### On Non-eliminability of the Cut Rule and the Roles of Associativity and Distributivity in Non-commutative Substructural Logics

*Ueno, Takeshi;Nakatogawa, Koji;Watari, Osamu*

Permalink : http://hdl.handle.net/2115/75401

#### Abstract

We introduce a sequent calculus FL', which has at most one formula on the right side of sequent, and which excludes three structural inference rules, i.e. contraction, weakening and exchange. Our formulations of the inference rules of FL' are based on the results and considerations carried out in our previous papers on how to formulate Gentzenstyle natural deduction for non-commutative substructural logics. Our present formulation FL' of sequent system for non-commutative substructural logic, which has no structural rules, has the same proof strength as the ordinary and standard sequent calculus FL (Full Lambek), which is often called Full Lambek calculus, i.e., the basic sequent calculus for all other substructural logics. For the standard FL (Full Lambek), we use Ono's formulation. Although our FL' and the standard formulation FL (Full Lambek) are equivalent, there is a subtle difference in the left rule of implication. In the standard formulation, two parameters Γ1 and Γ2 (resp.), each of which is just an finite sequence of arbitrary formulas, appear on the left and right side (resp.) of a formula appearing on the left side of the sequent on the upper left side the left rule ⊃ (which corresponds to ⊃' in FL'). On the other hand, there is no such parameter on the left side of the sequent on the upper left side in the left rule for ⊃' of our system FL. In our system FL', Γ1 is always empty, and only Γ2 is allowed to occur in the left rule for ⊃' (similar differences occur in the multiplicative conjunction, additive conjunction and additive disjunction). This subtle difference between our system FL' and the standard system FL (Full Lambek) matters deeply, for we are led to a construction of proof-figures in FL', which show how the associativity of multiplicative conjunction and the distributivity of multiplicative conjunction over additive disjunction are involved in the eliminations of the cut rule in those proofs. We clarify and specify how associativity and distributivity are related to the non-eliminability of an application of the cut rule in those proof-figures of FL'.

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