but we had entirely different background assumptions about how one makes a case for said position. There was a near-Kuhnian incommensurability between us.

This is very frustrating and when I realize it is happening, I stop the engagement. In my experience, rationalists are not that different from smart science or philosophy types because we agree on very basic things like the structure of an argument and the probabilistic nature of evidence. But in my experience normal people are very difficult to have productive discussions with. Some glaring things that I notice happening are:

a) Different definitions of evidence. The Bayesian definition of evidence is anything that makes A more likely than not A. But for many people, evidence is anything that would happen given A. For example a conspiracy theorist might say "Well of course they would deny it if were true, this only proves that I'm right".

b) Aristotelianism: the idea that every statement is either true or false and you can prove statements deductively via reasoning. If you've reasoned that something is true, then you've proved it so it must be true. Here is a gem from an Aristotelian friend of mine "The people in the US are big, it must be the food and they use growth hormones in livestock, therefore people in the US are big because of growth hormones".

c) Arguments that aren't actually arguments. Usually these are either insults or signals of tribal affiliation. For example "Good to know you're better than everyone else" in response to a critical comment. But insults can be more subtle and they can masquerade as arguments. For example in response to a call for higher taxes someone might say "If you love taxes so much then why aren't you sending extra money to the treasury?".

d) Arguments that just have nothing to do with their conclusion. An institute called Heartmath stated this gem (rough paraphrase): "The heart sends more information to the brain than the brain does to the heart therefore the heart is more important that the brain".

e) Statistical illiteracy. I want to grab a flamethrower every time the following exchange happens:

Salviati: "According to this study people who are X tend to be Y"

Simplicio: "Well I know someone who is X but isn't Y, so there goes that theory"

f) Logical illiteracy:

Example 1:

Salviati: " If A then B"

Simplicio: "But A isn't true therefore your argument is invalid"

Example 2:

Simplicio: "X is A therefore X is B"

Salviati: "Let us apply a proof by contradiction. 'A implies B' is false because Y is A, but Y is not B"

Simplicio: "How dare you compare X to Y, they are totally different! Y is only not B because ..."

Sorry if the symbolic statements are harder to read, I didn't want to use too many object level issues.