“Propositional density” refers to the relationship between design elements and the meanings they convey, with high propositional density corresponding to more pleasurable and interesting designs. Source.
In a non-visual-design context, one can understand “propositional density” as being like a double entendre, a word or phrase with multiple meanings. Designs, words or phrases with multiple layers of meaning are believed to be more interesting.
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I find the “Propositional Density” principle a great way for designer to have fun with their projects. It adds a second layer of meaning for the users to find. This two in one designs become more engaging and you’ll find the consumer spending more time observing your creation. I love the example of President Obamas logo you selected. The O looking like a sun setting in the back with the American flag paving the way forward. Simple yet meaningful on multiple levels. Here is a site that shows several examples of this principle.
https://brazendoots.com/2017/02/08/twone/