Abstract
This chapter is concerned with domain-specific alethic pluralism and domain-specific logical pluralism. If domain-specific alethic pluralism entails domain-specific logical pluralism, and vice versa, then in some sense we really only have one pluralism, not two. If, however, the two sorts of pluralism are independent of each other, then we truly have two distinct kinds of pluralism—that is, we have a plurality of pluralisms. The purpose of this chapter is to argue that domain-specific alethic pluralism does not entail domain-specific logical pluralism (contrary to arguments given by Lynch and Pedersen), nor does domain-specific logical pluralism entail domain-specific alethic pluralism, and hence we do have such a plurality of pluralisms. To accomplish this, in Sect. 2 I show how one can be a domain-specific logical pluralist while being a truth monist, and how one can be a domain-specific truth pluralist while being a logical monist. I will then, in Sect. 3, use the argument of Sect. 2 to identify the mistake in the arguments of Lynch and Pedersen. Section 4 will then further flesh out the model, distinguishing between different senses in which a domain might be epistemically constrained.
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Notes
- 1.
Further, other sorts of pluralisms—such as metaphysical pluralisms—also fall on one or the other side of the domain-specific/domain-independent divide.
- 2.
- 3.
- 4.
It is important to note that I am not arguing that we have a plurality of true pluralisms (see the previous footnote). The point is that the accounts in question—domain-specific alethic pluralism and domain-specific logical pluralism—are distinct positions, regardless of whether they are correct or not.
- 5.
I am, in what follows, going to concentrate on the argument as given in Lynch (2008, 2009). Pedersen (2014) characterizes itself as presenting a slightly more detailed and cleaned up version of Lynch’s argument (and extending the conclusions to metaphysical pluralism, which is interesting but orthogonal to our concerns here). I would like to note that Pedersen’s essay was extremely helpful in sorting out how, exactly, the argument in question is supposed to work.
- 6.
We need both quantifiers and all four propositional connectives in LPA+T(x) if we are to allow the logic in question to be full intuitionistic logic, since the familiar equivalences that allow for various (classical) definitions of one operator in terms of another fail intuitionistically.
- 7.
For an argument that the truth predicate is non-logical, see (Cook 2012). We will return to discuss the relevance of this fact in detail below.
- 8.
A brief explanation of the notation in these axioms:
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T0(x) is the arithmetically definable truth predicate for \( {\varDelta}_0^0 \) sentences
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AtomPA(x) is the (purely arithmetic) predicate expressing “x is the code of an atomic sentence of PA”.
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SentPA(x) is the (purely arithmetic) predicate expressing “x is the code of a sentence of PA”.
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FormPA(x) is the (purely arithmetic) predicate expressing “x is the code of a formula of PA with exactly one free variable”.
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\( \dot{\neg} \)is the recursive function that maps the code of a formula to the code of its negation (and similarly for \( \dot{\wedge} \), \( \dot{\vee} \), and \( \dot{\to} \)).
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\( \dot{\forall v} \) is the recursive function that maps the code of a formula with one free variable to the code of the universal quantification of that formula (and similarly for \( \dot{\exists v} \)).
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sub(x, y) is the recursive function that maps a number v and the code of a formula with one free variable to the code of the formula obtained by replacing the free variable with the numeral for that number.
Note that our presentation of CT↾ contains more axioms than in many standard presentations, since we are not treating the conditional as defined in terms of negation and disjunction.
-
- 9.
Nothing hinges on which particular theory of truth we chose, so long as combining that theory with Heyting Arithmetic HA (i.e. the axioms of PA plus the logic H) and combining that theory with classical Peano Arithmetic (i.e. the axioms of PA plus classical logic C) give extensionally distinct theories. The simplicity (and conservativeness) of CT↾ makes it a convenient choice, however.
- 10.
Thanks are owed to an anonymous referee for pressing this point.
- 11.
Further, the discussion of the next two sections serves to provide an explanation of the formal construction just given in terms of philosophical notions mobilized in Lynch’s account —in particular, in terms of the role played by epistemic constraint.
- 12.
We shall see the reason for the odd numbering of the domains shortly.
- 13.
This reconstruction involves a simplification—one harmless in the present context—namely that whatever logics we are taking to be candidate logics for our domains, if we supplement any of those logics with the law of excluded middle, we obtain C. This is, of course, not true for many logics (i.e. substructural logics), but it is true of H.
- 14.
Of course, non-logical vocabulary can be substituted for the metavariables in instances of the latter sort.
- 15.
In fact, this won’t work simply because the version of Bivalence we added to CT↾—that is, BivPA—does not entail excluded middle for all sentences in LPA+T(x). Let’s set aside this technical observation, however, since we have more important, more philosophical fish to fry.
- 16.
A slightly snarkier way of making the point: If the truth, or even the necessary truth, of a logical principle entailed its logical truth, and if second-order logic is logic, then any platonist would presumably also be some sort of logicist.
- 17.
I certainly find them interesting, even if, as I noted in endnote 3, I ultimately think neither of them is correct.
- 18.
Here LD is the language about domain D—that is, all sentences about D not including the truth predicate T(x), hence BivLD is Bivalence restricted to the sentences in LD.
- 19.
This, despite the fact that I think that the combination of domain-independent logical pluralism and alethic monism (i.e. option 6) is the right option. The point is that I don’t think the other options are incoherent—I just think that they are wrong.
- 20.
Thanks are owed to Nikolaj J. L. L. Pedersen, Nathan Kellen, and an anonymous referee for helpful comments on early versions of this material.
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Cook, R.T. (2018). Pluralism About Pluralisms. In: Wyatt, J., Pedersen, N., Kellen, N. (eds) Pluralisms in Truth and Logic. Palgrave Innovations in Philosophy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-98346-2_15
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