Most people don't argue within moral frameworks. Most people don't live logic-based lives. Most people prefer happiness to correctness, peace to progress, and quietude to justice. This is a very difficult thing, and not a very good thing, for those who want change.
Yes they do. You may contest aspects of the foundation of their logical and moral framework of choice, but most people on earth adhere to a moral or religious framework. Those are logic-based and that is what apologetics and natural theology and philosophy are for.
Like Bukowski said
those who preach peace do not have peace
beware the preachers
beware the knowers
beware those who are always reading books
I would extend it as
those who preach justice are not just
those who preach rationality are not rational
those who preach correctness are not right
And those who preach progress are against change
/edited for readability