Axiom systems we represented in this article, and a sequent calculus was introduced in (Artemov 1995, Artemov 2001).It has the curious disadvantage that it is cut free, but does not have the subformula property no version with the subformula property is known.
Consistency would then follow from the normalization of derivations and the subformula property.
Prawitz gave first a normalization theorem and subformula property for a system of natural deduction for classical logic.
This leads to the desirable subformula property for GL, because all formulas that appear in a cut-free proof are subformulas of the endsequent formulas.
Formulas in derivations are arranged in a linear succession, but Jaskowski's paper of 1934 remained fragmentary and without substantial results such as a subformula property.
We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut elimination.
By exploiting the subformula property a natural deduction automated theorem prover can drastically reduce its search space and bring the backward application of the elimination rules under control (Portoraro 1998, Sieg and Byrnes 1996).
This is hardly surprising since the subformula property which Cut Elimination secures itself guarantees that the rules for one connective conservatively extend the subsystem comprising the rules for the remaining connectives.
Fortunately, NK enjoys the subformula property in the sense that each formula entering into a natural deduction proof can be restricted to being a subformula of Γ ∪ Δ ∪, where Δ is the set of auxiliary assumptions made by the ~-Elimination rule.
Hacking shows that, given certain background semantic assumptions (bivalence, valid inference preserves truth), any introduction and elimination rules meeting certain proof-theoretic conditions (subformula property, provability of elimination theorems for Cut, Identity, and Weakening) will uniquely determine a semantics for the constant they govern (Hacking 1979, 311 314).
