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Should Anti-Realists be Anti-Realists About Anti-Realism?

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Abstract

On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful examination, the strongest Dummettian arguments for anti-realism of this sort fail to secure intuitionistic logic as the single, correct logic for anti-realist discourses. Instead, antirealists are placed in a situation where they fail to be justified in asserting monism (that intuitionistic logic is the unique correct logic). Thus, antirealists seem forced either to accept pluralism (i.e. one or more intermediate logic is at least as `correct’ as intuitionistic logic–an option I take to be unattractive from the anti-realist perspective), or they must be anti-realists about the realism/anti-realism debate (and, in particular, must refuse to assert the instance of excluded middle equivalent to logical monism or logical pluralism).

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Notes

  1. Given the typical anti-realist’s rejection of the equivalence of Φ and ¬ ¬ Φ, there are additional, more complicated positions that one might take with regard to Revision and H-Monism. For example, the anti-realist might accept Revision, refuse to explicitly accept Monism, but reject the rejection of H-Monism (in other words, accept the double negation of H-Monism). This sort of subtlety will play an important role in the arguments to follow.

  2. Here we understand a logic to be a language L (i.e. any set of objects plus recursive rules specifying which sequences of objects are well-formed) plus a relation ⇒ on ℘(L) × L. For our purposes here, it does not matter whether ⇒ is specified semantically or proof-theoretically.

  3. Note that quantification over appropriate interpretations “I” has been suppressed in these formulations for the sake of readability.

  4. There are a number of interesting views worthy of the term ‘logical pluralism’ that proceed via an underlying pluralism regarding the natural language itself (Carnap 1959; Shapiro 1997), the relation within natural language being codified (Beall and Restall 2000, 2001, 2006), or the logical/non-logical divide (Tarski 1936; Varzi 2002). While such views are worthy of consideration, they are not our main target here. For more discussion of such alternative formulations of pluralism, the reader is encouraged to consult Cook (2010).

  5. I thank Aaron Cotnoir for pointing out that arguments for pluralism can be easily transformed into arguments for nihilism.

  6. There is one other Dummettian argument for intuitionistic logic, one based on the idea that truth—in particular, mathematical truth—is indefinitely extensible, and further, that the proper logic for reasoning about indefinitely extensible domains is intuitionistic. I have omitted discussion of this argument for three reasons: First, space considerations would have made it impossible to do justice to this complicated idea as well as to handle all the other complex issues involved in anti-realist arguments for logical revision; second, I find the argument itself, as presented in, e.g. Dummett (1963, 1991), to be rather obscure [although see Williamson (1998) for some helpful discussion]; and third, I have presented a distinct account of how one should reason with indefinite extensible concepts elsewhere (see Cook 2008, 2009).

  7. For an attempt along these lines (although the target logic is not traditional intuitionistic logic, but the author’s intuitionistic relevant logic), see Tennant (1997).

  8. The reader who finds this assumption particularly troubling can reformulate all the arguments given below in terms of the lattice of logics with classical logic at the top and minimal logic at the bottom. The philosophical conclusions will remain the same.

  9. Logical analyticity, in the sense defined above, is outstripped by analyticity simpliciter. For example, the traditional intuitionist will believe that:

    All bachelors are unmarried men, or all bachelors are not unmarried men.

    is an analytic truth, but (given their refusal to accept the logical truth of excluded middle) will not think it is logically analytic. As a result, it is at least open whether the anti-realist could still claim that all logical truths are analytic in this more general sense.

  10. One possibility might be for the anti-realist to deny that logical truth and logical consequence are matters of form. I will not explore this strategy here, however, since it seems to me that to abandon the idea that logical truth and logical consequence are intimately tied up with form is, for all intents and purposes, to abandon the notions of logical truth and logical consequence altogether.

  11. Note that this formulation depends on our earlier charitable assumption that intuitionistic logic is the exact logic one obtains by assuming the introduction/elimination rule pairs and nothing else. The reader uncomfortable with this assumption can reformulate all of the arguments to follow in terms of minimal logic (or whatever system one believes contains all and only the meaning constitutive pairs of inference rules).

