A Modular Presentation of Modal Logics in a Logical Framework

We present a theoretical and practical approach to the modular natural deduction presentation of modal logics and their implementation in a logical framework Our work treats a large and well known class of modal logics including K KD T B S S S in a uniform way with respect to soundness and completeness for semantics and faithfulness and adequacy of the implementation Moreover it results in a pleasingly simple and usable implementation of these logics x Introduction Logical Frameworks such as the Edinburgh LF and Isabelle have been proposed as a solution to the problem of the explosion of logics and specialized provers for them However it is also acknowledged that this solution is not perfect these frameworks are best suited for encoding well behaved natural deduction formalisms whose metatheory does not deviate too far from the metatheory of the framework logic Modal logics in particular are considered di cult to implement in a clean direct way e g x and Encodings in both the LF and Isabelle have been proposed see section but they have been either Hilbert style or quite specialized and their correctness is subtle We present a method for encoding a large and useful class of propositional modal logics including K KD T B S S S in a natural deduction setting and show once and for all correctness for every encoding in the class We have implemented our work in Isabelle and the result is a simple usable and completely modular natural deduction implementation of these logics Let us consider in more detail the di culty with modal logics since the problem motivates the approach that we pursue The deduction theorem If by adding A as an axiom we can prove B then we can prove A B without A fails in modal logics A semantic explanation of this is that the standard completeness theorem for modal logics says that A i A is true at every world in every suitable Kripke frame hW Ri where W is the set of worlds and R is the accessibility relation Basically A means w W j w A and the deduction theorem states that w W j w A w W j w B w W j w A B where is implication in the meta language and is implication in the object language But this is false we have only w W j w A j w B w W j w A B Thus a naive embedding of a modal logic in a logical framework captures the wrong conse quence relation One solution to this problem is to turn to Hilbert presentations we reject this as it is well known that they are di cult to use in practice Instead motivated by the above semantic account we take the view of a logic as a Labelled Deductive System LDS proposed by Gabbay among others This approach pairs formulae with labels instead of proving A one proves w A where w represents the current world and w W w A i A Then it becomes possible to give a proof theoretic statement of the deduction theorem which is the analogue of the semantic version The same mechanism yields a direct formalization of modal operators like