The second argument for the preferability of ternicity is the argument from scoring inquiry: ternicity better suits ‘knows’ to its role in keeping score of the overall progress of inquiry.
Ternicity is the view that knowing is a three-place relation between a subject, a proposition, and a contrast class. Contextualism is the view that "knows" is a context-sensitive term, which designates different relations (presumably two-place) in different contexts of utterance/evaluation. Here's the argument Schaffer offers on the basis of scoring inquiry:
(10) One of the roles of ‘knows’ is to keep score of the overall
progress of inquiry;
(11) Indexicality precludes ‘knows’ from scoring the overall progress of inquiry, because indexicals cannot keep a consistent score across contexts; and
(12) Ternicity allows ‘knows’ to score the overall progress of inquiry, because the various stages of inquiry may be consistently logged under various values of q.
The nitpicky point concerns the second premise. Schaffer defends the premise by saying:
indexicality precludes ‘knows’ from keeping score of the overall progress of inquiry. This is because, with indexicality, the denotation of ‘knows’ is always warped to the
current context. As such ‘knows’ cannot keep consistent score across contexts. But scoring inquiry requires evaluating how a subject performs through a sequence of questions, and this requires a consistent score across contexts. (Imagine trying to score a baseball game if the denotation of ‘run’ changed with every inning!)
My nitpick is that, while Schaffer is right that baseball would be difficult to score if the denotation of "run" changed with every inning, I think he overstates the case here. It seems that we could devise a game which would be relatively simple to score, but where the denotation of key scoring terms shifted about throughout the game (at least, in the same sense as relevant for his argument). Here's such a game: there are six numbered buckets in a row. Players stand at one end of the row and take turns tossing differently colored ping pong balls into the buckets. To begin, getting a ball in any bucket is worth one point. After each round of play, a die is rolled. If the die comes up 6, only the points in the furthest bucket count (even from previous rounds). If the die comes up 5, only the points from the furthest two buckets count. And so on. In other words, points in the sixth bucket are "safer" points than points in the first bucket, because each round, the buckets that actually count for points can change, but the higher the bucket number, the less likely it is that those points are excluded. So, if I toss all my ping pong balls in the first bucket, I may end the first round in the lead, but enter the second round with no points.
Ok, so, you might be thinking this fictional bucket-toss game is too minor of a point to raise, even for a nitpick post. At the same time, this isn't entirely unrelated to Schaffer's point: If we think of the shifting standards for which buckets count as analogizing changes in the contextually set standards for knowledge, and the ping pong balls as beliefs, we can see how the contextualist might conceive of scoring inquiry for "knows"-ascriptions. Your score fluctuates from context to context, but some of your points are safer than others. Perhaps a better analogy would allow the players to attempt to influence the bucket-boundary for points (to better analogize the popular contextualist view that knowledge claims can be used to attempt to shift the standards for knowledge-ascriptions).
So, I think Schaffer overstates the case against the indexicalist when it comes to the claim above labeled (11). Of course this is just one small part of Schaffer's case, and I don't think anything crucial for Schaffer's larger project turns on it (hence the status as a nitpick).