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A note on generalized functional completeness in the realm of elementary logic

Abstract : 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 con-nectives (possibly infinitary ones), the first-order quantifiers and possibly some generalized quantifiers Q 1 , ..., Q n. Semantically, quantifiers are identified with classes of structures in a standard manner. 1 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. 2 Naturally associated with L, we have 1. an elementary equivalence relation between structures (≡ L), and 2. the class of elementary classes of L (El L). 3 1 See e.g. [9], p.235. For instance ∀ is the class of structures M, A such that M = A. 2 Note 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. 3 A class of structures is L-elementary in our terminology iff it is definable by a single sentence of L.
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  • HAL Id : hal-01992926, version 1

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Henri Galinon. A note on generalized functional completeness in the realm of elementary logic. Bulletin of the Section of Logic of the Polish Academy of Sciences, 2009, 38, pp.51 - 59. ⟨hal-01992926⟩

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