# Use Math to Make Better Logo Designs: The Propositional Density Principle

When most people think about evaluating the effectiveness of a design, they think in terms of aesthetics. They will scrutinize color schemes and layouts and typography and photographic acuity. And they will look for creative approaches to old ideas (like how awesome this cool Martin Luther King mural is, made entirely out of Rubik’s cubes). But what if I told you that you can determine the effectiveness of a logo or visual campaign with a simple math equation?

That’s a theory that underlies a well-documented principle in linguistics and design, propositional density.

Noam Chomsky is noted for having come up with the idea, a theoretical model that suggests words or phrases that have multiple meanings will create an added level of interest if used correctly than ones that don’t. Most of us are familiar with the concepts of puns (a play on words to create humor or irony) and double entendres (a similar play on words, where one meaning is a bit risque). Such phrases are said to have multiple “propositions,” where there is a surface-level meaning (or proposition) and then something else underneath, deep propositions.

Think, for example, of a Volkswagen marketing campaign during a period when gas prices were high: “If gas pains persist, try Volkswagen.” There is the surface level proposition, which suggests buying Volkswagen cars will save money by getting better gas mileage; then, there are deeper meanings, which compare gasoline to gastrointestinal problems, Volkswagen to a pain reliever medication, and the entire phrase in general to pharmaceutical advertising. The deeper propositions outweigh the surface proposition (there are, in other words, more deep propositions than surface propositions), which makes the phrase funny, memorable, and effective.

In design, propositional density is calculated by dividing the number of deep propositions (ideologies, cultural references, and so forth) by the number of surface propositions (like color, typography, and shapes). If, after dividing the two, you come up with a number larger than one, your design will have a high propositional density. The higher the density, the more effective your design is likely to become.

PD=Pd/Ps

PD is propositional density

Pd is the number of deep propositions

Ps is the number of surface propositions

In Lidwell, Holden, and Bulter’s *Universal Principles of Design*, they use President Obama’s logo during the 2008 campaign as a prime example of high propositional density.

According to the authors, as you look at the logo, you’ll notice about three (3) surface propositions: the logo contains a blue circle, the logo contains horizontal and slanted red and white lines, and the logo has a gradient applied in the middle. There are, however, at least nine (10) deep propositions: the circle represents the ‘O’ in Obama; the circle can represent unity; the circle represents stability; the blue represents the sky; the negative space in the middle represents a sun rising; a sun rising represents “change,” or a new day; the red and white lines represent the American flag; the red and white lines can represent waves of grain; the red and white lines represent a landscape or country; the red, white, and blue represent patriotism; the red white and blue represent freedom.

If you divide the 10 by 3, you get 3.333, which is a very high propositional density. Because of the many deep propositions, this logos is said to be incredibly memorable and effective. Consider these other very effective logo campaigns (I Love NY, the Apple Logo, the Starbucks Logo, the Twitter Logo, and the Miami Heat Logo), and calculate the propositional density:

wow

thats cool