The alethic modality is recognised by Cinque in the most evident case only. It is for such parenthetical words as “possibly,” “necessary,” etc. that his table contains the two rows
- [ - necessary]
- [ - possible]
The four values of these modalities are “necessary,” “not necessary,” “possible,” “not possible” (= none). One can see that this is nothing but the complete square of the generalized quantifier of ‹1, 1› type (some) in alethic modality.
It is interesting to verify against direct linguistic data whether “not necessary” appeared here as the true internal negation of “some” or as a misinterpretation of ‹1, 1, 1› type quantifying in alethic modality. At least, the data concerning the operator more than in alethic modality should be checked.
Alethic modality may be covered as well by what Cinque calls “mood irrealis/realis.” Here also the parentheses are meant, but, this time, such as “probably.” “Probably” could be used in either alethic or epistemic sense. Accordingly, Cinque’s “irrealis/realis” must be treated as a particular case of alethic or epistemic modality.
Let us add in margin, that the very place of Cinque’s “irrealis/realis” in the hierarchy of functional heads reflects its ambiguity—between epistemic and alethic modalities.
ASPprogressive with its value “generic” (default) and “progressive” (marked) is also alethic modal operator but, this case, quantifying only on the acts, not the whole phrase. “Generic” belongs to the general properties of an object, including those that are not revealed. “Progressive” means only the properties that are expressed in acts. It is easy to see that “generic” corresponds to the alethic operator all “possible,” and “progressive” to the alethic operator some “necessary.”