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mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that
first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
is the '' strongest logic'' (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property. Lindström's theorem is perhaps the best known result of what later became known as abstract model theory, the basic notion of which is an abstract logic; the more general notion of an
institution An institution is a humanly devised structure of rules and norms that shape and constrain social behavior. All definitions of institutions generally entail that there is a level of persistence and continuity. Laws, rules, social conventions and ...
was later introduced, which advances from a set-theoretical notion of model to a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
-theoretical one. Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers. Jouko Väänänen
Lindström's Theorem
/ref> Lindström's theorem has been extended to various other systems of logic, in particular
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
s by Johan van Benthem and Sebastian Enqvist.


Notes


References

* Per Lindström, "On Extensions of Elementary Logic", ''
Theoria Christian mysticism is the tradition of mysticism, mystical practices and mystical theology within Christianity which "concerns the preparation f the personfor, the consciousness of, and the effect of ..a direct and transformative pr ...
'' 35, 1969, 1–11. * Johan van Benthem, "A New Modal Lindström Theorem", '' Logica Universalis'' 1, 2007, 125–128. * * Sebastian Enqvist, "A General Lindström Theorem for Some Normal Modal Logics", '' Logica Universalis'' 7, 2013, 233–264. * * Shawn Hedman, ''A first course in logic: an introduction to model theory, proof theory, computability, and complexity'', Oxford University Press, 2004, , section 9.4 {{Mathlogic-stub Mathematical logic Theorems in the foundations of mathematics Metatheorems