Modal Logic as Metalogic KOSTA DOZEN Matemati~ki lnstitut, Knez Mihailova 35, 11001 Belgrade, p f. 367, Yugoslavia (Received 24 September, 1991; in final form 14 July, 1992) Abstract. The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's. First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are. However, though we do not always have the deduction theorem, we may always obtain a modal analogue of the deduction theorem for conservative modal extensions of the basic systems. Modal postulates which are necessary and sufficient for that are postulates of $4 plus modal postulates which mimic structural rules. For example, the modal postulates which Girard has recently considered in linear logic are necessary and sufficient for the modal analogue of the deduction theorem. All this may lead towards results about functional completeness in categories. When functional completeness, which is analogous to the deduction theorem, fails, we may perhaps envisage a modal analogue of functional completeness in a modal category, of which our original category is a full subcategory. Key words: modal logic, substructural logics, derived rules, deduction theorem, functional completeness in categories. 1. INTRODUCTION Logicians sometimes make a distinction between derived rules of a logical system and rules admissible for a logical system. A rule is derived if we can pass from the premises to the conclusion by using the axioms and primitive rules of the system. To say that a rule is admissible means that if the premises are theorems, then the conclusion is also a theorem; i.e., that adding this rule to the system would not increase the stock of theorems. An altemative way to say that a rule is admissible for a system is to say that the system is closed under this rule. Every derived rule is admissible, but not the other way round. A famous example of an admissible rule that is not derived is cut in Gentzen's cut-free sequent formulations of classical and intuitionistic logic. The distinction between derived and admissible rules is important when we Journal of Logic, Language, ana Information 1:173-201, 1992. (~) 1992 Kluwer Academic Publishers. Printed in the Netherlands. 174 KOSTADO~EN want to prove a deduction theorem. Roughly speaking, a deduction theorem says that a derived, but not any admissible role, can be recorded as an arrow in the system. A derived rule is one we wish to hold not only for our system but for all its extensions as well, whereas our system may just happen to be closed under an admissible rule, and this need not hold for all its extensions. The notion of an admissible rule should not cause misunderstanding here. We will take the 'if .... then ...' involved in the definition of admissible rules as a material implication. (There is a related notion of a constructively admissible rule, where from a proof of the premises we can construct a proof of the conclusion, the construction not necessarily taking place inside our system; cf. Anderson and Belnap (1975: w 54; w With this notion, 'if .... then ...' should better be taken as an intuitionistic, or even weaker implication. To prevent confusion, we shall not rely on this notion here.) On the other hand, we shall allow derived rules to make us perplexed. Suppose: is a derived rule. Is then: X also derived for any X? Classical and intuitionistic logic force us to say yes, because we have: ~(X~) X X ---~ 9~ But what if our logic lacks cp ---+ (X ~ ~), as relevant logic notoriously does? Should we then feel obliged to accept in the metalogic a principle corresponding to ~ ~ (X ~ ~P), which would make the second rule above derived? So it seems that depending on what logic we accept as our metalogic we may end up with different notions of derived rule. In Section 1 of this paper we explore various notions of derived rule. We first formulate a family of propositional logical systems, starting with a variant of the nonassociative Lambek calculus, and ending up with a system corresponding to intuitionistic logic. In between we have systems correspond- ing to the associative Lambek calculus, linear logic, relevant logics and BCK logic. These basic systems make a family because they may all be obtained by gradually adding structural rules in a suitable sequent formulation, whereas rules for the connectives are kept invariant as in Dogen (1988) (such logics are MODAL LOGIC AS METALOGIC 175 called substructural in Dogen (1992)). The nonassociative Lambek calculus, which is minimal in this family, is a good candidate for a minimal implicative logic. Inspired by Lambek and Scott (1986), we formulate these basic systems so that their theorems are sequents of the form A ~- t3. This we do because we have in mind possible extensions of our approach to categories, where A F-/3 will be a morphism from A to/3. In principle, we could have presented our basic systems as more orthodox Gentzen-type sequent systems, but the main things we want to say can also be said with our simpler sequents A ~- /3. When the turnstile t- in A ~-/3 is replaced by the connective of implication -% our basic systems look like ordinary Hilbert-type systems, but we should then bear in mind that the rules for these systems are rules for theorems, and that every implication A --+ B standing for A F- B is prefixed with F-. All primitive rules of our basic systems will be binary, i.e., with two premises. We require this because we then build, above our basic systems, sequent systems especially well suited to represent deductions with binary rules. These sequent systems have a binary comma between formulae on the left of their tumstile, rather than the n-ary commas of usual Gentzen formu- lations. (This type of sequent system was, so far as I know, first considered in Lambek (1961), and was further studied in Belnap (1982) and Dogen (1988).) We get various sequent systems by gradually adding structural rules, redu- plicating in this manner the family of the underlying basic systems. In these sequent systems we formalize various notions of derived rule for our basic systems. So, above a basic system we may assume, as its deductive metalogic, a sequent system weaker, equal in strength, stronger, or incomparable, with our basic system. The fact that there are various notions of derived rule has repercussions on the deduction theorem: with some notions we can prove it and with others we cannot. In Section 2 we then investigate what conditions are necessary and sufficient to prove the deduction theorem for our basic systems. These conditions also happen to be sufficient for translating the metalogic into the basic logic. Our deduction theorem is what is also in ordinary logical usage properly called a deduction theorem. It is a deduction theorem of higher level: one of its instances says that if from t- A we can derive k- B, then A t- B is provable in the basic system. On the other hand, an instance of a deduction theorem of lower level, better called introduction of implication, says that if we have A k-/3, then we have ~- A ~ B, which is already incorporated in our basic systems. These two types of deduction theorem are easy to confuse in ordinary versions of classical and intuitionistic propositional logic, where we have the higher-level deduction theorem and can translate the metalogic into the basic logic. But already in formulations of predicate logic with a rule of universal generalization, or in formulations of modal logic with a rule of 17 6 KOSTADO~EN necessitation, we can make the distinction, because the higher-level deduction theorem falls in full generality, whereas the lower-level one obtains with a proper understanding of F, like in Gentzen's sequent systems (cf. Kleene, 1952: w167 As usual, these sequent systems are interpreted as if they were about natural deduction. From F A we can derive F Vz A and F DA, but we shouldn't have A F Vx A and A F DA. We shall see with our weak basic systems that this distinction between the higher-level and lower-level deduction theorem should be drawn even in what amounts to implicational logic. Some specific problems for the deduction theorem arise if we allow our systems to be formulated with a primitive rule of substitution for variables. We shall not consider here this matter, which would get us involved in a discussion about the proper understanding of variables. For the propositional systems we shall introduce, substitution, though admissible, is not a derived rule. In Section 3 we extend all we have done previously to a richer language for our basic systems, with the usual connectives of propositional logic. Next, in Section 4, which is the central section of this paper, we show that though we don't always have the deduction theorem, we may always obtain a modal analogue of the deduction theorem for modal extensions of our basic systems.