while "knowledge" might mean different things to different philosophers, I'm not sure there's a philosophy which aims at understanding as opposed to knowledge. — Moliere
To make this more than a slogan, you'd need some sort of theory (hermeneutics would be an example), and I think what that theory would try to account for is, first of all, the "as opposed to science" part.
Williamson begins by claiming (uncontroversially) a shared lineage for science and philosophy, and he mentions the relation of science to philosophy at several points. (The other disciplines that can discipline philosophy; whether a theory can be used in empirical linguistics; etc.)
And this is as it should be, because Williamson wants to talk about rigor, and throughout the 20th century, at least, that discussion took this form: (1) Can philosophy be a science? (2) Should philosophy be a science?
(Williamson doesn't quite approach the issue this way, so his answer seems to be that philosophy can and should be science-ish.)
So we need to talk about science, and what the comparison to science might reveal about philosophy.
Here's where I thought to start, with the self-image of a toy version of science: in order to study and theorize the laws of nature, science breaks itself into one part that is by design subject to those laws, and another that is not. (There's a problem with this we'll get to, but it's not where you start.)
What I mean by that is simply that the data a scientist wants is generated by the operation of the laws of nature in action. You can observe events where those laws are operative; you can also conduct experiments to try to isolate specific effects, which you then observe. But the whole point of an experiment is to submit some apparatus or material to the forces of nature so that you can see what happens. This part of the work of science deliberately submits itself to nature at work.
But the two further steps, observing and theorizing, are intended to be separate, and not subject to the forces and constraints and whatnot under investigation. The weights fall from the tower and I observe the action of gravity upon them, but my watching them does not require that I too fall from the tower. I need not submit my process of observation to gravity to observe the effect of gravity on bodies.
Then I collect my observations and I work out a mathematical description. My mathematics describes the action of gravity, but is not subject to it, and need not be to describe it. My mathematics is not a theory of gravity, but provides constraints on the theories I produce. (By showing what it does to what, and how much it does it, and what it doesn't do because it's not part of that equation, and so on.) My theories of gravity are also not subjected to the work of gravity as the bodies I observed were.
Before getting to philosophy, I'll note that this self-division of science worked right up until it didn't, but also that when it stopped working, it didn't entirely stop working. It seems when you observe nature at very small scales the process of observation itself has effects on the observed big enough that they must be taken into account. We might wonder whether something similar happens in philosophy, but for now I'll just observe that we know more or less exactly why this happens at quantum scale, and could have predicted it would. (But we don't end up with the equations I write on a whiteboard changing the outcome of an experiment, for instance.)
The practice of science doesn't make a universal claim about not being subject to the laws it studies. The paper upon which my equations describing gravity are written is itself subject to the force of gravity, but not in a theoretically important way. The self-division of science is not absolute. (It is even plausible to claim that the division itself is not a posit of theory, but is itself found in nature -- right up until you hit the exception at quantum scale.)
Now what about philosophy?
Can it achieve this sort of self-division? Must it do so to achieve the same rigor as science? (Or can it be just as rigorous without doing so?)
--- I spent a few pages trying to answer these questions, but it was a mess, so here's just a couple obvious points:
1. If you think philosophy (or logic) studies the laws of thought or of reason, you're unlikely to think any of your work needs to separate itself from those laws
2. If you think philosophy studies norms of thought and behavior, neither making your work subject to the specific norms you're studying nor making it subject to different norms seems obviously satisfactory. Both present problems.
I think Williamson wishes to describe something like an experimental approach to philosophy, and that's what his whole competition between theories business is meant to be. Is it really similar to how science does this? If it's not, does it still make sense?