As either an election system design tool, or as a method of discriminating better from worse election systems, one can try demanding an election system have certain logical properties (or determining which properties various systems obey or disobey).
In my opinion the property-based approach for discriminating among single-winner election systems, was largely a wrong approach which is now largely obsolete, which led the whole political science community down the wrong path for decades. That is because it is not clear which properties are more important than others. So while understanding the properties of a voting system is good because it gives us more understanding, it is insufficient to make it clear that election system A is better than election system B. And indeed such results as "Arrow's impossibility theorem" (showing that no voting system with a certain combination of properties could exist) erroneously led political scientists to the wrong idea that there could not be any "best" single-winner voting system. And then everyone was reduced to eternal arguments of the form "my system satisfies the Transvestite Inversion Property and yours doesn't, so mine is better," versus "Oh yeah? But my system satisfies Super-Monotonic Angst and I consider that more important" versus "Well I don't!" etc etc. Not very productive.
The right path is ``Bayesian regret'' measurement. Although it is not directly defined this way, Bayesian regret in a sense automatically considers all possible "properties" or "paradox" (property failure) scenarios – whether or not any human has ever considered or named that property before – and automatically weights them all with the correct weights that depend both on the likelihood of that kind of paradox and on its severity (utility-decreasing effect for society).
Bayesian regret measurement can be automated. Several people (including me, although I came later than most) independently invented Bayesian regret and realized it could be used for automated election system comparison. But I did the most extensive such automated comparative study and mine was the only study that included "range voting." Therefore it was I who made the extremely important discovery of the clear superiority of range voting over all other voting systems in that study. This reversed decades of wrong views that no voting system was clearly "best."
(However, for multiwinner elections, nobody has proposed any concept like Bayesian regret, and hence we do not know a way to make a clear, automated comparison between different election systems. Therefore, we are forced back to the old-style property-based approach.)
Robert F. Bordley: A pragmatic method for evaluating election schemes through simulation, Amer. Polit. Sci. Rev. 77 (1983) 123-141.
Samuel Merrill: Making multicandidate elections more democratic, Princeton Univ. Press 1988.
Warren D. Smith: Range voting, #56 at /WarrenSmithPages/homepage/works.html.