 The article “Some Remarks on Logical Form” (1929) marks the end of this optimism. Wittgenstein now arrives at the view that some incompatibilities cannot, after all, be reduced to logical impossibilities. His change of heart appears to have been occasioned by a consideration of incompatibilities involving the attribution of qualities that admit of gradation — e.g., the pitch of a tone, the brightness of a shade of colour, etc. Consider, for example, the statements: “A has exactly one degree of brightness” and “A has exactly two degrees of brightness.” The challenge is to provide analyses of these statements that bring out the logical impossibility of their being true together. What Wittgenstein takes to be the most plausible suggestion — or at least a sympathetic reconstruction of it — adapts the standard definitions of the numerically definite quantifiers to the system described in the Tractatus, analysing these claims as respectively: “∃x(Bx &amp; A has x) &amp; ~∃x,y(Bx &amp; By &amp; A has x and A has y)” (“Bx” means “x is a degree of brightness”) and “∃x,y(Bx &amp; By &amp; A has x and A has y) &amp; ~∃x,y,z(Bx &amp; By &amp; Bz &amp; A has x &amp; A has y &amp; A has z).” But the suggestion will not do. The trouble is that this analysis — absurdly — makes it seem as though when something has just one degree of brightness there could be a substantive question about which (if any) of the three mentioned in the analysis of the second claim— x or y or z — it was—as if a degree of brightness were a kind of corpuscle whose association with a thing made it bright (cf. Klagge and Nordmann, 33). Wittgenstein concludes that the independence of elementary propositions must be abandoned and that terms for real numbers must enter into atomic propositions, so that the impossibility of something’s having both exactly one and exactly two degrees of brightness emerges as an irreducibly mathematical impossibility. This, in turn, contradicts the Tractatus’s idea that all necessity is logical necessity (6.37). 