It serves as emphasis. Most people have interpreted "caffeine has a half-life of X hours" to mean "oh so it's fine to have it just X hours before bed.

By explicitly calling out the quarter-life, it prompts people to think if a quarter of the amount would still affect them.

Depends on if it’s linear or a curve, right?

A constant half-life is necessarily an exponential decrease (linear curve on a logarithmic plot). So, yes, if the half-life is 6 hours without qualification, then the quarter life will be 12 hours.

Though, in real-life you get things like enzyme saturation, depletion of chemicals consumed in the metabolism of the chemical, etc. Real world pharmacokinetics are probably not exactly constant half-lives.

For example, IIRC, the average American adult male saturates his alcohol dehydrogenase at about 3/4 shot of 80 proof liquor per hour. Below saturation, I presume ethanol has something close to a constant (and short) half-life, but that exponential decrease is only observed at alcohol concentrations where the effects of alcohol aren't very obvious. So, for most practical purposes, the biological half-life of ethanol doesn't apply and its metabolism is better approximated as a constant rate process.

Are there any examples of this? If its half-life is 6 hours, then after six hours half of the caffeine is still there, and we know caffeine has a half life of 6 hours. I don't know of any molecules with a non-linear half-life but I might be completely wrong

There's a whole area for this - pharmacokinetics. See e.g. https://en.wikipedia.org/wiki/Biological_half-life#Rate_equa... for examples of e.g. first order models (exponential elimination rate, etc.) It has to do with protein binding dynamics among other factors I have no idea about. Remember that you are dealing with biological systems.

Some of these processes to do with elimination and clearance are very much non-linear and are modelled with diff equations. https://en.wikipedia.org/wiki/Physiologically_based_pharmaco...

There aren't many with zero-order pharmacokinetics (linear curve decay). The only notable exception is alcohol though.

Interesting, for a second I thought that had to be wrong because I was aware of aspirin and alcohol, but apparently they are the two common ones and the vast majority are a bit more complex, which also makes sense if you think about it logically I suppose.

It looks like in addition to those, others with zero-order pharmacokinetics include salicylates (salts of salicylic acid incl. aspirin), omeprazole (for indigestion), fluoxetine (antidepressant), phenytoin (for epilepsy), methanol (a poisonous alcohol) and cisplatin (chemo medication.)

Even if it was as simple as any physical system, it is still something one might not realize for never making the correct question.

Technically those two phrases mean the same, but people will draw very different conclusions from them.