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On When a Disjunction Is Informative

Ambiguous Connectives and a Realist Commitment to Pluralism

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The Realism-Antirealism Debate in the Age of Alternative Logics

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 23))

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Abstract

In this paper we investigate the topic of ambiguous connectives, as it was recently explored by Paoli (J. Philos. Logic 32:531–548,2003), from an informational perspective. That is, starting from the framework of informational pluralism Allo (Computing, Philosophy, and Cognitive Science.Cambridge Scholars Press, Cambridge,41–52,2007a) Allo (J. Philos. Logic 36:659–694,2007b), we ask what it means for a message of the form ‘A or B’, to be informative. Using these disjunctive messages as an example, we answer three traditional objections to substructural logic and logical pluralism, and eventually show that the linear or relevant logician’s road to unambiguous connectives is consistent with informational pluralism.

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Notes

  1. 1.

    This accounts differs from Hanson’s own, since he identifies the content of A with the consequence-set of A.

  2. 2.

    Actually, the symmetry of compatibility only requires \(s \sqsubseteq s^{\ast\ast}\) (which enforces double negation introduction) instead of \(s = s^{\ast\ast}\) (the semantic postulate for double negation equivalence).

  3. 3.

    Note that the situated content of A (\(\{s: s\not\Vdash A\}\)) does not express a proposition, and hence is not a persistent kind of content. Namely, it does not hold that if \(s \not\Vdash A\), then \(s' \not\Vdash A\) for all \(s \sqsubseteq s'\). Such issues regarding the non-persistence of properties should be kept in mind.

  4. 4.

    A second, more complicated, aspect of the relevant recapture of DS, namely the admissibility of rule γ, which states that from \(\vdash A \sqcup B\) and \(\vdash \sim A\) we may derive that ⊢ B, is left aside [3, §25].

  5. 5.

    Note, however, that the group-theoretical conjunction equally expresses a choice; namely, the choice the player himself has.

  6. 6.

    Remark in that light that rules (1.10) and (1.11) effectively turn some rules of the game into logical rules.

  7. 7.

    Let \(c = c_1;c_2\) where \(c_1 \in Log\). Assume for reductio that \(c \in Log\) too. By \(c_1 \sqsubseteq_{c_2} c\), we have that \(c_2 \sqsubseteq_{c_1} c\), and since \(c_1 \in Log\) we also have that \(c_2 \sqsubseteq c\). But then, given our assumption that \(c \in Log\), it must at least hold that \(c_2 \sqsubseteq c_2^\ast\). But since c 2 can be any element of S, the latter should not hold in general.

  8. 8.

    Remember that our appeal to multiple agents is itself an artefact we use to account for the real phenomenon under consideration: the distributed nature of information (on that topic, see also [7]).

  9. 9.

    As suggested by Greg Restall (pc), the model of Section 1.4 might in a sense be closer to the approach in his ‘Modelling Truthmaking’ [33].

  10. 10.

    An often-used symbol for the lattice-theoretical constant is ⊤ (top). I prefer to use t and keep ⊤ for the classical truth-constant.

  11. 11.

    If \(s \Vdash 1\) then s is consistent too, that is \(s \sqsubseteq s^\ast\). Consequently, \(s^\ast \Vdash 1\) holds too, and by the same token \(s^\ast \sqsubseteq s^{\ast\ast}\). Since \(s = s^{\ast\ast}\), \(s^\ast \sqsubseteq s\), and hence \(s = s^\ast\).

  12. 12.

    Traditionally, an enthymeme is an argument with an unstated or suppressed assumption. In the relevantist tradition it is common to recapture classical reasoning enthymematically by treating consistency (in some or other form) as an unstated assumption.

  13. 13.

    This paper was originally written in the Summer of 2006, and complements [2]. Both these papers argue for logical and informational pluralism in the same purely model-theoreric fashion; a method I would no longer rely on in the same way as I did. Yet, I’ve chosen not to actualise the present paper to reflect these changes, but to stick to the original version. For the same reason, references to more recent literature on the topic of logical pluralism haven’t been included either.

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Authors and Affiliations

  1. Postdoctoral Fellow, Fund for Scientific Research—Flanders (Belgium), Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels, Belgium

    Patrick Allo

  2. IEG, Oxford University, Oxford, UK

    Patrick Allo

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  1. Patrick Allo
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Correspondence to Patrick Allo .

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Editors and Affiliations

  1. , UFR Philosophie/ UMR-STL: 8163, Université de Lille, B.P. 60149, Villeneuve d'Asque, 59653, France

    Shahid Rahman

  2. Centre for Logic & Philosophy of Sc, Philosophy & Moral Science Dept., Ghent University, Blandijnberg 2, Gent, 9000, Belgium

    Giuseppe Primiero

  3. , Département de philosophie, Université du Québec à Montréal, Succursale Centre-Ville 8888 C.P., Montreal, H3C 3P8, Canada

    Mathieu Marion

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Allo, P. (2012). On When a Disjunction Is Informative. In: Rahman, S., Primiero, G., Marion, M. (eds) The Realism-Antirealism Debate in the Age of Alternative Logics. Logic, Epistemology, and the Unity of Science, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1923-1_1

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