A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already earlier for the combinatorial principle? 


Is there some (deeper) reason explaining how $\diamondsuit$ got its name? Perhaps because $\Box$ was already established as a combinatorial principle? 



Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?
I once asked Jensen this question, when we were at a conference at Oberwolfach. 


I told him that I had always assumed that the diamond $\Diamond$ principle was called $\Diamond$ because it expresses that there is an object exhibiting an elaborate degree of **internal reflection**. After all, if $\langle A\_\alpha\mid\alpha<\omega\_1\rangle$ is a $\Diamond$-sequence, then it very often happens for $\alpha<\beta$ that $A\_\beta\cap\alpha=A\_\alpha$, which is an instance of $A\_\beta$ reflecting to $A\_\alpha$. 


But alas, this was not his reason. The actual reason was less interesting, having mainly to do, he said, with him simply needing a new symbol that had not yet been used, and that one was available.


Meanwhile, I believe that we should all adopt my explanation anyway! Henceforth, let it be known that a $\Diamond$ sequence is one exhibiting beautifully captivating **internal reflections**!


