Rule-irredundancy and the sequent calculus for core logic.

*(English)*Zbl 1403.03117Summary: We explore the consequences, for logical system-building, of taking seriously (i) the aim of having irredundant rules of inference, and (ii) a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of Reflexivity, Thinning, and Cut.

Reflexivity survives in the minimally necessary form \(\phi:\phi\). Proofs have to get started.

Cut is subject to a Cut-elimination theorem, to the effect that one can always make do without applications of Cut. So Cut is redundant, and should not be a rule of the system.

Cut-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of Thinning. But Thinning, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without Cut or Thinning. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic.

Given any intuitionistic Gentzen proof of \(\Delta:\phi\), one can determine from it a Core proof of some subsequent of \(\Delta:\phi\). Given any classical Gentzen proof of \(\Delta:\phi\), one can determine from it a classical Core proof of some subsequent of \(\Delta:\phi\). In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is \(\phi:\phi\).

Reflexivity survives in the minimally necessary form \(\phi:\phi\). Proofs have to get started.

Cut is subject to a Cut-elimination theorem, to the effect that one can always make do without applications of Cut. So Cut is redundant, and should not be a rule of the system.

Cut-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of Thinning. But Thinning, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without Cut or Thinning. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic.

Given any intuitionistic Gentzen proof of \(\Delta:\phi\), one can determine from it a Core proof of some subsequent of \(\Delta:\phi\). Given any classical Gentzen proof of \(\Delta:\phi\), one can determine from it a classical Core proof of some subsequent of \(\Delta:\phi\). In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is \(\phi:\phi\).

##### MSC:

03F05 | Cut-elimination and normal-form theorems |

03F03 | Proof theory in general (including proof-theoretic semantics) |