  12. The point here is intimately connected to a commonly heard, but mistaken, claim regarding intuitionistic logic: That, for the intuitionist, the law of excluded middle is valid in decidable domains. Traditionally (and contrary to the position urged at the end of this paper), intuitionists are logical monists, and thus, if excluded middle were valid (i.e. a logical truth) in decidable discourses, it would be valid everywhere. The correct way of formulating the point is that excluded middle is true in decidable discourses (and presumably necessarily true in necessarily decidable discourses), even though it is not (in general) a logical truth. Put in these terms, the point made in the body of the text above is merely that the intuitionist is free to accept that every discourse is in principle decidable (or necessarily in principle decidable), and thus that every instance of excluded middle is true (or necessarily true). Intuitionism, merely requires that excluded middle not be a logical truth.

  13. The reader interested in intermediate logics in general, and in the various formal results regarding them reported here and below, is encouraged to consult the excellent survey by Chagrov and Zakharyaschev (1991).

  14. An anecdote: In graduate school, I read Dummett (1973) with my advisor, Stewart Shapiro. At the end of the discussion, aware of the existence of these intermediate logics, I pointed out to Shapiro that the argument did not seem to be a positive argument for intuitionistic logic, but merely a negative argument against classical logic. The response I received, typically for a defender of classical logic, was “Isn’t that bad enough?” This paper is the result of my continuing conviction that it isn’t enough, and more than a decade spent figuring out why it isn’t enough.

  15. Σ here represents generalized disjunction.

  16. Note that if we add the nth Gödel axiom:

    \( \sum {({\text{A}}_{\text{j}} \leftrightarrow {\text{A}}_{\text{k}} )} \, \quad \left( {0 \le {\text{j}} < {\text{k}} \le {\text{n}}} \right) \)

    to intuitionistic logic H, we do not obtain the Gödel logic Gn but rather the logic of bounded cardinality BCn. The Gödel logics Gn differ from the logics of bounded cardinality BCn in that the former, but not the latter, require that the semantic statuses assigned to sentences be linearly ordered.

  17. In actuality, the argument given in the text above actually may only justify the following slightly weaker constraint, which we can call the classical disjunction property:

    A logic L has the classical disjunction property iff, for any Φ and Ψ, if (Φ ∨ Ψ) is a logical truth in L, then either Φ is a logical truth in C or Ψ is a logical truth in C.

    Since neither version of the disjunction property singles out classical logic uniquely, I will ignore this complication.

  18. In fact, the logic:

    \( {\text{SKP}} = {\text{H}} + (\sim {\text{P}} \to ({\text{Q}} \vee R)) \to ((\sim {\text{P}} \to {\text{Q}}) \vee (\sim {\text{P}} \to {\text{R}})) + \, ((\sim \,\sim {\text{P}} \to {\text{P}}) \to ({\text{P}} \vee \sim {\text{P}})) \to (\sim {\text{P}} \vee \sim \,\sim {\text{P}}) \)

    that is, the result of adding both the Scott axiom and the Kreisel-Putnam axiom to intuitionistic logic, has the disjunction property (see Minari 1986).

  19. The situation is not made any better by the following fact, due to Chagrov and Zakharyaschev (1989): The class of intermediate logics with the disjunction property is undecidable.

  20. Tennant also points out that the idea that strictly classical principles are not logical truths, but instead are metaphysical theses, can be found in Dummett, although it remains undeveloped there. Tennant (1996) cites the following passage from Dummett (1975):

    … the second fundamental principle of classical semantics (is) that the condition for the truth of each sentence is, determinately either fulfilled or unfulfilled. We can regard this as a metaphysical assumption—an assumption of the existence of an objective reality independent of our knowledge. We can, equally, regard it as an assumption in the theory of meaning, namely that we succeed in conferring on our sentences a sense which renders them determinately true or false. (p. 122)

  21. A logic has the finite model property if it is the logic of some class of Kripke models, where all models in the class are finite. Kreisel-Putnam logic, for example, has the finite model property, but is not finitely bounded, since it is the logic of a class of Kripke models of arbitrarily large finite size.

