Cinque’s “grammar of functional heads” is translatable into the language of modal logic.
More specifically, this means that:
1. “Functional heads” represent a system of modal operators.
2. Each of these operators performs quantification in some modality.
3. The corresponding quantifiers’ set and modalities’ set are limited.
4. The set of modalities contains a grammatically defined subset of six modalities plus one variable modality (any feature other than those belonging to the grammatically defined subset of modalities; it has no grammatical value).
5. The set of quantifiers consists from two basically different (irreducible to each other) kinds of generalized quantifiers corresponding to either “absolute” modal logic or logic of preferences (“comparative” modal logic).
6. The grammatically defined six modalities are alethic, deontic, axiological, epistemic, volition, and spatiotemporal.
7. The generalized quantifiers are, according to Lindström’s classification, of the following types:
7.1. for “absolute” modal operators: ‹1›, ‹1, 1›, sometime ‹1, 2›, very rarely ‹1, 3›;
7.2. for modal operators of logic of preferences: ‹1, 1, 1›, very rarely ‹1, 1, 2›.
8. Within each type of generalized quantifiers, the whole square of each quantifier is at work, where “square of quantifier” is a set formed by the quantifier itself together with its negations of all the three possible kinds (external negation, internal negation, dual).
9. The volition modality so far has no properly logical treatment, but the logic of volition could be constructed after the same pattern as deontic and axiological modal logics.
10. The spatiotemporal modality is presented in either generalized form (when the time is nothing more than a dimension of the spatiotemporal continuum) or purely temporal form (when the temporal modal operators have their own formal structure which is inacceptable to the spatial operators).
11. The properly temporal operators deal with a specific topology of the temporal one-dimension universe where the set of Past is defined, the set of Present is the topological boundary of the set of Past, and the set of Future is topological complement of the set of topological closure of Past.
12. The “absolute” spatiotemporal modal operators are featuring the topology of the spatiotemporal continuum.
13. The spatiotemporal modal operators of logic of preferences are featuring the distances within the spatiotemporal continuum.
14. The ramification of the verbal tenses is obtained by combination of the “absolute” (topological) time modal operators with the operators of spatiotemporal preference.