A Proof-Theoretic Proof of Functional Completeness for Many Modal and Tense Logics

In what follows we shall use display logic to define a proof-theoretic semantics in terms of general introduction schemata. It will be shown that with respect to this semantics the set of connectives {[F], [P], ∧, ¬} is functionally complete for every displayable normal propositional tense logic and the set of connectives {[F], ∧, ¬} is functionally complete for every displayable normal propositional modal logic. It seems that there exists no other proof-theoretic characterization of modal operators (apart from intuitionistic implication ⊃ h ) in the literature.

[1]  Heinrich Wansing.  Tarskian Structured Consequence Relations and Functional Completeness , 1995 .

[2]  John P. Burgess,et al.  Basic Tense Logic , 1984 .

[3]  M. de Rijke.  Advances in intensional logic , 1997 .

[4]  D. Gabbay.  A General Theory of Structured Consequence Relations , 1995 .

[5]  Heinrich Wansing,et al.  A Full-Circle Theorem for Simple Tense Logic , 1997 .

[6]  Heinrich Wansing,et al.  The Logic of Information Structures , 1993, Lecture Notes in Computer Science.

[7]  Heinrich Wansing,et al.  Proof Theory of Modal Logic , 1996 .

[8]  Nuel Belnap,et al.  Linear Logic Displayed , 1989, Notre Dame J. Formal Log..

[9]  Franz Kutschera.  Ein verallgemeinerter Widerlegungsbegriff für Gentzenkalküle , 1969 .

[10]  Heinrich Wansing.  Strong Cut-elimination in Display Logic , 1995, Reports Math. Log..

[11]  Kazuo Matsumoto,et al.  Gentzen method in modal calculi. II , 1957 .

[12]  Heinrich Wansing,et al.  Sequent Calculi for Normal Modal Proposisional Logics , 1994, J. Log. Comput..

[13]  Franz Kutschera,et al.  Die Vollständigkeit des Operatorensystems {¬, ∨, ⊃} für die Intuitionistische Aussagenlogik im Rahmen der Gentzensematik , 1968 .

[14]  HEINRICH WANSING.  Functional completeness for subsystems of intuitionistic propositional logic , 1993, J. Philos. Log..

[15]  Peter Schroeder-Heister,et al.  A natural extension of natural deduction , 1984, Journal of Symbolic Logic.

[16]  M. Kracht.  Power and Weakness of the Modal Display Calculus , 1996 .