THE ALGEBRAIC INTERPRETATION OF CLASSICAL AND
                      INTUITIONISTIC QUANTIFIERS
                            Dana S. Scott
                      Carnegie Mellon University
                    Pittsburgh, Pennslyvania, USA

ABSTRACT: In their 1963 book, "The Mathematics of Metamathematics",
Rasiowa and Sikorski present an approach to completeness theorems of
various logics using algebraic methods.  This idea can of course be
traced back to Boole, but it was revived and generalized by Stone and
Tarski in the 1930s; however, the most direct influence on their work
came from their well-known colleague, Andrzej Mostowski, after WW II.
Mostowski's interpretation of quantification can as well be given for
intutionistic as classical logic.  The talk will briefly review the
history and content of these ideas and raise the question of why there
was at that time no generalization made to higher-order logic and set
theory.  Entirely new light on this kind of algebraic semantics has
more recently been thrown by the development of topos theory in
category theory.  Reasons for pursuing this generalization will also
be discussed.