They speculate tentatively that natural kind terms (such as 'water') might be seen as abbreviated A-involving descriptions (such as 'the actual waterish stuff around here'), in which case necessary a posteriori identities such as 'water is H2O' may also be []-necessary and FA-contingent.
As defined here, FA-intensions are closely tied to apriority for some sentences: especially for A-involving sentences, and for tacitly A-involving sentences such as those involving descriptive names and perhaps natural kind terms (if these are indeed tacitly A-involving).
In the case of Davies and Humberstone, it holds only for A-involving expressions and tacitly A-involving expressions such as descriptive names and perhaps some natural kind terms.
Thesis (T2) says that the primary and secondary intensions of a complex expression depends on the primary and secondary intensions of its parts according to the natural compositional semantics.
Thesis (T3) states a natural connection between the intensions and the extension of an expression token.
But it is crucial to the two-dimensionalist position that typical a posteriori identities involving proper names or natural kind terms, such as 'Mark Twain is Samuel Clemens' or 'water is H2O', have a primary intension that is false in some scenario.
In the foregoing, a qualitative vocabulary is, to a first approximation, a vocabulary that is free of terms (such as names and natural kind terms) that give rise to Kripkean a posteriori necessities and a priori contingencies.
The same goes for many or most other terms, plausibly including most names or natural kind terms.
As before, a qualitative vocabulary is one that excludes terms, such as names and natural kind terms that give rise to Kripkean a posteriori necessities.
Primary intensions may also vary between different tokens of the same name (especially by different speakers), for different tokens of the same demonstrative (e.g. 'this' or 'that'), and perhaps also for different tokens of the same natural kind term.