My chief aim has been to convey the thought that the application of model theoretic techniques to natural languages needn't force a distortion of intentional phenomena. I hope that at least I have succeeded in accomplishing this.

ABSTRACT This paper introduces a multi-valued variant of higher-order resolution and proves it correct and complete with respect to a variant of Henkin's general model semantics. This resolution method is parametric in the number of truth values as well as in the particular choice of the set of connectives (given by arbitrary truth tables) and even substitutional quantifiers. In the course of the completeness proof we establish a model existence theorem for this logical system. The work reported in this paper (...) provides a basis for developing higher-order mechanizations for many non-classical logics. (shrink)

§1. Introduction. This paper deals with aspects of my doctoral dissertation which contributed to the early development of model theory. What was of use to later workers was less the results of my thesis, than the method by which I proved the completeness of first-order logic—a result established by Kurt Gödel in his doctoral thesis 18 years before.The ideas that fed my discovery of this proof were mostly those I found in the teachings and writings of Alonzo Church. This may (...) seem curious, as his work in logic, and his teaching, gave great emphasis to the constructive character of mathematical logic, while the model theory to which I contributed is filled with theorems about very large classes of mathematical structures, whose proofs often by-pass constructive methods.Another curious thing about my discovery of a new proof of Gödel's completeness theorem, is that it arrived in the midst of my efforts to prove an entirely different result. Such “accidental” discoveries arise in many parts of scientific work. Perhaps there are regularities in the conditions under which such “accidents” occur which would interest some historians, so I shall try to describe in some detail the accident which befell me.A mathematical discovery is an idea, or a complex of ideas, which have been found and set forth under certain circumstances. The process of discovery consists in selecting certain input ideas and somehow combining and transforming them to produce the new output ideas. The process that produces a particular discovery may thus be represented by a diagram such as one sees in many parts of science; a “black box” with lines coming in from the left to represent the input ideas, and lines going out to the right representing the output. To describe that discovery one must explain what occurs inside the box, i.e., how the outputs were obtained from the inputs. (shrink)