The projectivist about e.g. judgements of tastiness explains how “X is tasty” (as an ordinary judgement made in the restaurant, not the philosophy class) is an assertion that can be correct or incorrect even though there is no such property-out-there as tastiness, so the assertion isn’t representationally-true (or correspondence-true, if your prefer). Or so the story goes.
The theory about fiction that Weir sketches explains how “Sherlock Holmes lived in Baker Street” (as an ordinary judgement made in discussing the stories, not the in history class) is an assertion that can be correct or incorrect even though there is no such person-out-there as Sherlock (or Meinongian substitute), so again the assertion isn’t representationally-true. Or so the story goes.
The projectivist line about tastiness or goodness or beauty, the theory about fiction, allow us to speak with the vulgar but think with the learned (assuming the learned have a naturalistic bent). We can legitimately talk as if there is a kosher property of tastiness or as if there are fictional beings such as Sherlock, while not being really ontologically committed to such things. If we use small caps to signal when we are making assertions in full-on, stick-by-it-even-in-the-metaphysics-classroom, genuine-representation mode, then we can say (ordinary conversation) “Marmite is tasty” even though (when in Sunday metaphysical mode) we can agree “Marmite IS NOT TASTY“; and likewise we can say (conversationally) “Holmes lived in London” while (on Sundays) agreeing that “Holmes NEVER EXISTED“. Hence the projectivist story and the story about fiction allow us to eliminate some of our ostensible ONTOLOGICAL commitments in talking with the vulgar (Weir calls this “ontological reductionism”, but I’ve grumbled before about that label). So the story goes.
What does Weir add to the story in this section, to further set the scene before trying to paint a comparable picture of mathematics as non-representational? (1) Some remarks about what makes for the difference between a representational mode of assertion and a non-representational one. (2) Some remarks about why the difference between “Holmes existed” and “Holmes EXISTED” shouldn’t be confused with a lexical or structural ambiguity. (3) Some remarks about what a projectivist should say about the likes of “If sentient beings had never existed, there would still have been beautiful sunsets”.
Concerning (1), I’d have thought the way to go is to illustrate the kind of basic semantic story that applies to canonical examples of “representational” discourse, and then say that non-representational discourse is whatever needs some different kind of semantic story (I’m not saying that’s easy to do! — but Weir’s p. 59 seems to go off in a slightly skew direction.)
Concerning (2), I agree. I’m not sure it is helpful then to go on to talk about ‘metaphysical ambiguity’ (but maybe that’s just complaining about Weir’s taste in labels again).
Concerning (3), Weir discerns a wrinkle, but also thinks that it doesn’t carry over his promised non-projectivist but analogously anti-realistic account of mathematics. So we needn’t pause over this.