  22. Proof sketch: Given any finitely bounded logic L, let n be the least upper bound on the size of models of L. Then:

    $$ \sum {({\text{A}}_{\text{j}} \leftrightarrow {\text{A}}_{\text{k}} )} \, \quad \left( {0 \le {\text{j}} < {\text{k}} \le {\text{n}}} \right) $$

    is valid in that logic, hence the logic fails to have the disjunction property.

  23. Wronsky (1973) provides examples of logics with the disjunction property but without the finite model property.

  24. One can easily see that the generalized disjunction property implies the simpler version discussed earlier by letting Bi = Ai, for each i.

  25. The fact that the class of intermediate logics with the disjunction property is undecidable (Chagrov and Zakharyaschev 1989) is of course relevant here.

  26. It is worth noting that the inconsistency of Logical Pluralism has no bearing on the claim that:

    Logical Nihilism ∨ Logical monism ∨ Logical Pluralism

    is a classical logical truth. Any disjunction of the form:

    ¬ (∃⇒)CP(⇒) ∨ (∃!⇒)CP(⇒) ∨ (∃⇒1)(∃⇒2)(⇒1 ≠ ⇒2 \( \wedge \) CP(⇒1) \( \wedge \) CP(⇒2))

    is a classical logical truth. In this case, however, since the third disjunct turns out to be inconsistent, the logical truth of the above turns out to follow from the logical truth of:

    ¬ (∃⇒)CP(⇒) ∨ (∃!⇒)CP(⇒)

    Thanks to an anonymous referee for helping to clarify this.

  27. The Kripke model given here is not meant to reflect any actual ‘modeling’ of what might be going on, but is merely a technical construction intended to demonstrate the formal consistency of Weak Logical Pluralism within these logics.

  28. I am developing a positive defense of such a view in related work currently in progress.

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Authors and Affiliations

  1. Department of Philosophy, University of Minnesota, 819 Heller Hall, 271 19th Ave S, Minneapolis, MN, 55455, USA

    Roy T. Cook

  2. Northern Institute of Philosophy, University of Aberdeen, The Old Brewery, High Street, Aberdeen, AB24 3UB, Scotland, UK

    Roy T. Cook

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Appendix

Appendix

Proof of the inconsistency of Logical Pluralism:

  1. 1.

    (∀Δ)(∀Φ)(I(Δ) » I(Φ) ↔ Δ ⇒1 Φ)  Given

  2. 2.

    (∀Δ)(∀Φ)(I(Δ) » I(Φ)↔ Δ ⇒2 Φ)  Given

  3. 3.

    (∃Δ)(∃Φ)(Δ ⇒1 Φ \( \wedge \) ¬(Δ ⇒2 Φ))  Given

  4. 4.

    P ⇒1 C \( \wedge \) ¬(P ⇒2 C) Assumption

  5. 5.

    I(P) » I(C) ↔ P ⇒1 C (1), ∀E

  6. 6.

    I(P) » I(C)  (4), (5), \( \wedge \)E, MP

  7. 7.

    I(P) » I(C) ↔ P ⇒2 C  (2), ∀E

  8. 8.

    ¬(I(P) » I(C))  (4), (7), \( \wedge \)E, MT

  9. 9.

    I(P) » I(C) \( \wedge \) ¬(I(P) » I(C))  (6), (8), \( \wedge \)I

  10. 10.

    (∃Δ)(∃Φ)(I(Δ) » I(Φ) \( \wedge \) ¬(I(Δ) » I(Φ)))  (9), ∃I

  11. 11.

    (∃Δ)(∃Φ)(I(Δ) » I(Φ) \( \wedge \) ¬(I(Δ) » I(Φ)))  (3), (4)–(10), ∃E

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Cook, R.T. Should Anti-Realists be Anti-Realists About Anti-Realism?. Erkenn 79 (Suppl 2), 233–258 (2014). https://doi.org/10.1007/s10670-013-9475-y

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