Definability and Undefinability in Logical Languages: Tools for the Monadic Case
Stanley Peters · 2008
Abstract To prove a positive definability result, one usually has to exhibit the defining sentence, and it helps if this sentence belongs to a well-defined language. To prove a negative undefinability result, on the other hand, one needs to verify that no sentence works as a definition. Then it is crucial that the eligible sentences are generated by a precise grammar and have precise truth conditions. This chapter examines the notion of relative expressive power for logical languages as well as two measures of the syntactic complexity of their formulas, which is used to prove claims about all formulas of a given logic by induction. It considers the appropriate notion of definability of a quantifier in a logic to show that relative expressive power can be cashed out in terms of definability. A number of examples of definable quantifiers are given, and the crucial role of isomorphism closure in logical languages is pointed out. The principal strategy for proving that a quantifier is not definable in a given logic is discussed.