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Non-Boolean classical relevant logics I

Abstract

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing.

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Notes

  1. 1.

    The scornfully intended passage is from Meyer’s A Farewell to Entailment (1985, p. 578). The Relevantist is the personification of the position portrayed in Belnap and Dunn (1981).

  2. 2.

    DS\(_\vee \) is identified as the inference schema ifAand\(\ulcorner \sim A \vee B \urcorner \)are true, then so isB

  3. 3.

    See Russell (1903, §38) where entailment is called “the notion of therefore”. The distinction is also drawn by Whitehead and Russell (1910, p. 9).

  4. 4.

    See for instance MacColl (1908) and Lewis (1912). According to Stephen Read, MacColl developed the modal system \(\mathbf {T}\) decades before Fey and von Wright did. See his essay Read (1999) for detail and also for information on how MacColl’s work relate to that of Lewis’.

  5. 5.

    The translation is my own. Note that Ackermann seems to think that holds in Lewis’ systems. This is true for normal modal logics such as \(\mathbf {S4}\), but it fails in Lewis’ preferred systems \(\mathbf {S2}\) and \(\mathbf {S3}\). Note, however, that Ackermann does not mention in the sequel-paper Ackermann (1958) in which he compares his logic to \(\mathbf {S2}\). See Hughes and Cresswell (1996, ch. 11) and Priest (2008, ch. 4) for more on strict implication and its paradoxes.

  6. 6.

    Ackermann designated that rule by ‘\(\gamma \)’, and so much literature on relevant logics will refer to the rule, and variants of it, as precisely this.

  7. 7.

    Read “the classical logician”.

  8. 8.

    The slight generalization is needed, as the proof below shows, for the proof of the admissibility of the necessitation rule to go through.

  9. 9.

    Since this difference between entailment and implication is not a widely accepted one, I will continue to use these two concepts interchangeably.

  10. 10.

    Incidentally, Lewis abandoned S3 in favor of S2 when it was pointed out to him that the suffixing axiom, (Ax9), was derivable in S3 [see Ballarin (2017) for references]. This is, presumably, why Ackermann mentions (Ax9) in the introduction to Ackermann (1956) quoted from above. It is, however, easy to verify that (Ax8\(^\flat \)) and (Ax9\(^\flat \)), with \(\rightarrow \) of course replaced by , are theorems of S2.

  11. 11.

    MaGIC finds a 12-element algebra which is a countermodel to (Ax16) for \({\Pi '}_{\!\!\!\mathbf {E}}\) minus (Ax16). MaGIC—an acronym for Matrix Generator for Implication Connectives—is an open source computer program created by John K. Slaney (Slaney 1995).

  12. 12.

    Are there algebras with fewer than eight elements which can be used to show the same thing? The smallest such, according to Slaney’s MaGIC, is the six-element algebra for R displayed in Fig. 2. In it \(\{2\}\) and \(\{3\}\) do the same job as \(\{-1, +1\}\) and \(\{-2, +2\}\) do in Belnap’s model: This model, however, is not a model for \({\Pi '}\) since it does not validate (\(\gamma \)): \(1 \wedge (\sim 1 \vee 0) \in \mathcal {T}\), but \(0 \not \in \mathcal {T}\).

  13. 13.

    That Mingle is not derivable in \(\mathbf {R}\) plus (Ax17) is easily verified by MaGIC.

  14. 14.

    This is easily seen by noting that \(\top \rightarrow (\bot \rightarrow \bot )\) is a theorem, where \(\bot =_{df} \sim \top \), and that therefore the instance \((A \wedge (\sim A \vee \bot )) \wedge (\bot \rightarrow \bot ) \rightarrow \bot \) suffices for deriving \(A \wedge \sim A \rightarrow B\).

  15. 15.

    This is in fact in effect how the logics Æ and M in the sequel to this paper are defined for which even Belnap’s original model suffice for proving the variable sharing property.

  16. 16.

