Continuous logic is a logic whose truth values can take continuous values in . The main variant used in model theory is motivated by the model theory of Banach spaces and similar structures. The language has connectives for each continuous function and the quantifiers are interpreted as infimum and supremum. The models of this logic are bounded complete metric structures equipped with uniformly continuous maps and -valued predicates.
As well as satisfying a form of completeness theorem,
…continuous first-order logic satisfies suitably phrased forms of the compactness theorem, the Löwenheim-Skolem theorems, the diagram arguments, Craig’s interpolation theorem, Beth’s definability theorem, characterizations of quantifier elimination and model completeness, the existence of saturated and homogeneous models results, the omitting types theorem, fundamental results of stability theory, and nearly all other results of elementary model theory. (Yaacov & Pedersen 10)
Building on the proposal of Lawvere to understand a form of metric space as a category enriched in the monoidal poset , there have been attempts to consider continuous logic as a similarly enriched logic. In (Albert & Hart), the authors develop a parallel for conceptual completeness via the notion of a metric pretopos.
Simon Cho argues that the object of truth values of continuous logic, , should be seen as a “continuous subobject classifier” in a similar sense to the subobject classifier of topos theory where subobjects are classified via pullbacks (Cho 19).
Continuous logic is introduced in
motivated by
A recent version is in
Model Theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Volume 2, Cambridge University Press, 2008, pdf
A discussion of conceptual completeness in the setting of continuous logic is found in
A treatment of metric space semantics for continuous logic as a variety of enriched categories is given in
See also his thesis
For syntax-semantics duality in the case of infinitary continuous logic, see
An older and rather different system also called continuous logic of a Russian school is surveyed
There is a version of abstract elementary classes in the setting of continuous logic, metric abstract elementary classes.
An analysis which avoids enriched logic is
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