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.



Is there a good list of nomenclature for modal axioms?
One of the most complete online summaries of modal logic systems is still [Halleck's list](http://home.utah.edu/~nahaj/logic/structures/systems/). 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](http://plato.stanford.edu/entries/logic-modal/) 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](http://books.google.com.br/books/about/Logic_and_Structure.html?id=4u9gQ6pctuIC) and [also here](http://books.google.com.br/books/about/Modal_Logic.html?id=gFEidNVDWVoC)), 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](https://books.google.com.br/books?id=oSNs8aRexYYC) 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](http://en.wikipedia.org/wiki/Newman%27s_lemma) of [ARS](http://en.wikipedia.org/wiki/Abstract_rewriting_system) 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.


