There is a theological explanation of the nature of the divine names. It is relatively well known, and so, we will recall it only briefly. It is based on the Cappadocian teaching of the uncreated energies of God as the tool of the divine revelation. In Dionysius’ wording, energies of God are mostly (although not exclusively) called διακρίσεις and πρόοδοι:
Ταύτας ἡμεῖς τὰς κοινὰς καὶ ἡνωμένας τῆς ὅλης θεότητος διακρίσεις εἴτ' οὖν ἀγαθοπρεπεῖς προόδους ἐκ τῶν ἐμφαινουσῶν αὐτὰς ἐν τοῖς λογίοις θεωνυμιῶν ὑμνῆσαι κατὰ τὸ δυνατὸν πειρασόμεθα τούτου, καθάπερ εἴρηται, προδιεγνωσμένου τὸ πᾶσαν ἀγαθουργικὴν θεωνυμίαν, ἐφ' ᾗπερ ἂν κεῖται τῶν θεαρχικῶν ὑποστάσεων, ἐπὶ τῆς ὅλης αὐτὴν ἐκληφθῆναι θεαρχικῆς ὁλότητος ἀπαρατηρήτως. (DN 2:11, 652A).
The divine names are referring to God in general and not to a specific hypostasis. However, the divine names, θεωνυμίαι, are far from being passive signs, but are themselves ἀγαθουργικαί, that is, the beneficent energies of God. To be ἀγαθουργική is, in Areopagite’s terminology, a specific feature of God himself, and so, the divine names are specifically divine names qua uncreated. The “beneficent energies” ensure that any name of God is referring to God. They ensure, moreover, the only effective way to know God (which is dealt with by Dionysius repeatedly but especially in MT), which consists in overcoming both assertion and negation.
This properly theological side of the Dionysius’ doctrine of divine names is much more studied than its properly philosophical side related to the divine names qua created terms (signs) referring to denotations which are created as well. What kind of the reference theory is implied in Dionysius?
The uncreated divine energies of God are responsible for the most radical form of the semantic externalism, where any singular or general term or negation of whatever of them, regardless of its direct denotation, is a sign of God. In some way, this is recalling to us the “new antisubjectivism” of Donald Davidson, but only in some way. In Davidson, the schemes necessary to understand our empirical experience are equally external, and so, are as “objective” as this experience itself. If we take, turning to Dionysius, the divine energies as analogous to such schemes, it would be right to state that they are responsible for our knowledge of God in a “Davidsonian” way: there is no, here, any “subjective” contribution to knowledge of God, which is not coming from outside of the human personality. However, Davidson’s negativist attitude toward the reference has have a purpose to explain the relations between the sign and the denotation. In Dionysian case, it is not a priori clear whether God is the denotation of the divine names or something else. At least, all these names are denotations of their own, which are not God.
When τὸ ἀνώνυμον (“the Nameless”: DN 1:7, 596 D etc., cf. above) is named, it remains unnamed. It is named not as a reference (= Fregean Bedeutung, denotation) but as a sense (= Fregean Sinn), or, in another terminology, not as extensional but as intensional. The latter terminology goes back to Rudolf Carnap, who is, too, of some help for understanding Dionysius.
In the most general sense, intensional is any kind of meaning that is not reduced to extensional, that is, to the individual objects and the true-values. It is hotly disputed topic, however, whether this part of meaning is reducible to the extensional or not. According to Carnap himself, it must be reducible, at least, in the language of science. He expressed his hope in his “Thesis of Extensionality”: “[A] universal language of science may be extensional.” Carnap put forward this thesis “only as supposition” but considered it as “fairy plausible.” Even in the natural sciences this thesis is not without problems, but, in the theology of Dionysius, it would become an exact opposition to the truth: Dionysius’ “universal language of theology” is irreducibly intensional. The divine energies, when they act in arranging things in the whole world (s. below), are legitimate heirs of the Platonic ideas, and so, they are a classical example of the “intensional entities.” Thus, the intensionality of Areopagite’s semantics and his corresponding language is irreducible.
