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Relevant and Substructural Logics Relevant and Substructural Logics GREG RESTALL∗ PHILOSOPHY DEPARTMENT, MACQUARIE UNIVERSITY [email protected] June 23, 2001 http://www.phil.mq.edu.au/staff/grestall/ Abstract: This is a history of relevant and substructural logics, written for the Hand- book of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods.1 1 Introduction Logics tend to be viewed of in one of two ways — with an eye to proofs, or with an eye to models.2 Relevant and substructural logics are no different: you can focus on notions of proof, inference rules and structural features of deduction in these logics, or you can focus on interpretations of the language in other structures. This essay is structured around the bifurcation between proofs and mod- els: The first section discusses Proof Theory of relevant and substructural log- ics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, Dunn's algebraic models [76, 77] Urquhart's operational semantics [267, 268] and Routley and Meyer's rela- tional semantics [239, 240, 241] arrived decades after the initial burst of ac- tivity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in lin- guistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore. Girard's linear logic is a different story: it was discovered though considerations of the categorical models of coherence ∗This research is supported by the Australian Research Council, through its Large Grant pro- gram. Thanks, too, go to Nuel Belnap, Mike Dunn, Bob Meyer, Graham Priest, Stephen Read and John Slaney for many enjoyable conversations on these topics. This is a draft and it is not for citation without permission. Some features are due for severe rhhevision before publication. Please contact me if you wish to quote this version. I expect to have a revised version completed before the end of 2001. Please check my website for an updated copy before emailing me with a list of errors. But once you've done that, by all means, fire away! ii 1The title, Relevant and Substructural Logics is not to be read in the same vein as “apples and oranges” or “Australia and New Zealand.” It is more in the vein of “apples and fruit” or “Australia and the Pacific Rim.” It is a history of substructural logics with a particular attention to relevant logics, or dually, a history of relevant logics, playing particular attention to their presence in the larger class of substructural logics. 2Sometimes you see this described as the distinction between an emphasis on syntax or se- mantics. But this is to cut against the grain. On the face of it, rules of proof have as much to do with the meaning of connectives as do model-theoretic conditions. The rules interpreting a formal language in a model pay just as much attention to syntax as does any proof theory. 1 http://www.phil.mq.edu.au/staff/grestall/ 2 spaces. However, as linear logic appears on the scene much later than rele- vant logic or the Lambek calculus, starting with proof theory does not result in too much temporal reversal. I will end with one smaller section Loose Ends, sketching avenues for fur- ther work. The major sections, then, are structured thematically, and inside these sections I will endeavour to sketch the core historical lines of develop- ment in substructural logics. This, then, will be a conceptual history, indicat- ing the linkages, dependencies and development of the content itself. I will be less concerned with identifying who did what and when.3 I take it that logic is best learned by doing it, and so, I have taken the lib- erty to sketch the proofs of major results when the techniques used in the proofs us something distinctive about the field. The proofs can be skipped or skimmed without any threat to the continuity of the story. However, to get the full flavour of the history, you should attempt to savour the proofs at leisure. Let me end this introduction by situating this essay in its larger context and explaining how it differs from other similar introductory books and es- says. Other comprehensive introductions such as Dunn's “Relevance Logic and Entailment” [81] and its descendant “Relevance Logic” [94], Read's Rel- evant Logic [224] and Troelstra's Lectures on Linear Logic [264] are more nar- rowly focussed than this essay, concentrating on one or other of the many relevant and substructural logics. The Anderson–Belnap two-volume Entail- ment [10, 11] is a goldmine of historical detail in the tradition of relevance logic, but it contains little about other important traditions in substructural logics. My Introduction to Substructural Logics [234] has a similar scope to this chapter, in that it covers the broad sweep of substructural logics: however, that book is more technical than this essay, as it features many formal re- sults stated and proved in generality. It is also written to introduce the subject purely thematically instead of historically. 2 Proofs The discipline of relevant logic grew out of an attempt to understand notions of consequence and conditionality where the conclusion of a valid argument is relevant to the premises, and where the consequent of a true conditional is relevant to the antecedent. “Substructural” is a newer term, due to Schroder¨ -Heister and Dosen.ˇ They write: Our proposal is to call logics that can be obtained . by restricting structural rules, substructural logics. [250, page 6] The structural rules mentioned here dictate admissible forms of transforma- tions of premises in proofs. Later in this section, we will see how relevant logics are naturally counted as substructural logics, as certain commonly ad- mitted structural rules are to blame for introducing irrelevant consequences into proofs. 3In particular, I will say little about the intellectual ancestry of different results. I will not trace the degree to which researchers in one tradition were influenced by those in another. Greg Restall, [email protected] June 23, 2001 http://www.phil.mq.edu.au/staff/grestall/ 3 Historical priority in the field belongs to the tradition of relevant logic, and it is to the early stirrings of considerations of relevance that we will turn. 2.1 Relevant Implication: Orlov, Moh and Church Dosenˇ has shown us [71] that substructural logic dates back at least to 1928 with I. E. Orlov's axiomatisation of a propositional logic weaker than classi- cal logic [207].4 Orlov axiomatised this logic in order to “represent relevance between propositions in symbolic form” [71, page 341]. Orlov's propositional logic has this axiomatisation.5 A ∼∼A double negation introduction ∼∼A A double negation elimination A ! ∼(A ∼A) contraposed reductio (A !B) (∼B ∼A) contraposition (A! (B !C)) (B (A C)) permutation (A ! B) ! ((C ! A) (C B)) prefixing A; A! B! B ! ! ! modus ponens ! ! ! ! ! The axioms and rule here form a traditional Hilbert system. The rule modus ponens is written! in)the form using a turnstile to echo the general definition of logical consequence in a Hilbert system. Given a set X of formulas, and a single formula A, we say that A can be proved from X (which I write “X A”) if and only if there is a proof in the Hilbert system with A as the conclusion, and with hypotheses from among the set X. A proof from hypotheses is)sim- ply a list of formulas, each of which is either an hypothesis, an axiom, or one which follows from earlier formulas in the list by means of a rule. In Orlov's system, the only rule is modus ponens. We will see later that this is not nec- essarily the most useful notion of logical consequence applicable to relevant and substructural logics. In particular, more interesting results can be proven with consequence relations which do not merely relate sets of formulas as premises to a conclusion, but rather relate lists, or other forms of structured collections as premises, to a conclusion. This is because lists or other struc- tures can distinguish the order or quantity of individual premises, while sets cannot. However, this is all that can simply be done to define consequence re- lations within the confines of a Hilbert system, so here is where our definition of consequence will start. These axioms and the rule do not explicitly represent any notion of rele- vance. Instead, we have an axiomatic system governing the behaviour of im- plication and negation. The system tells us about relevance in virtue of what 4Allen Hazen has shown that in Russell's 1906 paper “The Theory of Implication” his propo- sitional logic (without negation) is free of the structural rule of contraction [133, 243]. Only after negation is introduced can contraction can be proved. However, there seems to be no real sense in which Russell could be pressed in to favour as a proponent of substructural logics, as his aim was not to do without contraction, but to give an axiomatic account of material implication. 5The names are mine, and not Orlov's. I have attempted to give each axiom or rule its com- mon name (see for example Anderson and Belnap's Entailment [10] for a list of axioms and their ¡ ¢¤£¥ ¡ ¢¦ ¨§ names).
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