Jens Erik Fenstad

Situations, Language and Logic

I. Introduction.- 1. Prom linguistic form to situation schemata.- 2. Interpreting situation schemata.- 3. The logical point of view.- II. From Linguistic Form to Situation Schemata.- 1. Levels of linguistic form determining meaning.- 2. Motivation for the use of constraints.- 3. The modularization of the mapping from form to meaning.- 4. Situation schemata.- 5. The algorithm from linguistic form to situation schemata.- III. Interpreting Situation Schemata.- 1. The art of interpretation.- 2. The inductive definition of the meaning relation.- 3. A remark on the general format of situation schemata.- 4. Generalizing generalized quantifiers.- IV. A Logical Perspective.- 1. The mechanics of interpretation.- 2. A hierarchy of formal languages.- 2.1. Propositional logic.- 2.2. Predicate logic.- 2.3. Tense logic.- 2.4. Temporal predicate logic.- 2.5. Situated temporal predicate logic.- 3. Mathematical study of some formal languages.- 3.1. Definition of structure.- 3.2. The system L3.- 3.3. Modal operators.- 4. On the model theoretic interpretation of situation schemata.- 4.1. The basic correspondence.- 4.2. The correspondence extended.- V. Conclusions.- Appendices.- A. Prepositional Phrases in Situation Schemata.- by Erik Colban.- B. A Lyndon type interpretation theorem for many-sorted first-order logic.- C. Proof of the relative saturation lemma.- References.

Natural Language Systems

The formal study of natural languages has always had an algorithmic flavor. This is part of our inheritance from the pioneering work of N. Chomsky. His Syntactic Structures from 1957 marked a theoretical renewal of linguistic science. And with this renewal a bond was soon forged with symbolic logic, a part of which is formal language theory. Chomsky himself was one of the active participants in this development. This theoretical development soon joined forces with the emerging computer science and a vigorous field of computational linguistics was established. This science does not only have important applications to the study of natural languages, it is also an integral part of computer science, e.g. in compiler design.

THE AXIOM OF DETERMINATENESS

Publisher Summary This chapter presents axiom of determinateness and discusses certain infinite games with perfect information. It is a mathematically interesting problem to decide which games are determinate. An unrestricted assumption of the type that games Gx(A) are always determinate, seems like cheating. It is well known that the uncountable axiom of choice is inconsistent with determinateness. It does not seem to have been noticed earlier that the BPI also fails. In this connection, a remark of interest to the logician is added. To prove the general completeness theorem for first order logic one needs the BPI. In addition, without the general completeness theorem little will be left of model theory. It is a standard fact of measure theory that there are subsets of the real line that are not Lebesgue measurable.

Models for Natural Languages

“The structure of every sentence is a lesson in logic.” This often quoted statement of John Stuart Mill (1867) leads directly into a controversial area of theoretical grammar.

On Axiomatizing Recursion Theory

Publisher Summary This chapter focuses on axiomatizing recursion theory. This chapter focuses on how to choose the “correct” primitives for axiomatic recursion theory. New results, partly proved, partly conjectural, within the (modified) Moschovakis framework, are indicated in the chapter. Moschovakis framework emphasized the fact that whatever computations may be, they have a well-defined length, which always is an ordinal, finite or infinite. The basic definition of a computation theory is presented. Some basic facts about computation theories are listed. The basic result, due to Moschovakis in the axiomatic setting, is the first recursion theorem. The definition of regular computation theories are discussed in the chapter. Such theories have selection operators, hence a reasonable theory for the computable and semicomputable relations is proposed. For this class of theories, an adequate notion of finiteness is presented. The chapter addresses a problem of how to strengthen regularity. Of several possibilities two are emphasized—(1) the idea that a “computation” should be a finite object in the sense of the theory and (2) the theory should satisfy the prewellordering property.

A Logical Perspective

In this part of the monograph, we are going to use a hierarchy of logical formal languages to learn more about the preceding semantic enterprise. As was explained in the general introduction, our purpose is not only the traditional search for complete systems of inference, but also to investigate a variety of semantic notions appearing in situation semantics. More specifically, Chapter IV. 1 contains a general discussion of our semantic format, with a preview of the later model theory. The language hierarchy is developed in Chapter IV.2, with matching enrichments of the model structures at each step-introducing a variety of logical notions, questions and results. The main technical contribution is presented in Chapter IV.3, consisting of completeness and persistence theorems for several languages in our hierarchy, up to an expressive two-sorted one. Finally, Section IV.4 explains how the meaning relation, d, c〚SIT.ϕ〛s, of Chapter III is related to the model-theoretic approach of this chapter.

Tarski, truth and natural languages

The (rst part of the paper traces the history of the relationship between logic and linguistics with particular emphasis on the contributions of Tarski and Ajdukiewicz. In the second part we give a brief review of current work on formal semantics for natural language systems and argue for the need for a richer geometric structure on the semantic model space. c

Science between freedom and responsibility

The post Second World War period was a good time for modern science-driven technology; it had played a decisive part in the allied victory and now it was to be harnessed to the task of postwar reconstruction, promoting increased welfare, better health and improved security. But there were also misgivings related to the freedom in the conduct of science. Could science be freely pursued under the terms of a social contract so inextricably intertwined with national security concerns? After the end of the Cold War, new concerns emerged. The security element in the old contract had acquired a new meaning and was now understood in the sense of a protected environment, safe living conditions and future sustainability. Previously, science was the problem solver. Now science came to be seen as a major source of the problems. We have seen a shift from issues of freedom and trust to questions of responsibility and accountability. How should science respond?

ON THE COMPLETENESS OF SOME TRANSFINITE RECURSIVE PROGRESSIONS OF AXIOMATIC THEORIES

The well-known incompleteness results of Godel assert that there is no recursively enumerable set of sentences of formalized first order arithmetic which entails all true statements of that theory. It is equally well known that by introducing sufficiently nonconstructive rules, such as the ω-rule of induction, completeness can be re-established. Starting from the work of Turing [4] Feferman in [1] developed another method, viz. the study of transfinite recursive progressions of theories, for closing the gap between Godel (recursively enumerable sets of axioms yield incompleteness) and Tarski (number-theoretic truth is not arithmetically definable).

How Mathematics is Rooted in Life

Mathematics is almost always an insider’s affair. But sometimes things happen within the mathematical community that have a relevance, and perhaps also an interest, beyond the tribe itself. The Grundlagenstreit of the 1920s is such an example. In this review essay we tell this story with focus on the main actors involved, David Hilbert in Gottingen and L. E. J. Brouwer in Amsterdam. We shall see how fine points concerning the existence of mathematical objects, the question of the editorship of the Mathematische Annalen, and the attempts to resume normal scientific contacts between French and Germans scientists after the First World War led to an unusual bitter conflict within the tribe and beyond. But even if the effects of the fight were at the time negative, the long range outcome was positive. Hilbert’s work on the foundation of mathematics is still a powerful influence on current research, and Brouwer’s view on the constructive foundation of mathematics, which at the time inspired both Husserl and Wittgenstein, is today of increasing importance in the evolving science of logic and computing.