# Restall’s Proof-Theoretic Pluralism and Relevance Logic

## Abstract

Restall (Erkenntnis 79(2):279–291, 2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall (Aust J Philos 78(4):475–493, 2000) and in Beall and Restall (Logical pluralism, Oxford University Press, Oxford, 2006). What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of Restall’s pluralism is that the relevance logic that gets added must have connectives which mean the same thing as the connectives in the already admitted logic. This is what I will show is not possible.

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## Notes

1. 1.

The positive form is that from a contradiction, one can infer anything. The negative for is that from a contradiction, one can infer any negated sentence.

2. 2.

The lessons drawn for Restall here will certainly have an impact on the debate on the possibility of defining logical connectives inferentially. See, for example, Hjortland (2014) and Paoli (2014), who discuss the possible meaning-constituting laws of the logical connectives.

3. 3.

Dual intuitionistic logic is a paraconsistent logic. It has the same sentential theorems as classical logic, but not the same counter-theorems. In fact, it does not disprove all contradictions. See Urbas (1996) for more details.

4. 4.

The two weakening rules are not given explicitly in Restall (2014). They have been adapted from Restall (2000).

5. 5.

I am not quite sure what term to use here, and have settled on “notion” (with thanks to a blind reviewer for the suggestion). It is something like a reading, a sharpening, an interpretation or a precisification. I will use the word “notion” in order to remain neutral on this this topic, as the results I present only require that any “notions”/readings/sharpenings/etc. of the same term have something in common, which all of these ways to spell out what a “notion” is.

6. 6.

Though Beall and Restall (2006) speak of logical consequence relations, and I am here speaking of validities, I take these two notions to be similar enough. A logical consequence relation can, for example, be thought of as a set/class of validities satisfying certain closure properties.

7. 7.

I take it that this requirement on a logic is inherently plausible, and hold that Beall and Restall (2006) would agree. As they note, their system does not license logics which are not reflexive; logics which do not license the inference from A to itself. They refer to such systems as “logics by courtesy and by family resemblance” (Beall and Restall 2006, p. 91). Their dismissal of non-reflexive systems suggests that they would agree that the logic developed by removing the identity rule from the sequent calculus ought not to be one of the right logics, and that they hold a right logic needs to license at least some unconditional inferences.

8. 8.

To my knowledge, this is the only method that has taken hold for this particular proof theory. At the very least, Restall (2000) claims this is a method which is natural to use. It is due to Tennant (1984, 1992).

9. 9.

There are other options if we remove the CUT rule, see Tennant (forthcoming) for a relevance logic without CUT. Restall, however, has made claims that logics must be unrestrictedly transitive (see Beall and Restall 2006, p. 91), and so I will not consider the possibility here.

10. 10.

This will work if the meta-language is classical. It is plausible to think that Beall and Restall (2006) take it to be classical, and they are criticized for doing so by Read (2006).

11. 11.

Thanks to Marcus Rossberg and Nathan Kellen for pointing this out to me. Restall’s response to this version of the argument would presumably be to read the sequent intentionally, and thus the same reasoning as in Sect. 4 will apply here as well.

12. 12.

The move to intensional connectives actually precludes the meta-language being classical. It would require a much broader change to Restall’s position than I will address here.

13. 13.

Thanks to a blind reviewer for suggesting this issue.

## References

1. Anderson, A. R., & Belnap, N. D. (1975). Entailment: The logic of relevance and neccessity (Vol. I). USA: Princeton University Press.

2. Beall, J., & Restall, G. (2000). Logical pluralism. Australian Journal of Philosophy, 78(4), 475–493.

3. Beall, J., & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press.

4. Belnap, N. (1982). Display logic. Journal of Philosophical Logic, 11, 375–417.

5. Dunn, J. M., & Restall, G. (1992). Relevance logic 1. Handbook of Philosophical Logic, 6, 1–192.

6. Hjortland, O. T. (2014). Verbal disputes in logic: Against minimalism for logical connectives. Logique et Analyse, 227, 463–486.

7. Paoli, F. (2014). Semantic minimalism for logical constants. Logique et Analyse, 227, 439–461.

8. Read, S. (1982). Disjunction. Journal of Semantics, 1(3), 275–285.

9. Read, S. (2006). Monism: The one true logic. In D. Devidi & T. Kenyon (Eds.), A logical approach to philosophy (pp. 193–209). Netherlands: Springer.

10. Restall, G. (2000). An introduction to substructural logic. New York: Routledge.

11. Restall, G. (2014). Pluralism and proofs. Erkenntnis, 79(2), 279–291.

12. Tennant, N. (1984). Perfect validity, entailment and paraconsistency. Studia Logica, 43(1–2), 181–200.

13. Tennant, N. (1992). Autologic. Edinburgh information technology series. Edinburgh: Edinburgh University Press.

14. Tennant, N. (forthcoming). Core Logic.

15. Urbas, I. (1996). Dual-intuitionistic logic. Notre Dame Journal of Formal Logic, 37(3), 440–451.

## Acknowledgments

I owe a debt of gratitude to Stewart Shapiro very carefully read and commented on several earlier drafts. I am grateful to comments from several people, including three blind reviewers, Scott Brown, Nathan Kellen, Chris Pincock, Marcus Rossberg, Kevin Scharp, Matt Souba and Neil Tennant. I am also grateful to audiences at the Dubrovnik Conference on Pluralism (2015), the Pacific APA (2015), PhiloSTEM (2014), the Society for Exact Philosophy (2013) and The University of Western Ontario Philosophy of Logic, Math and Physics Graduate Conference (2013).

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Correspondence to Teresa Kouri.

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Kouri, T. Restall’s Proof-Theoretic Pluralism and Relevance Logic. Erkenn 81, 1243–1252 (2016). https://doi.org/10.1007/s10670-015-9792-4

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### Keywords

• Intuitionistic Logic
• Logical Rule
• Proof Theory
• Structural Rule
• Sequent Calculus