The property of being true in almost all finite, initial segments of the standard model for arithmetic is a Σ 0 2-complete property. Thus, it admits a kind of a weak truth definition. We define such an arithmetical predicate. Then we define its modal logic SL and prove a completeness theorem with respect to finite models semantics. The proof that SL is the modal logic of a weak truth definition for finite arithmetical models is based on an extension of SL by a fixpoint construction.

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