FEEDBACK FUNCTIONS IN TEACHING MACHINES

The efficiencies of two ways of providing feedback in teaching machines were studied. The task was to learn paired associates by choosing one alternative out of three to match a stimulus. Feedback was provided in one machine by immediately presenting a new item in the case of a correct response. In the other machine the correct associates were shown for 2 sec. after a correct choice had been made. The latter method was superior. The results are discussed in relation to theories concerning the effects of exposure time on rate of learning.     

Extensions of the Finitist Point of View

Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ω -rule in his 1931 paper ‘Die Grundlegung der elementaren Zahlenlehre’. The main question we discuss here is whether the finitist (meta-)mathematician would be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ‘Die Widerspruchsfreiheit der reinen Zahlentheorie’ (1936). As to Hilbert and Bernays 1939, we end on a ‘critical’ note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism.     

Nonmonotonicity in (the Metamathematics of) Arithmetic

This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic logic through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, U are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by U -closed theories and U -fixed points. The last section contains results on V-closed theories which are specific for V.     

Structuralism, Model Theory and Reduction

In this paper, the (possible) role of model theory forstructuralism and structuralist definitions of ``reduction' arediscussed. Whereas it is somewhat undecisive with respect tothe first point – discussing some pro's and con's ofthe model theoretic approach when compared with a syntacticand a structuralist one – it emphasizes that severalstructuralist definitions of ``reducibility' do not providegenerally acceptable explications of ``reducibility'. This claimrests on some mathematical results proved in this paper.     

On The Logic Of Reducibility: Axioms And Examples

This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T'. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal logic.