:''This page is about the concept in mathematical logic. For the concepts in

sociology Sociology is a social science that focuses on society, human social behavior, patterns of Interpersonal ties, social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of Empirical ...

, see

Institutional theory In sociology and organizational studies, institutional theory is a theory on the deeper and more resilient aspects of social structure. It considers the processes by which structures, including schemes, rules, norms, and routines, become establishe ...

and

Institutional logic Institutional logic is a core concept in sociological theory and organizational studies, with growing interest in marketing theory. It focuses on how broader belief systems shape the cognition and behavior of actors. Friedland and Alford (1991) w ...

''. In

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

, institutional model theory generalizes a large portion of

first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

to an arbitrary

logical system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...

Overview

The notion of "logical system" here is formalized as an

institution Institutions are humanly devised structures of rules and norms that shape and constrain individual behavior. All definitions of institutions generally entail that there is a level of persistence and continuity. Laws, rules, social conventions a ...

. Institutions constitute a model-oriented meta-theory on logical systems similar to how the theory of

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

s and

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

s constitute a meta-theory for classical

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

. Another analogy can be made with

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...

versus

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

s 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 * elementary diagrams *

elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often ...

s *

ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors ...

s, Los' theorem *

saturated model In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \al ...

s * axiomatizability *

varieties Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...

, Birkhoff axiomatizability *

Craig interpolation In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable sy ...

* Robinson consistency *

Beth definability In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicit definability. Specifically Beth definability states that the two senses of definability are equivalent. First-order logic has th ...

* Gödel's

completeness theorem Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

For each concept and theorem, the infrastructure and properties required are analyzed and formulated as conditions on institutions, thus providing a detailed insight to which properties of first-order logic they rely on and how much they can be generalized to other logics.

References

* Răzvan Diaconescu
Institution-Independent Model Theory
Birkhäuser, 2008. . * Răzvan 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 The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zentra ...

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 The ''Journal of Logic and Computation'' is a peer-reviewed academic journal focused on logic and computing. It was established in 1990 and is published by Oxford University Press under licence from Professor Dov Gabbay Dov M. Gabbay (; born Oc ...

16(6), p. 713-735, 2006. * Till Mossakowski,

Joseph Goguen __NOTOC__ Joseph Amadee Goguen ( ; June 28, 1941 – July 3, 2006) was an American computer scientist. He was professor of Computer Science at the University of California and University of Oxford, and held research positions at IBM and SRI In ...

, Rãzvan Diaconescu, Andrzej Tarlecki: What is a Logic?. In

Jean-Yves Beziau Jean-Yves is a French masculine given name. Notable persons with that name include: * Jean-Yves André (born 1977), Mauritian footballer * Jean-Yves Anis (born 1980), French footballer * Yves Jean-Bart (born 1947), Haitian football executive * Je ...

, editor, Logica Universalis, pages 113-133. Birkhauser, 2005. * Andrzej Tarlecki: Quasi-varieties in abstract algebraic institutions.

Journal of Computer and System Sciences The ''Journal of Computer and System Sciences'' (JCSS) is a peer-reviewed scientific journal in the field of computer science. ''JCSS'' is published by Elsevier, and it was started in 1967. Many influential scientific articles have been publishe ...

33(3), p. 333-360, 1986.

External links

Răzvan Diaconescu's publication list
- contains recent work on institutional model theory {{Mathematical logic Mathematical logic Model theory Theoretical computer science