Normally, when one says “intensional language,” it is partly intensional and partly extensional language (very probably, reducible to an extensional language, in conformity with Carnap’s Thesis of Extensionality) what is meant. However, in Dionysius, the divine names form an absolutely intensional language, which is irreducible to any extensional one. In this language, even the every act of naming provides that the named, although became named, remains unnamed. It is impossible to invent a logical way to avoid this always presenting namelessness, because it is a theological (ontological) fact.
God is the intensional of the divine names in the sense that he is not their extensional (denotation), whereas these names are still names of God. God is the intensional of the divine names in the sense that he is “implied” by them. Of course, the mechanism of such implication is quite specific: he is really present within the names. However, from the rational point of view, one can say that God is implied. This theology of “implication by real presence” will be elaborated much further in the eighth-, ninth-, and eleventh-century discussions on the holy icons and other Church symbols.It is always difficult to explain, what extensionality is all about. I think there is no ready formal logical framework suited to explain the thought of Dionysius about God as, so-to-say, the intensional of the divine names. Nevertheless, the modern analytical philosophy elaborated some ideas, which may be used in explaining Dionysius. Let us continue.
 On the notions of διακρίσεις and πρόοδοι in God, see especially a classical article: Lossky, Vladimir. “La notion des analogies chez Denys le pseudo-Areopagite.” Archives d’histoire doctrinale et littéraire du Moyen Age 5 (1930): 279-309. Cf. also Golitzin, Alexander. Et introibo ad altare Dei. The Mystagogy of Dionysius Areopagita, with Special Reference to Its Predecessors in the Eastern Christian Tradition. Ἀνάλεκτα Βλατάδων 59. Θεσσαλονίκη: Πατριαρχικὸν Ἵδρυμα Πατερικῶν Μελετῶν, 1994, p. 54-61. These terms cover both “energies” of God and divine hypostases, without, however, confusing them with each other.
 Rolt’s translation: “For in divine things the undifferenced Unities are of more might than the Differentiations and hold the foremost place and retain their state of Undifference even after the One has, without departing from Its oneness, entered into Differentiation. These Differentiations or beneficent Emanations of the whole Godhead—whereby Its Undifferenced Nature is shared in common—we shall (so far as in us lies) endeavour to describe from the Divine Names which reveal them in the Scriptures, having now made this clear beforehand (as hath been said): that every Name of the Divine beneficent Activity [or “energy,” that became a more common term today. — B. L.] unto whichever of the Divine Persons it is applied, must be taken as belonging, without distinction, to the whole entirety of the Godhead.”
 Davidson, Donald. “The Myth of the Subjective” , in: Davidson, D. Philosophical Essays. Vol. 3: Subjective, Intersubjective, Objective. Oxford: Clarendon Press, 2001, p. 39-52.
 In his book Carnap, Rudolf. Meaning and Necessity. Chicago: University of Chicago Press, 1947.
 Carnap, Rudolf. The Logical Syntax of Language. Tr. Amethe Smeaton (Countess von Zeppelin). International Library of Psychology. London: Routledge, 2001, p. 245-247 (first published in 1937).
 See especially a recent study: Shani, Itay. “The Myth of Reductive Extensionalism.” Axiomathes 17 (2007): 155-183.
 S., on the intensional entities and the various attempts to reduce them to the extensional ones in the intensional logic, a succinct but informative entry by Bealer, George. “Intensional entities,” in: Craig, Edward (ed.). Routledge Encyclopedia of Philosophy 4 (2000), pp. 803-806.
 Cf. Bealer, George. Quality and Concept.Clarendon Library of Logic and Philosophy. Oxford: Clarendon Press, 1982 (repr. 2002), esp. 147-157. This author argues that the ideal of the Thesis of Extensionality can be achieved, but only in the case if the intensional entities are accepted and, then, treated in a similar way to the extensional ones: “The moral is that those who wish to be extensionalists in logic may be so, but only if they are intensionalists in ontology” (p. 157). In the case of Areopagite, the intensional entities (divine energies considered as intensionals arranging the whole world; s. below) do not allow such logical operations, because they remain “unnamed” (not explicated).
 Cf. Lourié, Basile. “Le second iconoclasme en recherche de la vraie doctrine.” Studia Patristica 34 (2000): 145-169 , and idem. “Une dispute sans justes: Léon de Chalcédoine, Eustrate de Nicée et la troisième querelle sur les images sacrées.” Studia Patristica 42 (2006): 321-339.