Is there a good list of nomenclature for modal axioms? 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.
 A: 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' (Hilbert-style) tradition on modal logic is clearly losing way to the 'profane' (Kripke-style) tradition (see here and also here), it would not seem advisable to choose names that focus on old-school 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.
A: Apparently $\Box\Box\alpha\rightarrow\Box\alpha$ is known as C4 (Converse of (4)).
Source: Stanford Encyclopedia of Philosophy.
