In the famous TOS episode "Mirror, Mirror", we see Uhura, Kirk, Scotty & McCoy transported to the mirror universe. If this was a true mirror universe, then why would Spock be logical and not purely emotional? All the other characters seem to be the reverse of themselves except for Spock's logic as far as I can tell.
It's just an alternate universe. The "Mirror" universe is just a name, based on the title of the original episode. The Point of Departure (PoD) is when Zefram Cochrane murdered the Vulcan who made first contact with Earth, leading the humans to steal the ship and subjugate Vulcan en route to creating the Terran Empire. As such, the Vulcans were still logical beings, merely subjects of the TE rather than citizens of the Federation. That Spock eventually rose to rule the Terran Empire is likely based on his half-human heritage.
The Mirror Universe's Earth was even more barbaric than our Earth. Human nature was less kind. This was reflected throughout the history of Mirror Earth. Recall that in one of the episodes, one of Shakespeare's works viewed barbarism in a good light.
"Mirror" is a loosely used term. Not everyone or thing was exactly the opposite of our universe. Vulcans were still logical. Their history was affected by meeting humans.
The true mirror between the "prime" universe and the "mirror" universe isn't a physical inversion. If it were, you might expect to see Kirk as a female comm officer and Uhura as a male captain. Perhaps the Enterprise would be a seafaring vessel rather than a spacefaring one.
Instead, the true surprise and shock of the crossovers come not from the physical nature of the opposing universe but by the inversion of its moral impetus. When characters on either side meet one another, they are not horrified by how unrecognizable their counterparts are, but by how recognizable they are and the resulting discord that comes of their terrible actions. The characters don't want to be faced with the appalling nature of the person they could have become, had they been born into the wrong universe.
Outside the canon, it's also a mirror of the rest of Star Trek as a whole; While most "standard" episodes of the entire Star Trek franchise go out of the way to show humanity as we could be in our best possible future, the "mirror" episodes subvert that and show humanity as we could be in our worst possible future. Instead of a hopeful bellwether, it's a dire warning, especially as we get into
the Terran empire's fall and subjugation of humanity by the Cardassian/Klingon alliance.
Let's compare the universes.
In both, J. Kirk was a competent leader, Spock a science officer, M. Scott an Engineer, H. Sulu a helmsman with a thing for Uhura, Uhura a communications officer. It's clear their aptitudes are much the same, even as their tolerance for both violence and the use of force are not.
Looking at the DS9 mirror episodes, likewise: The basic aptitudes remain, and many but not all the personalities are recognizable. The propensity to use sex and violence to accomplish goals is again higher, and we see more clearly that selfishness is higher, but, really, the basic personality is close. Especially close parallels are Garrik, Bashir and O'Brien; the least close are Kira and Sisko - and yet, some elements are still present. Kira remains a sesualist - in the prime universe, she's constantly fighting it, while in the mirror, she's reveling in it; likewise, she's a practical rules-give-way-to-needs kind of leader in both. Sisko, in the mirror, is still a ship driver - but isn't a military officer in the mirror, and so has a considerably different skill set.
We see that Spock is actually possessed of a high aptitude for logic - As a Kohlinaru, he's pushed well past the point most Vulcans even desire, and to a point where he's recognized for his logic. And, while he forgoes the title, he's still extremely logical in both the prime and as old Spock in the Alternate. Logic is inherently something all three Spocks are good at.
Bottom line: Logic is both a skill and a reflection of an Aptitude - Spock has an aptitude for logic, both as a Vulcan and an individual.