Institutional model theory generalizes a large portion of first-order model theory to an arbritary logical system. The notion of "logical system" here is formalized as an institution. Institutions constitute a model-oriented meta-theory on logical systems similar to how the theory of rings and modules constitute a meta-theory for classical linear algebra. Another analogy can be made with universal algebra versus groups, rings, modules etc. By abstracting away from the realities of the actual conventional logics, it can be noticed that institution theory comes in fact closer to the realities of non-conventional logics. Institutional model theory analyzes and generalizes classical model-theoretic notions and results, like It has been suggested that Predicate calculus be merged into this article or section. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... The notion of institution has been created by Joseph Goguen and Rod Burstall in the early 1980s in order to deal with the population explosion among the logical systems used in computer science. The notion tries to capture the essence of what a logical system is. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

• elementary diagrams
• elementary embeddings
• ultraproducts, Los' theorem
• saturated models
• axiomatizability
• (quasi-)varities, Birkhoff axiomatizability
• Craig interpolation
• Beth definability
• Gödel's completeness theorem

For each concept and theorem, the infrastructure and properties required are analyzed and formulated as conditions on institutions, thus providing a detailed insight on which properties of first-order logic they rely and how much they can be generalized to other logics. In mathematical logic, given models and in the same language , a function is called an elementary embedding if is an elementary substructure of . ... An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones, and in model theory, a branch of mathematical logic. ... In mathematical logic, and in particular model theory, a saturated model M is one which realizes as many complete types as may be reasonably expected given its size. ... In mathematics, an axiomatizable class is a class of mathematical structures which are all models of a fixed set of sentences in formal (typically first order) logic. ... In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ... Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician. ... Evert Willem Beth (July 7, 1908 – April 12, 1964) was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics. ... Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... Gödels completeness theorem is a fundamental theorem in mathematical logic proved by Kurt Gödel in 1929. ...

References

• Marc Aiguier and Fabrice Barbier: An institution-independent proof of Beth definability theorem. Studia Logica, to appear.
• Daniel Gǎinǎ and Andrei Popescu: An institution-independent proof of Robinson's consistency theorem. Studia Logica, to appear.
• Razvan Diaconescu: Jewels of Institution-Independent Model Theory. In: K. Futatsugi, J.-P- Jouannaud, J. Meseguer (eds.): Algebra, Meaning and Computation. Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday. Lecture Notes in Computer Science 4060, p. 65-98, Springer-Verlag, 2006.
• Marius Petria and Rãzvan Diaconescu: Abstract Beth definability in institutions. Journal of Symbolic Logic 71(3), p. 1002-1028, 2006.
• Daniel Gǎinǎ and Andrei Popescu: An institution-independent generalisation of Tarski's elementary chain theorem, Journal of Logic and Computation, to appear, 2006.
• Razvan Diaconescu: Proof systems for institutional logic. Journal of Logic and Computation 16(3), p. 339-357, 2006.
• Till Mossakowski, Joseph Goguen, Rãzvan Diaconescu, Andrzej Tarlecki: What is a Logic?. In Jean-Yves Beziau, editor, Logica Universalis, pages 113-133. Birkhauser, 2005.
• Rãzvan Diaconescu, Petros Stefaneas: Possible Worlds Semantics in arbitrary Institutions. IMAR Preprint 7-2003, ISSN 250-3638.
• Razvan Diaconescu: Elementary diagrams in institutions. Journal of Logic and Computation. 14(5):651-674, 2004.
• Razvan Diaconescu: Herbrand Theorems in arbitrary institutions. Information Processing Letters. 90:29-37, 2004.
• Razvan Diaconescu: An institution-independent proof of Craig Interpolation Property. Studia Logica, 77(1):59-79, 2004.
• Razvan Diaconescu: Interpolation in Grothendieck Institutions. Theoretical Computer Science, 311:439-461, 2003.
• Razvan Diaconescu: Institution-independent Ultraproducts. Fundamenta Informaticae, 55(3-4):321-348, 2003.
• Andrzej Tarlecki: On the existence of free models in abstract algebraic institutions. Theoretical Computer Science 37, p. 269-304, 1986.
• Andrzej Tarlecki: Quasi-varieties in abstract algebraic institutions. Journal of Computer and System Sciences 33(3), p. 333-360, 1986.
• Razvan Diaconescu's publication list - contains recent work on institutional model theory