Searching for Deep Disagreement in Logic: The Case of Dialetheism


According to Fogelin’s account of deep disagreements, disputes caused by a clash in framework propositions are necessarily rationally irresolvable. Fogelin’s thesis is a claim about real-life, and not purely hypothetical, arguments: there are such disagreements, and they are incapable of rational resolution. Surprisingly then, few attempts have been made to find such disputes in order to test Fogelin’s thesis. This paper aims to rectify that failure. Firstly, it clarifies Fogelin’s concept of deep disagreement and shows there are several different breeds of such disagreements. Thus, to fully assess Fogelin’s thesis, it will be necessary to seek out cases of each breed to evaluate their rational irresolvability. Secondly, it begins this task by looking at a significant debate within the logical literature over the truth of contradictions. We demonstrate that, while the debate exemplifies a breed of deep disagreement, the parties involved can supply one another with rationally compelling reasons.

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  1. 1.

    There are some exceptions here, notably Adams (2005), which highlights the practical difficulties parties face in recognising when a disagreement is deep.

  2. 2.

    Though he also refers to such propositions as constituting the “scaffolding” of our thoughts (Wittgenstein 1975, § 211), and our “picture of the world” producing a background for reasons (Wittgenstein 1975, § 94).

  3. 3.

    Indeed, Wittgenstein (1975, §§ 611–3) seems to consider this possibility.

  4. 4.

    While Fogelin’s discussion was undoubtedly inspired by Wittgenstein’s talk of “hinges”, there are significant complications facing anyone attempting to identify Wittgenstein’s “hinges” with Fogelin’s framework propositions, such as Wittgenstein (1975, § 110 and § 204) ultimately talking in terms of acting, rather than accepting fundamental propositions. Unfortunately, a consideration of these complications is beyond this paper’s scope. See Godden and Brenner (2010) for a detailed discussion of Fogelin’s thesis in light of Wittgenstein’s claims in On Certainty.

  5. 5.

    I thank an anonymous referee for pushing me on this point.

  6. 6.

    Indeed, if we were so required to sketch out the conditions under which one party would be rationally required to accept their opponent’s position, then not only would the target deep disagreements fail the test of rational resolvability, but so would most (if not all) complex debates between research programmes.

  7. 7.

    Note that, on the assumption that the parties agree on the truth of the auxiliary propositions, a distant clash can always be transformed into a weak clash by making one of the framework propositions the target proposition under dispute. In reality, of course, this does not always happen.

  8. 8.

    Indeed, while at first sight it may seem obvious that strong clashes are the proper cases of deep disagreement that Fogelin had in mind, once we take Fogelin’s own example of the abortion debate seriously this type of clash seems the least relevant!

  9. 9.

    For precedent that we ought to treat partial clashes as constituting deep disagreements, see Davson-Galle (1992).

  10. 10.

    Of course, dialetheists and classical logicians are not the only parties involved in the debate over the self-referential paradoxes—gappy logicians disagree with both. However, this isn’t a concern for us here. The debate between dialetheists and classical logicians itself constitutes a disagreement, and thus is sufficient to test Fogelin’s Rational Resistance. But, we are confident that what we have to say about this debate is transferrable to the wider debate including other non-classical logicians.

  11. 11.

    Given that untruth entails falsity in classical semantics: ¬T(ϕ¬) → F(ϕ¬).

  12. 12.

    While these conditions constitute the core requirements of a dialetheist’s logic, they are not the only criteria used by dialetheists to evaluate a logic’s appropriateness. For example, dialetheists often wish for their logic to respect the normal semantics for the Boolean connectives. For this reason, Priest’s LP is preferred to da Costa’s (1974) Ci(1 ≤ i ≤ ω) logics. See Martin (2018).

  13. 13.

    One can also provide a truth-functional semantics for LP, however there are good reasons for preferring a relational semantics; see Martin (2018).

  14. 14.

    With the material conditional AB defined in the usual way, ¬AB.

  15. 15.

    In fact, LP has the same set of theorems as classical logic (Priest 2006b, p. 76).

  16. 16.

    For countermodels to explosion, modus ponens and disjunctive syllogism let 1 and 0 but only 0, and for reductio let 1 and 0 but only 1.

  17. 17.

    See, for example, Kirkham’s (1992, Ch. 9) criticisms of various non-dialetheic solutions along these lines.


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I am grateful to colleagues at the University of Bergen for their comments on a draft of this paper, particularly Pål Antonsen, Ole Hjortland, Tore Øgaard, and Sindre Søderstrøm. I would also like to thank three anonymous referees for TOPOI, and the editors of this issue, for their detailed comments on a previous version of this paper.


Research for this paper was supported by a European Research Council (ERC) Grant (No: 797507), under the European Union’s Horizon 2020 research and innovation programme.

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Correspondence to Ben Martin.

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Martin, B. Searching for Deep Disagreement in Logic: The Case of Dialetheism. Topoi (2019).

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  • Deep disagreements
  • Fogelin
  • Framework propositions
  • Logical disputes
  • Dialetheism
  • Contradictions