Trichotomous comparisons and small-improvement arguments[edit]

In terminology due to Ruth Chang, the three trichotomous comparisons are betterness, worseness, and equal goodness. For example, one artist, drawing, or cup of coffee might be better or worse than another, or precisely equally as good as it.^[1]


When two items are incomparable, none of the trichotomous comparisons holds between them (or at least it seems that way).


The clearest way of arguing that two options are incomparable is a small-improvement argument.


The purpose of such examples is to show that none of the trichotomous comparisons apply. Here is an example. Suppose that (for you, taking everything into account) a certain job as a professor and a certain job as a banker are such that neither seems better than the other. The professor job offers more freedom and security, and the banking job offers more money and excitement. But we might say that though they are good in different ways, they are just too different to be compared with one of the trichotomous comparisons.


Let's suppose that this means that the banking job is not better, and the professor job is not better. This seems to rule out two of the three trichotomous comparisons.


But what about the third? Might the jobs be exactly equally good? The small improvement argument is supposed to show that they could not. Suppose for the sake of argument that they are precisely equally good.


Suppose also that in order to tempt you, the bankers offer you a tiny pay rise, perhaps 5 cents a year. This new banking job (often called 'banking+') is clearly better than banking, albeit only by a tiny amount. You could (under normal circumstances) never rationally choose banking over banking+: they are the same in every way, except that the latter pays more.


Here is the crux of the small-improvement argument: if banking is exactly equally as good as professor, and banking+ is better than banking, then banking+ must be better than professor. But this seems very implausible: if banking and professor were so different that we could not say that professor is better, and we could not say that banking is better, then how could adding 5 cents a year to a huge salary make the difference?


This seems to show that one of our assumptions was incorrect. Defenders of incomparability will say it is most plausible that it is the assumption that banking and philosophy are equally good that is incorrect. So they conclude that this assumption as false, and thus that none of the trichotomous comparisons apply.