Equations, fixed points, and nonclassical logics

A fixed point is a solution of an equation of the form $p=\Phi(p,q,r,\dots)$, where $\Phi$ is an operator, $p$ is a variable, and $q,r,\dots$ are parameters. The nature of the operator $\Phi$ and the relation "$=$" may be different. In the case of modal logics, $\Phi$ is a propositional formula with modal operators, and the relation "$=$" turns into a logical equivalence $\lra$. The expression $p\lra\Phi(p,q,r,\dots)$ itself is understood as a theorem of some modal logic or as a formula true on some class of Kripke models. A fixed point is called determinable if the solution of the modal equation is expressible by a formula independent on $p$. The central line of research of S. I. Mardaev, a bright representative of the Novosibirsk school of non-classical logic, is the creation of the theory of determinability of fixed points of modal operators.
The talk will give an accessible introduction to this problematic. A general definition of logic as a closure operator on a completely free algebra will be given, the notion of equivalent algebraic semantics will be introduced, and Kripke semantics as a representation of a special kind for algebraic models will be introduced. We will conclude with examples of the most important results of S. I. Mardaev.

^* The report is dedicated to Sergey Mardaev (06.04.1962-10.04.2013)