The Fourth Scandinavian Logic Symposium and the First Soviet-Finnish Logic Conference were held in JyvaskyIa, Finland, June 29-July 6, 1976. The Conferences were organized by a committee which consisted of the editors of the present volume. The Conferences were supported financially by the Ministry of Education of Finland, by the Academy of Finland, and by the Division of Logic, Methodology, and Philosophy of Science of the International Union of History of Science. The Philosophical Society of Finland and the Jyvaskyla Summer Festival gave valuable help in various practicalities. 35 papers by authors representing 10 countries were presented at the two meetings. Of those papers 24 appear here. THE EDITORS v TABLE OF CONTENTS PREFACE v PART 1/ PROOF THEORY GEORG KREISEL / Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory 3 DAG PRAWITZ / Proofs and the Meaning and Completeness of the Logical Constants 25 v. A. SMIRNOV / Theory of Quantification and tff-calculi 41 LARS SVENONIUS/Two Kinds of Extensions of Primitive Recursive Arithmetic 49 DIRK VAN DALEN and R. STATMAN / Equality in the Presence of Apartness 95 PART II / INFINITARY LANGUAGES VEIKKO RANTALA / Game-Theoretical Semantics and Back-and- Forth 119 MAARET KAR TTUNEN / Infinitary Languages N oo~.

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InhaltsangabeI Proof Theory.- Some Facts from the Theory of Proofs and Some Fictions from General Proof Theory.- Proofs and the Meaning and Completeness of the Logical Constants.- Theory of Quantification and Promille-calculi.- Two Kinds of Extensions of Primitive Recursive Arithmetic.- Equality in the Presence of Apartness.- II Infinitary Languages.- Game-Theoretical Semantics and Back-and-Forth.- Infinitary Languages N?? and Generalized Partial Isomorphisms.- III Set Theory and Model Theory.- Generalizing Set-Theoretical Model Theory and an Analogue Theory on Admissible Sets.- Hierarchies of Model Theoretic Definability - An Approach to Second Order Logics.- Open Problems in the Theory of Ultrafilters.- IV Generalized Quantifiers.- The Reals Cannot Be Characterized Topologically with Strictly Local Properties and Countability Axioms.- On the Expressive Power of the Language Using the Henkin Quantifier.- Remarks on Free Quantifier Variables.- V Recursion Theory.- Recursion in 3E and a Splitting Theorem.- Retracts of Post's Numbering and Effectivization of Quantifiers.- VI Logic and Natural Language.- Quantifiers in Natural Languages: Some Logical Problems, I.- Models for Natural Languages.- Backwards-Looking Operators in Tense Logic and in Natural Language.- VII Philosophical Logic.- Paradoxes in a Semantic Perspective.- Hintikka's Possible Worlds and Rigid Designators.- On the Content Analysis of Two Normative Notions.- Singular Terms, Existence and Truth: Some Remarks on a First Order Logic of Existence.- VIII Truthlikeness.- On Distance From the Truth as a True Distance.- Truthlikeness in First-Order Languages.- Index of Names.