Abstract
Jc Beall and Greg Restall (Logical Pluralism. Clarendon Press, Oxford, 2006) propose a logical pluralism where the corresponding connectives in each logic mean the same thing. They contrast this with a Carnapian pluralism, where different logics have corresponding connectives which do not share meanings. I will show that due to the manner in which connectives are given their meaning by Beall and Restall, relevant negation and intuitionistic negation cannot mean the same thing. Thus, their pluralism is at least partly Carnapian, as not all the logics involved can have their corresponding connectives share meanings.
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Notes
- 1.
I argue elsewhere (see Kouri (2018)) that this is not the best way to understand the Carnapian picture. For the purposes of this chapter, though, it suffices.
- 2.
It is not clear that Restall is right in such an assessment of Carnap . At the very least, Restall ought to grant that Carnap’s pluralism is a pluralism of logics and languages. Restall’s position, on the other hand, is just a pluralism of logics. Additionally, it is the case that Carnap could claim “A and ¬A together, classically entail B, but A and ¬A together do not relevantly entail B”. He would claim this is the case precisely because the connectives mean something different in each logic. In some sense, what this quote does for Restall is to show that Restall disagrees with the second half (“A together with its classical negation entail B, but A together with its relevant negation need not entail B”.) and not that Carnap disagrees with the first. Thanks to Hannes Leitgeb for pushing me on this issue.
- 3.
There is a reason to think that the metalanguage Beall and Restall are using is also classical. Though they never explicitly make a claim about which metalanguage they are working with, it seems their description of stages in constructions and situations as subclassical suggests that classical logic is the strongest logic admissible on their picture, and their rejection of contra-classical logics suggests the same. If this is the case, then it seems that it would make a good choice for a metalanguage. For a criticism of their (probable) choice in metalanguage, see Read (2006). It should be pointed out here that Beall and Restall never explicitly make the claim that their metalanguage is classical. A referee has suggested to me that in personal correspondence, Restall has gone so far as to claim that there is no metalanguage associated with logical pluralism. I am not entirely sure what to make of this, as presumably, we need some language in which to discuss the logics which are admissible. Whether the metalanguage is classical, though, or even if there is no metalanguage, most of what follows can be re-stated accordingly, and so I will not pursue this complication further here.
- 4.
Unfortunately, very little about the compatibility relation is committed to by Restall. This poses some problems for Beall and Restall’s project. In particular, it will turn out that it is important whether compatibility is primitive, or somehow determined by negation.
- 5.
There is another way we can view this (if Beall and Restall’s metalanguage is classical, these will be equivalent). It might be rather than a disjunction, we have a conjunction of conditionals. The maximal truth condition for negation would then be
(if M is a Tarski model then M, s ⊨ ¬A if and only if it is not the case that M, s ⊨ A) AND (if w is a node in a Kripke structure then w ⊨ ¬A if and only if, for all u such that w ≤ u, u ¬ A) AND (if s is a situation then s ⊨ ¬A if and only if, ∀t such that sCt, t ¬ A)
Since I take it these are equivalent in their classical metalanguage (see endnote 3), I will focus on the disjunctive form.
- 6.
This is in part necessary to rule out “junky” connectives, formed by disjoining the “wrong” clauses to the maximal truth condition. For example, it is required to prevent having to claim that the disjunction of the clause for conjunction in Tarski models with the clause for the arrow in situations is a legitimate connective meaning. Without this, the maximal truth condition is not a good candidate condition for the meaning of any connective, and we can dismiss Beall and Restall on those grounds.
- 7.
There is an immediate response available on behalf of Beall and Restall here (thanks to Alexandru Radulescu for the suggestion). Beall and Restall claim that what they are doing is providing a precisification of the vague notion of logical consequence. So why not also assume that they are giving a precisification of a vague account of negation? Then, we would not need such a precise definition of negation, as I have suggested with maximal truth conditions and (*), and the examples I give in Sect. 4 can all be dismissed as borderline cases or outliers. However, I think there will still be a problem here, though it is different from any I will address here. There will be a tension between their firm claim that the connectives must mean the same thing, and the vague meaning of those connectives. In order to make the first claim, we must assume that there is something precise about the nature of the connectives, while to make the claim about vagueness, we must dismiss this preciseness. In effect, I think that if we want to pursue the claim that the connective meanings are vague, we are better to adopt something like the tentative conclusion I give in Sect. 7.
- 8.
Additionally, of course, the logical consequence relationship will have to satisfy GTT and be necessary, normative and formal. For Beall and Restall, necessity is truth preservation in all cases, normativity is the ability to go wrong if you accept the premises and reject the conclusion of a valid argument, and formality is either schematicity or one of: providing norms for thought as such, indifference to identities of objects, or contentlessness (Beall and Restall 2006, Chap. 2).
- 9.
