‘S’ but still knows, solely on the basis of testimony, that it expresses a necessary truth.

  7. Note: questioning extreme fragmentation is still compatible with engaging Stalnakerians on their home turf. For, as we argue later, Stalnakerians will want to capture the thought that agents are logically competent, but if extreme fragmentation is permitted, they will not be able to do so.

  8. Thanks to an anonymous reviewer for suggesting how a proponent of extreme fragmentation might respond to some of the issues we raise for them.

  9. In addition to Lewis (1982), Stalnaker (1991), and Elga and Rayo (2022), the list of people who accept minimal standards on rationality is long; for some of the people on that long list, see Smithies et al. (2022).

  10. There is a trivial sense in which agents are minimally rational—and far more—in the Stalnakerian framework. After all, in this framework, agents are logically omniscient with respect to each fragment: they know every logical truth and every logical consequence of what they know in every fragment. But for our purposes, the interesting notion of minimal rationality is one that relates to mathematical and logical knowledge and reasoning (understood metalinguistically).

  11. We will return to Elga and Rayo (2022) on logically non-omniscient but logically competent agents in Sect. 4. For further motivation of why fragmentation-based strategies do not sit well with everything-goes views on rationality, see Kinderman and Onofri (2021).

  12. In addition to Elga and Rayo (2022), there are many in the literature who share the thought that models of ordinary, non-omniscient agents should be able to capture the intuitive sense in which such agents remain capable of performing logically competent deductions; see, for instance, Bjerring and Skipper (2019, 2020), Cherniak (1986), Jago (2014a, b), Smets and Solaki (2018), Solaki (2021), and Weirich (2004).

  13. For similar considerations of what a proper solution to the problem of logical omniscience requires, see Bjerring and Skipper (2019), Jago (2014a, b), and Weirich (2004).

  14. For earlier statements of this result, see Bjerring and Schwarz (2017).

  15. Cf. Elga and Rayo (2022), p. 718.

  16. The table is taken from Elga and Rayo (2022), p. 719.

  17. Elga & Rayo (2022) explicitly deny that “an agent is confident in a claim if and only if some row of her access table assigns high probability to that claim”, for the biconditional “entails that a puzzlist with the access table [above] is confident not just in the claim that dreamt is a word of English spelled D-R-E-A-M-T, but also in the negation of that claim” (p. 733).

  18. For further details, see Elga and Rayo’s (2022) discussion of how they can “do justice to Frege’s logical and semantic competence while respecting his lack of logical and semantic omniscience” (p. 720).

  19. Clearly, the specific claims here are dependent on the characterization of an obvious entailment as involving only a single application of a basic inference rule. Yet, the general point is not. For by chaining together obvious entailments—whether characterized as above or not—we eventually arrive at a non-obvious one. That is, there will have to be a specific step (or steps) in the deduction where we move from an obvious to a non-obvious entailment.

  20. Cf. Elga & Rayo (2022), p. 720.

  21. Note also: when Stalnaker in an earlier quote talks about agents having knowledge of the “contingent fact that each axiom sentence expresses a necessary truth (however the descriptive terms are interpreted)”, he seems to grant that agents can have schema knowledge of axioms and inference rules (Stalnaker 1987, p. 76; our italics).

  22. Instead of capturing schema knowledge of inference rules in terms of knowledge of contingent propositions such as (R1), one might suggest that we capture it by appealing to a kind of rule-following behavior that generally—but not always—respects the inferential patterns suggested by the rule in question. For instance, as suggested by a referee for this journal, we might say that an agent has schema knowledge of modus ponens when, generally, the following obtains: when ‘A’ is known relative to some fragment F1, and when ‘If A, then B’ is known relative to some other fragment F2, then ‘B’ is known relative to some fragment F3. Setting aside whether such an account of schema knowledge would avoid the problems that we have isolated for Elga and Rayo—we doubt that it will—it requires a story about how different fragments combine and interact to generate the knowledge that ‘B’ in F3 as a result of applying modus ponens on what is already known in F1 and F2. But note: even if this story is available, it still does nothing to suggest that agents cannot also have schema knowledge of inference rules in the way we have suggested. And that is strictly all we need to make our case against Elga and Rayo.


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For valuable feedback and comments, we would like to thank two anonymous referees for this journal, the participants in the LOGOS Epistemology Workshop II at the University of Barcelona, the faculty at the Department of Philosophy and Religious Studies at Utrecht University, as well as members of the NUS work-in-progress reading group, including Zach Barnett, Ben Blumson, Michael Pelczar, Lavinia Picollo, Abelard Podgorski, Neil Sinhababu, Mattias Skipper, and Daniel Waxman.


Jens Christian Bjerring’s work on this article was supported by a Carlsberg Foundation Young Researcher Fellowship Grant (grant number CF20-0257).

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Bjerring, J.C., Tang, W.H. Fragmentation, metalinguistic ignorance, and logical omniscience. Philos Stud 180, 2129–2151 (2023).

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