On the existence of a modal-logical basis for monadic second-order logic

摘要

Kamp (PhD Thesis, University of California, LA) proved that the tense logic of the connectives Until and Since is expressively complete over the class DCLO of Dedekind complete linear orders in the sense that this logic can express exactly the same conditions over DCLO as first-order logic. In the present article a modification of the question of expressive completeness is considered-the question of whether there exists a basis consisting of a finite number of modal-logical connectives for monadic second-order logic. The notion of κ-dimensional basis that Gabbay (1981, Aspects of Philosophical Logic, 91-117) defined relative to FO is generalized to arbitrary abstract logics, and it is shown that a finite 2-dimensional basis exists for MSO on the class FLO of all finite linear structures. Beauquier and Rabinovich (2002, J. Logic. Comput., 12, 243-253) have proven that there is no finite 1-dimensional basis for MSO on FLO. Thus, the result yielding a 2-dimensional basis cannot be improved.