The cosmic story of five Titanes--four holding the corners of heaven--may be Phoenician in origin.
What Phoenician story could be the basis for the Titanes story?
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As far as I can tell, the modern-day theory for the connection between Phoenician gods and the Greek Titans coms from Atlantis, the Antediluvian World, written in 1882 by Ignatius Donnelly. Donnelly attempted to draw parallels between several different mythologies and historical accounts, and in doing so prove that the lost continent of Atlantis actually existed. As should be obvious, this makes most of his conclusions dubious, at best.
Not alone were the gods of the Greeks the deified kings of Atlantis, but we find that the mythology of the Phœnicians was drawn from the same source.
For instance, we find in the Phœnician cosmogony that the Titans (Rephaim) derive their origin from the Phœnician gods Agrus and Agrotus. This connects the Phœnicians with that island in the remote west, in the midst of ocean, where, according to the Greeks, the Titans dwelt.
Note that "Rephaim" appears to be an attempt to connect the Titans with the Rephaites, giants which appear in Hebrew texts. This is another indication of the book's attempt to find similarities between different religions and mythologies. For this reason, I'd be doubtful of the connection.
Atlantis, the Antediluvian World is not listed among the sources on the page you link to, but I honestly can find no other source of the Titan-Phoenician connection (not that there isn't one, just that most mentions are exact quotes of Donnelly). There are definitely parallels between Canaanite mythology and ancient Greek mythology, but I can't find any more information on this specific case.
As an interesting note, I've seen some references in passing to Targhizizi and Tharumagi, twin mountains which supposedly hold up the sky, and I suppose those could be connected with Agrus and Agrotus.