Algebraic logic

With a long and distinguished history, this began perhaps with the work of Boole and De Morgan in the nineteenth century, attempting to characterise logical properties in an algebraic manner. The well-known boolean algebras correspond to propositional logic, and equally to unary (1-place) relations, and they work very successfully. A field of unary relations can be viewed as a boolean algebra; and conversely, every boolean algebra can be represented as unary relations. This representation (or completeness) theorem was proved by Stone in 1936.

Attempts were made by Peirce (also see the Peirce entry in the St. Andrews history of mathematicians) and Schroder around the turn of the twentieth century to extend the study to binary relations. Great problems arose, and it seemed empirically that no simple algebraic characterisation would be found here.

The work was taken up again by Tarski from the 1940s. Tarski defined an approximation to a full calculus of binary relations, which he called relation algebras. The definition evolved and emerged slowly, culminating in a joint paper with Jónsson in 1948. It turned out that these relation algebras did not embody the whole story of binary relations: in 1964, Monk proved that no finite axiomatisation of all the properties of binary relations was possible, so confirming the earlier suspicions.

A large group of people worked with Tarski on algebras of binary and higher relations, including Chin, Givant, Henkin, Jónsson, Lyndon, Maddux, Monk, Nemeti. The study proceeded in several directions. One, that of giving an algebraic foundation to mathematics, led to the book A Formalization of Set Theory Without Variables by Tarski and Givant that appeared in 1987. Another line of work concerned finer degrees of approximation to binary (and higher) relations, and is reflected in papers of Maddux from the 1980s and 1990s connected with finite-variable logics, among other places. The question of finding a transparent (albeit infinite) axiomatisation of fields of relations also attracted attention.

Algebraic logic has recently become of great interest in modal logic: both fields are often studying the same phenomena, by different names. Influential figures here are van Benthem and the ILLC of University of Amsterdam, and the Hungarian school. Students having worked in both groups include Yde Venema, Maarten Marx, Szabolcs Mikulás. The last two, and also Robin Hirsch, Agnes Kurucz, and Mark Reynolds, are or were part of a circle of researchers in pure and applied algebraic (and modal) logic in London.
 
 

Links above are to the person's home page, if I could find one, and otherwise to some information about them.  Links updated March 2006.