We provide a mathematical analysis of an election system proposed by the eminent logician Gottlob Frege (1848--1925) . His proposal was written presumably in 1918, was (re)discovered around the turn of the millennium, and published for the first time in the original German in 2000. A remarkable feature of Frege's proposal is its concern for the representation of minorities and its sensitivity to past election results. Frege's proposal is based on some highly original and relevant ideas; his core idea is that the votes of unelected candidates are carried over to the next election . All candidates thus accumulate votes over time and eventually each candidate is elected at some point. We provide a mathematical formulation of Frege's election system and investigate how well it achieves its aim of a fair representation of all political opinions in a community over time. We can prove that this goal is fulfilled remarkably well. However, we also show that, in other aspects, it falls short of Frege's high ambition that no voter's vote be lost. We propose a slight modification of his voting rule, the modified Frege method, that remedies these shortcomings. We analyse both methods from the perspective of modern social choice and apportionment theory, and can show that they are novel contributions with noteworthy proportionality properties over time .
In 1998 a long-lost proposal for an election law by Gottlob Frege (1848--1925) was rediscovered in the Th\"uringer Universit\"ats- und Landesbibliothek in Jena, Germany. Frege's method for the election of representatives of a constituency features a remarkable concern for the representation of minorities . Its core idea is that votes cast for unelected candidates are carried over to the next election , while the elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a proportional representation of political opinions over time. We find that through a slight modification of the original voting method even stronger forms of proportionality can be achieved. This modified version of Frege's method can also be seen as providing a novel solution to the apportionment problem. We prove that it is distinct from all of the best-known apportionment methods, while it still possesses noteworthy proportionality properties .
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