nLab continuous logic

Idea

Continuous logic is a logic whose truth values can take continuous values in $[0,1]$. 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 $c: [0,1]^n \to [0,1]$ 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 $[0,1]$-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)

In terms of enriched category theory

Building on the proposal of Lawvere to understand a form of metric space as a category enriched in the monoidal poset $([0, \infty], \geq)$, 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, $[0,1]$, 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).

References

Continuous logic is introduced in

• C. W. Henson, J. Iovino, Ultraproducts in analysis in: Analysis and Logic, London Math. Soc. Lecture Note Series 262, Cambridge University Press 2002.

motivated by

• Chen-Chung Chang, Jerome H. Keisler, Continuous Model Theory, Annals of Mathematics Studies 58, 1966.

A recent version is in

• Itaï Ben Yaacov, Uncountable dense categoricity in cats, J. Symb. Logic 70, 829–860, 2005
• Itaï Ben Yaacov, Continuous first-order logic and logical stability, pdf
• Itaï Ben Yaacov, Arthur Pedersen, A proof of completeness for continuous first order logic, Journal of Symbolic Logic 75, 168–190, 2010, (arXiv:0903.4051).
• Itaï Ben Yaacov, Alex Usvyatsov. Logic of metric spaces and Hausdorff CATs
• Itaï Ben Yaacov, Alexander Berenstein, Ward C. Henson, Alexander Usvyatsov,

  Model Theory for metric structures, in Model Theory with Applications to Algebra and Analysis, Volume 2, Cambridge University Press, 2008, pdf

• Alexander Berenstein, Andrés Villaveces, Hilbert spaces with random predicates, pdf
• Jerome Keisler, Model Theory for Real-valued Structures, (arXiv:2005.11851)

A discussion of conceptual completeness in the setting of continuous logic is found in

• Jean-Martin Albert, Bradd Hart. Metric logical categories and conceptual completeness for first order continuous logic, (arXiv:1607.03068)

A treatment of metric space semantics for continuous logic as a variety of enriched categories is given in

• Simon Cho, Categorical semantics of metric spaces and continuous logic, (arXiv:1901.09077)

See also his thesis

• Simon Cho, Continuity in enriched categories and metric model theory, (thesis)

For syntax-semantics duality in the case of infinitary continuous logic, see

• Ruiyuan Chen, Representing Polish groupoids via metric structures, (arXiv:1908.03268)

An older and rather different system also called continuous logic of a Russian school is surveyed

• Vitaly I. Levin, Basic concepts of continuous logic, Studies in logic grammar and rethoric, 11 (24) 2007, pdf

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

• Daniel Figueroa, Benno van den Berg, A topos for continuous logic (arXiv:2107.10543)