A Note on Generalized Functional Completeness in the Realm of Elementary Logic

In “Logicality and Invariance” (2008), Denis Bonnay introduced a generalized notion of functional completeness. In this note we call attention to an alternative characterization that is both natural and elementary. 1. Maximizing the expressive power of a logic Let L be a logic whose logical vocabulary contains truth-functional connectives (possibly infinitary ones), the first-order quantifiers and possibly some generalized quantifiers Q1, ..., Qn. Semantically, quantifiers are identified with classes of structures in a standard manner. Satisfaction in a structure is defined in conformity with the meaning of the logical constants chosen, e.g., M Qxφ(x) iff 〈M, {a : M φ(a)}〉 ∈ Q. Given those constraints, a logic L can be identified with a set of logical constants. Naturally associated with L, we have 1. an elementary equivalence relation between structures (≡L), and 2. the class of elementary classes of L (ElL). 1See e.g. [9], p.235. For instance ∀ is the class of structures 〈M,A〉 such thatM = A. 2Note that this definition of a logic is not purely semantic, and thus is less general that the one usually found in abstract model theory. See for instance [4], Definition 1.1.1. for the general definition. 3A class of structures is L-elementary in our terminology iff it is definable by a single sentence of L.