From Hilbert proofs to consecutions and back

Restall set forth a "consecution" calculus in his An Introduction to Substructural Logics. This is a natural deduction type sequent calculus where the structural rules play an important role.  This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules

From Hilbert proofs to consecutions and back

Restall set forth a "consecution" calculus in his "An Introduction to Substructural Logics." This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules

Ternary relations and relevant semantics

Modus ponens provides the central theme. There are laws, of the form A→C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theorie

www.elsevier.com/locate/apal Ternary relations and relevant semantics

Modus ponens provides the central theme. There are laws, of the form A → C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theories L; T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product (or fusion) operation ◦ on theories (or other collections of formulas) L and T, whence RLTU i L◦T ⊆ U. These ideas have been expressed, especially with Routley, as (Kripke style) worlds semantics for relevant and other substructural logics. Worlds are best demythologized as theories, subject to truth-functional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians ’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates. Each logic L gives rise to a Calculus of L-theories, on which particular candidate logica