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

Bulletin of the Section of Logic 38 (1/2):1-9 (2009)
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We can think of functional completeness in systems of propositional logic as a form of expressive completeness: while every logical constant in such system expresses a truth-function of finitely many arguments, functional completeness garantees that every truth-function of finitely many arguments can be expressed with the constants in the system. From this point of view, a functionnaly complete system of propositionnal logic can thus be seen as one where no logical constant is missing. Can a similar question be formulated for quantified first-order logics ? How to make sense of the question whether, e.g., ordinary first-order logic is "functionaly" complete or have no logical constant missing ? In this note, we build on a suggestive proposal made by Bonnay(2006) and shows that it is equivalent to the criterion that a first-order logic L be functionaly complete if and only if every class of structures closed under L-elementary equivalence is L-elementary. Ordinary first-order logic is not complete in this sense. We raise the question whether any logic can be.



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The Bounds of Logic: A Generalized Viewpoint.Gila Sher - 1994 - British Journal for the Philosophy of Science 45 (4):1078-1083.
Elementary Logic.G. B. Keene - 1966 - Philosophical Quarterly 16 (65):400-401.

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