It is known that a number of inference principles can be used to trivialise the axioms of naïve comprehension - the axioms underlying the naïve theory of sets. In this paper we systematise and extend these known results, to provide a number of general classes of axioms responsible for trivialising naïve comprehension.

In this paper, constructions of free ordered algebras on one generator are given that correspond to some one-variable fragments of affine propositional classical logic and their extensions with n-contraction (n ≥ 2). Moreover, embeddings of the already known infinite free structures into the algebras introduced below are furnished with; thus, solving along the respective cardinality problems.

Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of the substructural logics (...) for which syntactic methods don't work well. (shrink)

We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.

We introduce structural rules mingle, and investigatetheorem-equivalence, cut- eliminability, decidability, interpolabilityand variable sharing property for sequent calculi having the mingle.These results include new cut-elimination results for the extendedlogics: FLm (full Lambek logic with the mingle), GLm(Girard's linear logic with the mingle) and Lm (Lambek calculuswith restricted mingle).

In this paper, constructions of free algebras corresponding to multiplicative classical linear logic, its affine variant, and their extensions with -contraction () are given. As an application, the cardinality problem of some one-variable linear fragments with -contraction is solved.

We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.