Summary of the talk given to the 22nd Conference on the History of Logic, Cracow (Poland), July 5–9, 1976.

This note concerns the ultraproduct construction. It is observed that in the class of SCI models ultraproducts may be constructed by a method apparently dierent from the standard one. Since the standard models of veryday model theory form a subclass of SCI models, one obtains a new view of ultraproducts. Aside from a few remarks, the paper is self-contained.

This report brings out a simple observation on the close connection of lters with algebraic closure systems. In [1], Orrin Frink gave a general denition of ideals in ordered sets. Here, we use the dual notion of lter and apply it to preordered sets. When referring to nite sets fc1; : : : ; ckg we often omit the brackets. The symbol ; denotes the empty set and, X f Y means that X is a nite subset of Y.

This note calls attention to the fact that the natural extensions of standard logics appear to be an application of the method of inductive generation of logics . One can generalize our observations beyond the range of algebraic logics under suitable conditions on cardinals of certain involved sets.