What is meant by living one's life by reason and logic?
When I hear someone say this the (almost) immediate pattern I sense is that all of my interlocutor's interactions are mediated by way of the unquestioned articles of faith driven by a passion for a simultaneous bloodlessness and hostility toward the very sensuousness through which such mediation is at all even possible. It bespeaks a demand for both a neatness of what is messy and a judgement that this messiness somehow 'proves' some type insufficiency on the part of this sensuousness. Any, if not every, moment or instance of sensuousness strong-armed into a role of redundancy and secondary importance at best...as 'barely existing,' paraphrasing Plato, in comparison to its representation; its name and image.
This role is a perennial one-liner vainly declaring itself-as-itself repeatedly, from start to finish (as start and finish) and with increased vigor when any attempt to get a word in edge-wise is made by another. All questioning is taken as protest and brusquely handled as such. How can one dare question the authority of logic, after all!
It's paradoxical, then, that the whole insipid, anti-climactic farce within which this role 'in itself' may even begin to count as half-coherent, cannot come to its own conclusion; the ending must be forever deferred in this sad, lifeless theater since it is impermissible to have a story to tell. Only endless points of disembodied positions within the imperative of three-dimensional space overlaying the world, perhaps even believed as 'transcending' our world, if not an improved version of it. All the singing, humming, drumming and dancing sensuality grounding, and thus allowing for, this screeching command will never be admitted into these spectre-filled catacombs, excepting when useful as the servant-girl named 'Evidence,' after which she's promptly sent back to her miserable quarters by her man-handler.
All the fleshy succulence of possibility sings to us against this rather narcissistic article of faith whenever we silence our chatter and simply allow the shining forth of appearance. Logic is the will to never shut-up, however. The Word is always primary.
Now, none of the above is meant to indicate my 'rejection' of using reason/logic. It's not a question of belief in regard to reason/logic at all...at least for me. As others have already said, reason and logic may be considered tools, and like all tools they have a use and a process of evolution, which is rarely, if ever, questioned by those purporting to live according to reason and logic. My experience has been, if the so-called origins of reason and logic are allowed to surface in conversation, they are nearly always placed within a mythological framework we might call Progress, with all of the latter's attendant yet unexamined assumptions, implications and consequences.