I would like to see what names that has been suggested for useful modal axioms. By name here I mean some abbreviation such as $T$, $K$, $4$, $.2$, $E$ and so on. In particular I am interested in suggestions that would amount to a good name for the useful modal axiom $\Box\Box\alpha\rightarrow\Box\alpha$ which corresponds with density of frames.
2 Answers
One of the most complete online summaries of modal logic systems is still Halleck's list. Even though the axiom corresponding to density is not explicitly identified in the latter list, it is a particular example of the Chellas/Lemmon/Geach G$(a,b,c,d)$axiom $\lozenge^a\Box^b\varphi\to\Box^c\lozenge^d\varphi$, where $\circ^n$ denotes an $n$long string of the modality $\circ$. Each G$(a,b,c,d)$axiom corresponds to an appropriate confluence property $(\forall x,y,z\in W)(xR^ay\land xR^cz\to\exists w(yR^bw\land zR^dw))$, where $R^m$ means accessible in $m$ steps. Accordingly, one option would be to simply call the density axiom G(0,2,1,0).
At SEP one may find the density axiom called by the alternative name C4 standing for 'converse of axiom 4', which could also be called G(0,1,2,0) but you will probably not consider C4 a 'good name', for the density property is nothing like the converse of transitivity. I have also seen the converse implication of axiom 4 called 4$^{1}$ a name that looks equally bad as C4, for it has a purely syntactical motivation, but not a semantical one. At any rate, as the 'sacred' (Hilbertstyle) tradition on modal logic is clearly losing way to the 'profane' (Kripkestyle) tradition (see here and also here), it would not seem advisable to choose names that focus on oldschool axioms rather than the thereby induced modal properties.
Finally, let me point out that in the literature on labeled deductive systems the density axiom is nowadays standardly referred to as axiom X, while the particular confluence axiom G$(1,1,1,1)$ representing the property in the diamond lemma of ARS is called 2. This terminology seems to have been widely adopted in recent publications that deal with uniform useful proof theories for modal logics. Accordingly, KX would the name given to the extension of the basic normal modal logic K by the addition of the density axiom.

$\begingroup$ Isn't that the Sahlqvist formula or condition, João? $\endgroup$ Mar 21, 2015 at 19:37

1$\begingroup$ No, Frode, the Gaxiom (a.k.a. CatachSahlqvist incestual schema) is only a very particular example of the Sahlqvist schema. $\endgroup$– J MarcosMar 21, 2015 at 19:42

$\begingroup$ OK. I think of these names as names of the axiomatic schemas, and not as names of the corresponding properties. So I do not see a problem with a purely syntactical criterion for the use of the notion converse here. What do you think of my idea to turn the "4" upside down? $\endgroup$ Mar 21, 2015 at 19:48

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Apparently $\Box\Box\alpha\rightarrow\Box\alpha$ is known as C4 (Converse of (4)).
Source: Stanford Encyclopedia of Philosophy.

$\begingroup$ Yes, I saw that suggestion in the article by James Garson. To my mind it is not a good name as it uses two characters. I like names with only one character (or a character with a glyph) so that we can identify useful combined logics by a series of characters such as KD4.2 or KD45. Garson's term "C4" is, if I remember correctly, motivated by it being the converse of 4. Perhaps by using that idea we could instead name the scema by rotating the numeral "4" 180 degrees? To me that would be an aesthetically more pleasing baptismal. $\endgroup$ Mar 21, 2015 at 18:48

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$\begingroup$ I think I will settle for \begin{sideways}\begin{sideways}4\end{sideways}\end{sideways} with the rotating package for the text I am writing up. The instruction will turn the "4" upside down, and we will have a general scheme for naming converses. $\endgroup$ Mar 21, 2015 at 19:33