This logical consequence relationship is also necessary, normative and formal, and satisfies the GTT. It satisfies the GTT since the cases in question are just whichever situations are “blind”. The empty-R relation is just a sublogic of what Beall and Restall refer to as relevant logic, which we had already assumed was necessary and formal, and so these characteristics are preserved. Additionally, it is normative, since we “go wrong” by assuming that the situations are compatible with something.
- 10.
Thanks to Graham Leach-Krouse for suggested an example of this type.
- 11.
These restricted domains of situations still abide by the necessary-normative-formal requirements on being an admissible logic (see definition in endnote 9), and satisfy the GTT since the cases are simply situations of a specific type. These situations produce a consequence relation which is formal (since it is in a model) and normative, since we “go wrong” by assuming that the situations we are considering are not restricted to not making true a certain set of sentences. Finally, it is necessary. Beall and Restall define necessity as “the truth of the premises necessitates the truth of the conclusion” (p. 14), in other words, as long as it is not possible for the premises to be true and the conclusion to be false (p. 40). But in this case, this is not possible, since the possibilities in play are a subset of the original domain, so there are no possible cases where things are different.
- 12.
One might think at this point that we might be able to accomplish the Beall and Restall project if we framed everything in terms of nodes in Kripke structures rather than situations. After all, terminal nodes in Kripke structures are classical models (see Beall and Restall 2006, p. 98). However, we find we have the same problem once again. The default instantiation of (*) would then be D=nodes in Kripke structures, R =≤. We would have to expand ≤ so that it could be symmetric, since compatibility is symmetric. However, Kripke structures give us no clues as to how to expand ≤ to appropriately capture compatibility, and so we can proceed as we wish, and expand it to any relationship. Then we simply take R to be the universal relation, and re-run the previous counterexample. Another option would be to consider cases as classes of pointed frames (thanks to Shay Logan for this suggestion). Then, we could capture the various logics by restricting what gets admitted to the class of pointed frames in question. Moreover, the two counter examples presented above would be less odd, since one ought to expect that odd frames come equipped with odd negations. However, we will still have a problem here (thanks to Beau Madison Mount for this suggestion). It is still the case that we will have “overlap” (e.g., pointed Kripke frames for propositional logic with the null signature will overlap with classical frames with a unary relation), and thus we will still be able to develop sentences which are true when the frame is considered as one kind of model, and not true when considered as the other.
- 13.
Thanks to Aaron Cotnoir for suggesting this option.
- 14.
Thanks to Ethan Brauer, Steven Dalglish and Giorgio Sbardolini for help in constructing this example.
- 15.
They will be necessary, normative and formal on the basis of Beall and Restall’s definitions. Formality here seems relatively simple. Necessity and normativity, on the other hand, are a bit odd, and this is the result of Beall and Restall’s definitions of them. For Beall and Restall, any sublogic of classical logic will be necessary, since they hold that relevant logic is necessary because worlds are a special type of situation. But here, since, on Thomas’s system, classical logic can be recovered as a special type of model, and since no model we are concerned with proves anything contra-classical, all models ought to be truth preserving, and so necessary on Beall and Restall’s picture. This reasoning mimics their discussion of relevant logic, where they say “The question is, in other words, whether truth-preservation over all situations guarantees truth-preservation over all possible worlds. The answer is ‘yes’ if possible worlds count as (perhaps special) situations” (Beall and Restall 2006, p. 54). Here, we have swapped “models” for situations, and “classical models” for possible worlds, but the reasoning is similar. Normativity, for Beall and Restall, comes down to there being some sort of mistake associated with reasoning invalidly on this type of logic. But they are not terribly specific about what types of things count as mistakes. The mistake associated with relevant logic is irrelevant reasoning, and the mistakes associated with constructive logic are “mistakes of constructivity” (Beall and Restall 2006, 70). In this case, then, I claim that not reasoning properly according to this new logic is reasoning compositionally when one shouldn’t. As one referee has suggested, there is a good reason for denying that all subclassical logics are normative and necessary. Though this seems plausible (and many commentators agree that Beall and Restall’s definitions of necessary and normative are problematic, see, e.g., Bueno and Shalkowski (2009) and Keefe (2013)), strictly speaking here all we only need is the Thomas system to match Beall and Restall’s definitions, and it seems to. On the other hand, if the Thomas system does not produce necessary, normative and formal logics, then it is not an option for Beall and Restall to begin with.
- 16.
- 17.
Acknowledgments: I would like to thank Stewart Shapiro, Shay Logan, Chris Pincock, Kevin Scharp, Neil Tennant and a referee for this volume for feedback on earlier drafts. Additionally, audiences at the 2016 Central APA, the UConn Conference on Logical Pluralism and Truth Pluralism, the Munich Center for Mathematical Philosophy and the 2017 Canadian Philosophical Association meeting provided incredibly helpful comments.
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Kouri Kissel, T. (2018). Connective Meaning in Beall and Restall’s Logical Pluralism. 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_10
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