Abstract
Relevance logic is ordinarily seen as a subsystem of classical logic under the translation that replaces arrows by horseshoes. If, however, we consider the arrow as an additional connective alongside the horseshoe, then another perspective emerges: the theses of relevance logic, specifically the system R, may also be seen as the output of a conservative extension of the relation of classical consequence. We describe two ways in which this may be done. One is by defining a suitable closure relation out of the set of theses of relevance logic; the other is by adding to the usual natural deduction system for it further rules with ‘projective constraints’, whose application restricts the subsequent application of other rules. The significance of the two constructions is also discussed.
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Notes
- 1.
For axiomatizations of R and neighbouring systems as sets of formulae, see Anderson and Belnap (1975) Sects. 21.1 and 27.1, summarized in Anderson et al. (1992) Sect. R2; also e.g. Dunn (1986) Sect. 1.3 repeated in Dunn and Restall (2002) Sect. 1.3; Mares (2004) appendix A and Mares (2012). Following a tradition going back to Prior (1955), we use the term thesis for what is also commonly called a ‘theorem’ of the logic, reserving the latter word for whatever one can prove about the logic (instead of the rather cumbersome ‘meta-theorem’).
- 2.
To be specific, Meyer (1974) showed that if we introduce as further primitives a non-classical two-place ‘fusion’ connective and propositional constants f, t, writing \(\sim \) \(\alpha \) as an abbreviation for \(\alpha \rightarrow f\), then we can give a straightforward axiomatization of a conservative extension of R that neatly contains a classical axiom set for \(\wedge \), \(\vee , \lnot \) alone.
- 3.
- 4.
- 5.
The description of such detours as ‘funny business’ was popularized by R.J. Fogelin in the years before publication of Anderson and Belnap (1975), whose section 8.21 welcomes the epithet.
- 6.
For presentations of the Anderson/Belnap system of natural deduction for R (and neighbouring systems) see Anderson and Belnap (1975) Sect. 27.2, summarized in Anderson et al. (1992) Sect. R3, and Mares (2004) appendix A. Unfortunately, the presentations in Dunn (1986) Sect. 1.5 and Dunn and Restall (2002) Sect. 1.5 lack the rules for negation.
- 7.
It doesn’t really matter what label is placed on the conclusion of an application of a rule from Group II. We have used the natural label X \(\cup \) Y, but we could equally well have chosen any other subset of the currently undischarged assumptions, even \(\emptyset \). For labels are needed only to constrain subsequent applications of \(\rightarrow \!\!+\), and once we flag a line, as we do in the conclusions of the rules of Group II, such applications are barred anyway by the new proviso on \(\rightarrow \!\!+\) and the flagging inheritance regime. One could even drop the label on the conclusions of Group II rules, keeping just the flags; but that would have the inconvenience of requiring notational adjustments to the labelling subscripts of the Anderson–Belnap rules in Group I, which we prefer to keep intact, and would complicate the proof of Corollary 9.
- 8.
There seems to be little prospect for recuperating cumulative transitivity by choosing different classical rules in Group II of our system, so long as they are flagged and the flags are used to block subsequent applications of \(\rightarrow \!\!\!+\). Indeed, it would appear that, apart from degenerate cases, any introduction of projective constraint rules into a natural deduction system will create difficulties for cumulative transitivity in the induced consequence relation.
- 9.
Prawitz (1965) Chap. VII, Sect.2 presented a natural deduction system for relevance logic by modifying a straightforward system for classical logic (the modification uses global constraints on derivations rather than labels or flags). Unfortunately, however, the system does not output the system R or any of its well-known neighbours. As remarked by Dunn and Restall (2002) p. 26, for example, it outputs too little compared to R, since it does not yield the first-degree entailment of distribution of conjunction over disjunction. More seriously (because less easy to rectify) and apparently less well known (the author has not seen it in the literature), Prawitz’ system also yields too much. On the one hand, the formula p \(\rightarrow \)(\(\lnot \) p \(\rightarrow \) q) is not a thesis of R or sister relevance logics. But on the other hand, in Prawitz’ system, negation is defined from his primitive connectives \(\bot \),\(\rightarrow \) by treating \(\lnot \alpha \) as shorthand for \(\alpha \rightarrow \bot \), so that p \(\rightarrow \)(\(\lnot \) p \(\rightarrow \) q) abbreviates p \(\rightarrow \) \(((p\rightarrow \bot )\rightarrow q)\). That formula can be derived from the empty set of premises using Prawitz’ rules for the connectives occurring in it, as follows.
$$\begin{aligned} \begin{array}{lll} 1.&{} p_{1} &{} {\text {assumption}}\\ 2.&{} p\rightarrow \bot _{2} &{} {\text {assumption}}\\ 3.&{} \bot _{1,2} &{} {\text {from 1, 2 by}} \rightarrow \!-\\ 4.&{} q_{1,2} &{} {\text {from 3 by Prawitz' rule for}} \bot {\text {-elimination}}\\ 5.&{} (p\rightarrow \bot )\rightarrow q_{1} &{} \rightarrow \!+, {\text {unobstructed by Prawitz' global constraints}}\\ 6.&{} p\rightarrow ((p\rightarrow \bot )\rightarrow q)_{\emptyset } &{} \rightarrow \!+, {\text {unobstructed by Prawitz' global constraints}}\\ \end{array} \end{aligned}$$
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Acknowledgments
The author is very much indebted to Lloyd Humberstone, whose comments on a badly crashed early text helped the author to start again, and whose remarks on later drafts were also valuable. Thanks also to Diderik Batens, Michael Dunn, Jim Hawthorne, João Marcos and Peter Verdée for helpful discussions, as well as two anonymous referees for their suggestions.
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Makinson, D. (2014). Relevance Logic as a Conservative Extension of Classical Logic. In: Hansson, S. (eds) David Makinson on Classical Methods for Non-Classical Problems. Outstanding Contributions to Logic, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7759-0_17
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