A nonempty sequence 〈T1,...,Tn〉 of theories is tolerant, if there are consistent theories T 1 + ,..., T n + such that for each 1 ≤i ≤n, T i + is an extension of Ti 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 TOLω. It is shown that TOL (resp. TOLω) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if ◊(A1,..., An) is understood as (a formalization of) “〈 PA+A1, ..., PA+An〉 is tolerant”.
KeywordsMathematical Logic Computational Linguistic Decidable Logic Consistent Theory Propositional Language
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