On interpreting Gödel's Second Theorem

Summary In this paper I have considered various attempts to attribute significance to G2.²⁵ Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are literally false. Two others (BCR and Resnik's Interpretation), I have argued, are groundless.I would like to thank Dale Gottlieb, Stephen Barker, Tim McCarthy, Philip Kitcher, Michael Resnik and Richmond Thomason for extensive and helpful discussion of this work.     

Acquisition and forgetting in monkeys' memory of informational object-reward associations

The first experiment showed the monkeys could recall whether an object had been rewarded with peanuts or with sultanas, two equally preferred foods. The second investigated the effect of rewarded trials with an object on monkeys' ability to recall a nonrewarded trial with the same object. The third demonstrated that monkeys could use the memory of reward to predict nonreward and the memory of nonreward to predict reward, in a Win-Shift Lose-Stay paradigm. The fourth found differences between Win-Shift Lose-Stay and Win-Stay Lose-Shift in the rate at which associations between objects and reward events were forgotten. These results are discussed in relation to D. L. Medin's (In A. M. Schrier, Ed., Behavioral primatology, Hillsdale, N.J.: Erlbaum, 1977, Vol. I, pp. 33–69) distinction between informational and hedonic effects of reward in monkeys. It is argued that the association between an object and a reward event is represented in memory by many independent traces, different traces recording the object's association with different attributes of the reward event.     

Retention of conditioned inhibition in a bar-press suppression paradigm

In three experiments, the retention of conditioned inhibition in rats was examined over long intervals. Retention deficits of conditioned inhibition were found at 21 days, although conditioned excitation was relatively stable over time. This loss of conditioned inhibition was demonstrated using both the summation and the retardation of learning techniques in a bar-press suppression paradigm. Furthermore, retention differences were found between within-groups and between-groups generalization tests of conditioned inhibition, using the summation procedure. These differences were interpreted within a retrieval-oriented model of memory.     

Reinforcement magnitude and the inhibiting effect of reinforcement

In a two-key concurrent variable-interval schedule (using pigeons), if the reinforcement frequency for one response is held constant while that for the other is increased, the rate of response on the constant key decreases. The immediate reinforcement for key pecking can usually be conceptualized as the change from a condition in which the key light is on and the food hopper light is off to one in which the key light is off and the hopper light is on. The prechange condition is associated with a delay to food of one-half the average interreinforcement interval in effect during this condition. The postchange condition is associated with a delay to food of about .5 seconds. The programming of additional reinforcement results in a decrease in the delay to food associated with the prechange stimulus condition, and thus a decrease in the value of the improvement that results from the change. This would appear to be analogous to a decrease in the amount of reinforcement, and thus sufficient explanation for the decrease in the rate of the response.     

Relevant Robinson's arithmetic

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.This paper has been greatly influenced by the (largely unpublished) work of E. K. Meyer (cf. [7]) on relevant arithmetic, and I wish to thank him, and also N. D. Belnap, Jr. and D. Cohen for helpful advice. In fairness to Meyer it must be said that he finds my axioms 13 and 13(1) too strong (they are not theorems of his system R# - cf. §4 below). Meyer tells me be finds vindication for his view in my chief theorem of §2. For myself, I find the insights behind Meyer's work on R# to be both stable and fruitful, and if I now had to make a choice, I would follow Meyer in his rejection of my axioms. However, the systems I explore in this paper themselves have a surprising amount of internal consistency of motivation (cf. §5). Let a hundred formal systems bloom.     

A theorem in 3-valued model theory with connections to number theory,type theory,and relevant logic

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development).     

The role of some definite references in linguistic integration

Definite reference increases thematic integration for semantically related sentences (DeVilliers, 1974). Therefore, definite reference should increase linguistic integration in the Bransford-Franks integrated recognition paradigm. Two recognition and two recall studies demonstrated that, if anything, indefinite reference increases linguistic integration effects, as measured by existing measures of integration. However, definite reference increases topical clustering in recall. Results cast doubt on the validity of existing measures of linguistic integration and indicate that the tally model of performance in integrated recognition tasks (Reitman & Bower, 1973) is not totally adequate, either. Results are compatible with the associative model of Anderson and Bower (1973). Topical clustering appears to be a better measure of integrated memory than are existing measures of linguistic integration. Retention of information about the propositional form in which semantically related sentences are presented and integrated memory are not necessarily incompatible.     

Response and encoding factors in “ignoring” irrelevant information

Subjects classified either the numerosity or numeric value of elements in successive stimulus displays. In separate experiments, responses were indicated by oral naming, card sorting, manual tapping, and oral “tapping.” Incongruent levels of numeric value slowed naming and sorting, but not tapping, when numerosity was the cue for responding. Incongruent numerosity slowed tapping, but not naming and sorting, when numeric value was the cue. Changes in stimulus response mapping may thus critically alter the ability to ignore an irrelevant stimulus dimension.     

Values of British political extremists and potential extremists: A discriminant analysis

Studied the value systems of political extremists and potential extremists, comparing them with the value systems of centrist activists and supporters. Samples of political activists from the Labour, Conservative, Communist and National Front parties were obtained, as well as samples of non-active supporters. The non-active supporters were defined as Potential Extremists, if they supported a centrist party as first choice, but either Communist or National Front as second choice. All subjects completed the Rokeach Value Survey. Discriminant analysis showed that the four groups of activists could be clearly distinguished on the bask of their values. However the values of the Potential Extremists did not especially resemble the values of actual National Front or Communist activists. There were value differences between the Potential Extremists and the centrist supporters; nevertheless these two groups tended to be distinguished by very different values from those which distinguished between the activists. The appeal of value symbols for different types of political involvement was discussed.