Abstract
This paper uses the notion of Galois-connection to examine the relation between valuation-spaces and logics. Every valuation-space gives rise to a logic, and every logic gives rise to a valuation space, where the resulting pair of functions form a Galois-connection, and the composite functions are closure-operators. A valuation-space (resp., logic) is said to be complete precisely if it is Galois-closed. Two theorems are proven. A logic is complete if and only if it is reflexive and transitive. A valuation-space is complete if and only if it is closed under formation of super-valuations.
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Hardegree, G.M. Completeness and super-valuations. J Philos Logic 34, 81–95 (2005). https://doi.org/10.1007/s10992-004-6302-6
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DOI: https://doi.org/10.1007/s10992-004-6302-6