The logic of linear tolerance

A nonempty sequence T₁,...,T_n of theories is tolerant, if there are consistent theories T₁⁺,..., T_n⁺such that for each 1 i n, T_i⁺is an extension of T_i in the same language and, if i n, T_i⁺interprets T_i+1⁺. We consider a propositional language with the modality , the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOL. It is shown that TOL (resp. TOL) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if (A₁,..., A_n) is understood as (a formalization of) PA+A₁, ..., PA+A_n is tolerant.