# Probability Relations within Response Sequences under Ratio Reinforcement.

01 Apr 1958-Journal of the Experimental Analysis of Behavior (Society for the Experimental Analysis of Behavior)-Vol. 1, Iss: 2, pp 109-121

About: This article is published in Journal of the Experimental Analysis of Behavior.The article was published on 1958-04-01 and is currently open access. It has received 257 citation(s) till now. The article focuses on the topic(s): Reinforcement.

Topics: Reinforcement (50%)

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##### Citations

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TL;DR: Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio.

Abstract: Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.

1,223 citations

### Cites result from "Probability Relations within Respon..."

...Number discrimination depends on the ratio that distinguishes the two set sizes, in accord with Weber's Law, both for animals (e.g. Mechner, 1958) and for humans tested 0010-0277/00/$ - see front matter q 2000 Elsevier Science B.V....

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TL;DR: The preverbal system of counting and arithmetic reasoning revealed by experiments on numerical representations in animals is described and a model of the fact retrieval process accounts for the salient features of the reaction time differences and error patterns revealed by experiment on mental arithmetic.

Abstract: We describe the preverbal system of counting and arithmetic reasoning revealed by experiments on numerical representations in animals. In this system, numerosities are represented by magnitudes, which are rapidly but inaccurately generated by the Meck and Church (1983) preverbal counting mechanism. We suggest the following. (1) The preverbal counting mechanism is the source of the implicit principles that guide the acquisition of verbal counting. (2) The preverbal system of arithmetic computation provides the framework for the assimilation of the verbal system. (3) Learning to count involves, in part, learning a mapping from the preverbal numerical magnitudes to the verbal and written number symbols and the inverse mappings from these symbols to the preverbal magnitudes. (4) Subitizing is the use of the preverbal counting process and the mapping from the resulting magnitudes to number words in order to generate rapidly the number words for small numerosities. (5) The retrieval of the number facts, which plays a central role in verbal computation, is mediated via the inverse mappings from verbal and written numbers to the preverbal magnitudes and the use of these magnitudes to find the appropriate cells in tabular arrangements of the answers. (6) This model of the fact retrieval process accounts for the salient features of the reaction time differences and error patterns revealed by experiments on mental arithmetic. (7) The application of verbal and written computational algorithms goes on in parallel with, and is to some extent guided by, preverbal computations, both in the child and in the adult.

1,139 citations

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Brown University

^{1}TL;DR: The conclusion was that the same internal mechanism is used for counting and timing that can be used in several modes: the "event" mode for counting or the "run" and the "stop" modes for timing.

Abstract: The similarity of animal counting and timing processes was demonstrated in four experiments that used a psychophysical choice procedure. In Experiment 1, rats initially learned a discrimination between a two-cycle auditory signal of 2-sec duration and an eight-cycle auditory signal of 8-sec duration. For the number discrimination test, the number of cycles was varied, and the signal duration was held constant at an intermediate value. For the duration discrimination test, the signal duration was varied, and the number of cycles was held constant at an intermediate value. Rats were equally sensitive to a 4:1 ratio of counts (with duration controlled) and a 4:1 ratio of times (with number controlled). The point of subjective equality for the psychophysical functions that related response classification to signal value was near the geometric mean of the extreme values for both number and duration discriminations. Experiment 2 demonstrated that 1.5 mg/kg of methamphetamine administered intraperitoneally shifted the psychophysical functions for both number and duration leftward by approximately 10%. Experiment 3 demonstrated that the magnitude of cross-modal transfer from auditory signals to cutaneous signals was similar for number and duration. In Experiment 4 the mapping of number onto duration demonstrated that a count was approximately equal to 200 msec. The psychophysical functions for number and duration were fit with a scalar expectancy model with the same parameter values for each attribute. The conclusion was that the same internal mechanism is used for counting and timing. This mechanism can be used in several modes: the "event" mode for counting or the "run" and the "stop" modes for timing.

918 citations

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TL;DR: The number domain is a prime example where strong evidence points to an evolutionary endowment of abstract domain-specific knowledge in the brain because there are parallels between number processing in animals and humans.

Abstract: There is evidence to suggest that animals, young infants and adult humans possess a biologically determined, domain-specific representation of number and of elementary arithmetic operations. Behavioral studies in infants and animals reveal number perception, discrimination and elementary calculation abilities in non-verbal organisms. Lesion and brain-imaging studies in humans indicate that a specific neural substrate, located in the left and right intraparietal area, is associated with knowledge of numbers and their relations ('number sense'). The number domain is a prime example where strong evidence points to an evolutionary endowment of abstract domain-specific knowledge in the brain because there are parallels between number processing in animals and humans.The numerical distance effect, which refers to the finding that the ability to discriminate between two numbers improves as the numerical distance between them increases, has been demonstrated in humans and animals, as has the number size effect,which refers to the finding that for equal numerical distance,discrimination of two numbers worsens as their numerical size increases.

742 citations

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TL;DR: The present results, together with related research, suggest that the ratio of time spent in two activities equals the ratios of the "values" of the activities.

Abstract: When pigeons' standing on one or the other side of a chamber was reinforced on two concurrent variable-interval schedules, the ratio of time spent on the left to time spent on the right was directly proportional to the ratio of reinforcements produced by standing on the left to reinforcements produced by standing on the right. The constant of proportionality was less than unity for all pigeons, indicating a bias toward the right side of the chamber. The biased matching relation obtained here is comparable to the matching relation obtained with concurrent reinforcement of key pecks. The present results, together with related research, suggest that the ratio of time spent in two activities equals the ratio of the "values" of the activities. The value of an activity is the product of several parameters, such as rate and amount of reinforcement, contingent on that activity.

673 citations

##### References

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340 citations

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TL;DR: I am indebted to Professor Lighthill for some further illuminating remarks regarding this point and his comments on Heisenberg's Theory of Isotropic Turbulence are highly illuminating.

Abstract: 1 G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge: At the University Press, 1954). 2 G. K. Batchelor and A. A. Townsend, \"Decay of Turbulence in the Final Period of Decay,\" Proc. Roy. Soc. London, A, 194, 527-543, 1948. 3 W. Heisenberg, \"Zur statistischen Theorie der Turbulenz,\" Z. Physik, 124, 628-657, 1948. 4W. H. Reid, \"Two Remarks on Heisenberg's Theory of Isotropic Turbulence,\" Quart. Appl. Math. 14, 201-205, 1956. 6 Cf. M. J. Lighthill, Nature, 173, 746, 1954. I am indebted to Professor Lighthill for some further illuminating remarks regarding this point.

133 citations