    In order for this to work for E one seems to need to add \(A \Vdash 0 = 0 \rightarrow A\) as an additional arithmetical rule, where, then, \(0 = 0\) gets the same logical properties as Restall’s truth-constant t in Restall (1994), namely that both \(t \rightarrow A \vdash ^h A\) and \(A \vdash ^h t \rightarrow A\) are derivable. It is shown in Restall (1994, thm. 11.27) that \(\mathbf {L}^\#\vdash ^h_{\mathbf {L}}\sim t \vee A\) holds for classical Peano theorems A for a variety of contraction-free logics \(\mathbf {L}\) and so easily carries over to to \(\mathbf {E}^\#\) provided the extra arithmetical rule is added.

  17. 17.

    The following theorem is an easy consequence of Anderson and Belnap’s Entailment Theorem which we’ll get back to later. For more on deduction theorems in relevant logics, see Dunn and Restall (2002, §1.4).

  18. 18.

    Their argument could in fact easily be extended to also cover any theo of \(\mathbf {R}\).

  19. 19.

    It is somewhat strange that they never tried to show that the variable sharing property follows from the Entailment Theorem. We’ll see in the next section why this would have been doomed to fail.

  20. 20.

    The notion of “taking together” here is the extensional-conjunctive one. There is also a stricter intensional-conjunctive sense of “taking together” which gives rise to a stricter notion of relevant deduction in which all the premises in \(\varGamma \) need to be used. Anderson and Belnap, however, prefer the notion where it is sufficient that some of the premises be used to obtain the conclusion (Anderson and Belnap 1962, p. 36). The next subsection deals with this notion of premise use.

  21. 21.

    Mares (2004, p. 176) claims that \(\mathbf {R}\) so strengthened yields classical logic. This is not so: the weakening axiom \(A \rightarrow (B \rightarrow A)\) fails in the model for RX displayed in Fig. 4.

  22. 22.

    Both reflexivity and weakening drop quite immediate from the definition of both \(\vdash ^h\) and \(\vdash ^r\). That cut holds is easily seen by the usual replacing proof: simply replace the assumption A in the proof of B from \(\{A\}\cup \varDelta \) by the proof of A from \(\varTheta \). This will evidently be a \(\vdash ^h\)-proof of A from \(\varTheta \cup \varDelta \). That this also yields a \(\vdash ^r\)-proof is seen by noting that since A is #-marked if it is used in the \(\vdash ^r\)-proof of B from \(\{A\}\cup \varDelta \) and it is marked in the \(\vdash ^r\)-proof of A from \(\varTheta \), it follows that so replacing A allows every succeeding inference step to be justified by the same #-rules as the rest of the \(\vdash ^r\)-proof of B from \(\{A\}\cup \varDelta \).

  23. 23.

    The minimal conditions on \(\circ \) are the residuation rules: \(A \circ B \rightarrow C \Vdash A \rightarrow (B \rightarrow C)\) and \(A \rightarrow (B \rightarrow C) \Vdash A \circ B \rightarrow C\). \(A \circ B\) is not definable in E, but is definable as \(\sim (A \rightarrow \sim B)\) in strong logics like R.

  24. 24.

    In the case of T and E, further restrictions on the use of modus ponens are required.

  25. 25.

    There is much more that ought to be said about suppression and how it relates to intuitions behind relevant logics. This will, however, have to wait for another occasion.

  26. 26.

    This is also how Brady (2006, p. 4) defines relevant logic.

  27. 27.

    The quote is from The Book of Beasts. Here quoted from Anderson and Belnap (1975, p. 297).

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Acknowledgements

I am very grateful to the participants of the Bergen Logic group, the audience of the Bergen Workshop on Logical Disagreements as well as the LanCog Workshop on Substructural Logics for constructive feedback. Special thanks go to Ole Thomassen Hjortland, David Makinson and Ben Martin for comments on earlier drafts. I would also very much like to thank the two anonymous referees for their comments and suggestions which helped improve this paper considerably.

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Correspondence to Tore Fjetland Øgaard.

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In honor of Robert Meyer.

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Øgaard, T.F. Non-Boolean classical relevant logics I. Synthese (2019). https://doi.org/10.1007/s11229-019-02507-z

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Keywords

  • Disjunctive syllogism
  • Entailment
  • Modality
  • Paraconsistency
  • Relevant logics
  • Suppression