The problem is with the definition of logic. Logic is "the science of reasoning, proof, thinking, or inference..." (OED, 1986) Logic allows you to examine a problem, but if your internal set of data begins with errors, those errors will effect the outcome of the processed data.
The cult, depending on the input they use (terrible stuff done by other races, the inevitability of death, whatever) might lead them to believe that a certain group(s) needs detroyed. Like most Vulcans, the logic is probably top-notch...but their data is flawed or incomplete.
I think the VUlans would try to get to the group first, before Starfleet. I think they would try to reason with them, and barring success there, they wold destroy them -- logically, it would be necessary to preserve the safety of the masses, as well as the honor of the Vulcans. Besides, there's that privacy thing -- Vulcans like to keep things in-house. So you might have the V'Shar up against the PC's as well...same goal, stop these guys, but working at cross-purposes. (Welcome to bureaucratic politics...)
"War is an ugly thing but not the ugliest of things; the decayed and degraded state of moral and patriotic feelings which thinks that nothing is worth war is much worse. A man who has nothing for which he is willing to fight, nothing which is more important than his own personal safety, is a miserable creature and has no chance of being free unless made and kept so by the exertions of better men than himself."
John Stuart Mill