Universal logic
Encyclopedia
Universal logic is the field of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
that is concerned with giving an account of what features are common to all logical structures. Universal logic aims to be to logic what universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
is to algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
; currently there is no universally accepted notion of logic (or logical system). Several frameworks have been proposed.
The term 'universal logic' was introduced in the 1990s by Swiss logician Jean-Yves Béziau
Jean-Yves Béziau
Jean-Yves Béziau is a professor and researcher of the Brazilian Research Council - CNPq - at the Federal University of Ceara, Brazil. Béziau is a dual citizen of France and Switzerland...
, but the field has arguably existed for many decades. Some of the works of Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
in the early twentieth century, for example, can be regarded as fundamental contributions to universal logic.
Three model-theoretic
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
directions for universal logic have been explored to some depth: abstract model theory
Abstract model theory
In mathematical logic, abstract model theory is a generalization of model theory which studies the general properties of extensions of first-order logic and their models....
axiomatized by Jon Barwise
Jon Barwise
Kenneth Jon Barwise was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used....
, a topological / categorical approach based on sketches
Sketch (mathematics)
A sketch is a category D, together with a set of limit cones and a set of colimit cones. A model of the sketch in a category C is a functorM:D\rightarrow C...
(sometimes called categorical model theory), and yet another categorical
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
approach based on Goguen
Joseph Goguen
Joseph Amadee Goguen was a computer science professor in the Department of Computer Science and Engineering at the University of California, San Diego, USA, who helped develop the OBJ family of programming languages. He was author of A Categorical Manifesto and founder and Editor-in-Chief of the...
and Burstall
Rod Burstall
Rodney Martineau Burstall is one of four founders of the Edinburgh Laboratory for Foundations of Computer Science.He was an early and influential proponent of functional programming, pattern matching, and list comprehension, and is known for his work with Robin Popplestone on POP, an innovative...
's notion of institution
Institution (computer science)
The notion of institution has been created by Joseph Goguen and Rod Burstall in the late 1970sin order to deal with the "population explosion among the logical systems used incomputer science"...
.
The First World Congress and School on Universal Logic took place in Montreux, Switzerland in early 2005. Participants included Béziau
Jean-Yves Béziau
Jean-Yves Béziau is a professor and researcher of the Brazilian Research Council - CNPq - at the Federal University of Ceara, Brazil. Béziau is a dual citizen of France and Switzerland...
, Dov Gabbay
Dov Gabbay
Dov M. Gabbay is Augustus De Morgan Professor of Logic at the Group of Logic, Language and Computation, Department of Computer Science, King's College London . He has authored over four hundred and fifty research papers and over thirty research monographs...
, Saul Kripke
Saul Kripke
Saul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center...
, and David Makinson
David Makinson
David Clement Makinson, D.Phil, , is an Australian mathematical logician living in London, England.- Career :Makinson began his studies at Sydney University in 1958 and was an associate of the Libertarian Society and Sydney Push...
. A journal dedicated to the field, Logica Universalis
Logica Universalis
Logica Universalis is a peer-reviewed academic journal which covers research related to universal features of logic. Topics covered include tools and techniques for studying existing systems of logic and constructing new ones, the study of the various classes of logical systems, the scope of...
, with Béziau as editor-in-chief started to be published by Birkhäuser Basel (an imprint of Springer
Springer Science+Business Media
- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
) in 2007. Springer also started to publish a book series
Book series
A book series is a sequence of books having certain characteristics in common that are formally identified together as a group. Book series can be organized in different ways, such as written by the same author, or marketed as a group by their publisher....
on the topic, Studies in Universal Logic, with Béziau as series editor.
The term 'universal logic' has also been used by some logicians (e.g. Richard Sylvan
Richard Sylvan
Richard Sylvan was a philosopher, logician, and environmentalist.- Biography :Sylvan was born Francis Richard Routley in Levin, New Zealand, and his early work is cited with this surname...
and Ross Brady) to refer to a new type of (weak) relevant logic.