Is it possible to have a deep, meaningful relationship with someone based purely on being logical? (I'm thinking about friendships, but the same reasoning should apply to romantic relationships)

In my opinion, the answer is a resounding no. The deepest relationships are built on common experiences, trust, and the unique personality of each person which makes spending time together feel special. Our personalities become intertwined – you and I change slightly based on what the other is like. While it's possible to squeeze these components of relationships into some logical framework ("well, trust is just the expectation that the other person's future behavior won't affect me negatively, and it makes sense given the person's track record and personality to have that expectation..."), they are, in my view, fundamentally incongruent with being hyperlogical.

I don't claim that people in deep relationships shouldn't value logical discourse – quite the opposite, it's great to be able to have conversations and grow one's intellect and knowledge of the surrounding world. But I don't want to feel that the relationship could end because at some point, the other person may – through a perfectly logical train of thought – realize that the friendship simply doesn't "pay off". There are many reasons why relationships should end, but this one is a sign of shallowness, not depth.