I'm perfectly mentally capable of rigorous logic, I just question its value and the claim to objectivity. As I see it, the purely rational way of understanding the world is just as much based on irrational assumptions as a purely emotional one - the axiomata on which logic is based are not self-evident if one does not accept self-evidence. They're just as consistent as one another, with all of the Ti knowledge base cohering according to the rules of logic and all of the Fi knowledge base cohering according to principle and integrability with the ideal. As a dominant Ti type can dismiss the Fi knowledge base as irrelevant, so can an Fi dominant type dismiss the Ti knowledge base as irrelevant: e.g., I'm an INFP and an apatheist, meaning that if someone makes an objective and logically defended claim about the existence of God, I'm more likely to respond, "Why should I care? I know right from wrong on my own and it wouldn't change how I behave or what I believe," than to pick apart their argument. I could do the latter, but it's not necessary. Conversely, an INTP may need to pick apart their argument, because it would be insufficient to challenge that they wouldn't care even if it were true: they're interested in the truth for its own sake.
Both seek to achieve a qualitative understanding, but they prioritise different qualities. That's the ultimate difference, I suspect.