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Manipulating Sources of Information: Towards an Interpretation of Linear Logic and Strong Relevance Logic

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J. Michael Dunn on Information Based Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 8))

Abstract

Relevance Logics are interpreted in terms of agents’ comprehending and constructing sources of information. The rules governing these constructions are formulated in a natural deduction system. Two different sorts of interpretation are developed. On the productive interpretation, implications keep track of the number of times sources are to be applied to one another to produce a particular result. On the functional interpretation, only what is doable in principle (with whatever number of applications) is represented. The productive interpretation is used to understand the contraction-free logics, linear logic and RW. The functional approach is used to understand the logics LR and R.

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Notes

  1. 1.

    Perhaps the “opinion tetrahedron” that Dunn (2010) sets out can be extended to treat the full vocabulary of relevance logic, including the intensional connectives. At any rate, I do not know how to do that at this time. My only point here is to support a strong distinction between sources and their contents and the topic of uncertainty and risk helps to do that.

  2. 2.

    See Sect. 12 for more discussion of the denotation of sources. The issue is similar to the relationship between relevance subscripts in the Anderson–Belnap natural deduction system and indices in the Routley–Meyer semantics. A set of numerals \(\alpha \) picks out a collection of indices that are related to those denoted by the numerals in \(\alpha \).

  3. 3.

    There are other ways of generating the distribution rule than this. Anderson and Belnap add a primitive rule. Ross Brady adopts a structural connective that corresponds in some sense to extensional disjunction. Dunn incorporates the mechanism that is found in his and Mints’ sequent systems of having conjunctive hypotheses (Dunn and Restall 2002, Sect. 1.5). Dunn’s proposal is particularly interesting and it might be illuminating to provide a source of information reading of it.

  4. 4.

    The semantics for linear logic created by Allwein and Dunn (1993) might also be a candidate for a source reading, but I do not have the room here to discuss its complexities.

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Acknowledgments

I presented a very early draft of this paper to the Pukeko Logic Group. I am grateful to Jeremy Seligman, Zach Weber, and Patrick Girard for their helpful comments. I am also grateful for the two anonymous referees’ helpful comments.

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  1. Philosophy Programme, Victoria University of Wellington, Wellington, New Zealand

    Edwin Mares

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  1. Edwin Mares
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Correspondence to Edwin Mares .

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  1. Department of Philosophy, University of Alberta, Edmonton, Alberta, Canada

    Katalin Bimbó

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Mares, E. (2016). Manipulating Sources of Information: Towards an Interpretation of Linear Logic and Strong Relevance Logic. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_7

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