From the point of view of non-classical logics, Heyting's implication is
the smallest implication for which the deduction theorem holds. This
book studies properties of logical systems having some of the classical
connectives and implication in the neighbourhood of Heyt- ing's
implication. I have not included anything on entailment, al- though it
belongs to this neighbourhood, mainly because of the appearance of the
Anderson-Belnap book on entailment. In the later chapters of this book,
I have included material that might be of interest to the intuitionist
mathematician. Originally, I intended to include more material in that
spirit but I decided against it. There is no coherent body of material
to include that builds naturally on the present book. There are some
serious results on topological models, second order Beth and Kripke
models, theories of types, etc., but it would require further research
to be able to present a general theory, possibly using sheaves. That
would have postponed pUblication for too long. I would like to dedicate
this book to my colleagues, Professors G. Kreisel, M.O. Rabin and D.
Scott. I have benefited greatly from Professor Kreisel's criticism and
suggestions. Professor Rabin's fun- damental results on decidability and
undecidability provided the powerful tools used in obtaining the
majority of the results reported in this book. Professor Scott's
approach to non-classical logics and especially his analysis of the
Scott consequence relation makes it possible to present Heyting's logic
as a beautiful, integral part of non-classical logics.
