The effect of cardinality in the pigeonhole principle

Abstract

The pigeonhole principle is a well-known mathematical principle and is quite simple to understand. It goes as follows: If n items are placed into m containers, and if $m<n$, then there must be at least one container with more than one item in it. A previous study by Mercier et al. yet found that participants tasked to solve a problem akin to this were unable to correctly solve it when the number n is understood as a quantity (cardinal) instead of an identifier (nominal). They suggest quantities first need to be turned into categories. We believe this difference can instead be explained by pragmatic factors not taken into account in their study. We disambiguate the problems given to participants and study the effects of their disambiguation and the influence of different kinds of quantities: countable and mass. We show that using a problem with countable quantities instead of hair, usually understood as non-countable, increase the performance of participants, no longer distinguishable from that of the nominal problem.

Keywords:

• problem solving
• pigeonhole principle
• reasoning
• pragmatics
• count and mass numbers

Acknowledgements

We would like to give special thanks to Guy Politzer who helped us disambiguate some statements of the problems and for his advice. We would also like to thank Andrew Hromek for proofreading the paper. Finally, we would also like to thank the reviewers for their comments which have significantly improved the quality of the paper: Valerie Thompson, Laura Macchi, and one anonymous reviewer.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

All the data associated with this paper is available at https://doi.org/10.17605/OSF.IO/5PJCD

Notes

1. It is necessary that two men would have the same number of hair, of gold, and anything else. Quote from (Leurechon, 1622).

2. A cardinal number is defined by a scale which has a direction (an order) and the same interval between each of its values. For example, counting a number of items is a cardinal problem. 4 items are twice as many items as 2, and half as many as 8. An ordinal number is a number that only has an order, but no set interval between them. For example, a subjective trait measured by a Likert scale from 1 to 7. While we know that 4 on that scale is higher than 2, it is impossible to be certain that someone answering 4 on that scale has exactly twice the strength of that trait compared to someone answering 2. Finally, a nominal number has no order and no set interval between each value, it is only a label. For example, bus lines (Bus#32). The number is only an identifier and does not represent any order or any interval. Bus#28 could follow Bus#36 and it would not be surprising.

3. The themes of the different problems were: the number of cows for cardinal low, the number of hairs for cardinal high, the number of days available for the visit of a health inspector for nominal low and the number of lottery tickets for nominal high.

4. Was considered valid an entry which had answers for the entire questionnaire, who were native French speakers and who did not know this type of problem before. All other entries were removed.