The Paradox of Conviction

Here's a puzzle I've been thinking about (with qualifiers, especially about the rife usage of the term reasonable below):

1. Most of us are, more or less, reasonable.
2. Most of us have some convictions about controversial issues (and reasonable people have conflicting convictions; hence, the controversy).
3. When our beliefs conflict with the beliefs of other (equally) reasonable people we should weaken our confidence in our own belief (tantamount to withdrawing our conviction, possibly adopting an agnostic stance on the issue).
4. Thus, it would seem that reasonable people should not have convictions (about controversial issues).
5. But reasonable people do have such convictions.
6. Thus, reasonable people are not reasonable!

Qualifiers:
A. By reasonable, I mean susceptible to evidence, disinclined to take radical skepticism seriously (or a pragmatic reason for disbelief), not inclined to the preposterous view that we are infallible (or even less fallible than our peers), etc.
B. I limit "controversial issues" to disagreements between reasonable persons. So, whether the Holocaust occurred is not a controversial issue.

Thoughts:
- Maybe 1 is false? Let's hope not! (If we aren't, to some degree, reasonable, how will we get out of this mess?)
- I'm inclined to think there is something wrong with 3, even though it looks...reasonable. But denying 3 seems tricky. One possibility is that not all reasonable people are "epistemic peers"--that is, our background beliefs might be sufficiently different to give us each independent justification for holding the particular convictions we have. Nevertheless, if I have to assume that your background beliefs are just as prima facie reasonable as mine, then when confronted with a genuine, persisting controversy, I seem to have some reason to weaken my belief.

The puzzle, in part, has to do with a way I'm proposing we understand the notion of a conviction: we have convictions precisely about those things that are controversial in the sense above. If an issue is not controversial, then it does not, on my account, count as a "matter of conviction." (It's not my conviction that the Holocaust occurred, or that slavery is wrong...but I do, of course, believe these things and am completely convinced of them.) (NB: Convictions in my sense are not "blind": the person of conviction has reasons for her beliefs.)

So the puzzle is that it looks like it is never reasonable to have convictions on the very issues on which it is possible to believe with conviction (i.e. controversial issues). So, either we're not being reasonable in having convictions or, roughly, 3 above must go. (Or maybe my working account of conviction is the trouble-maker here?) To be continued...