Logical and Praxeological Impossibilities
The use of logical impossibilities makes rational discourse impossible. A and non-A cannot both be true. Everybody cannot have an income higher than the median or the average. Nobody can consume if nobody produces (including do-it-yourself). Everybody cannot consume more if everybody produces less. You can’t be inclusive without admitting the non-inclusive in your inclusive set. And so on.
There also exist praxeological impossibilities which make any rational discourse about society impossible. I take “praxeology” to mean the logic of human action in relation to individual incentives. For example, you cannot consume something that you want but cannot produce yourself, except if somebody else is motivated to produce it for you through exchange or out of benevolence.
Another example of praxeological impossibility—this one relevant to the current predicaments of a certain American governor: If anybody accused of “inappropriate behavior” by somebody must immediately resign, it would be impossible for anybody to hold a “position” for any predetermined length of time longer than the time it takes to formulate an accusation. The reason is that anybody who wants somebody to resign would only have to accuse him of “inappropriate behavior.” (On the other hand, of course, if charges of real assault, as opposed to the chameleonic “inappropriate behavior,” were not investigated and, if revealed founded, prosecuted, potential criminals would have incentives incompatible with a free and peaceful society.)