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Hmmm, I am not sure whether I am talking about a certain kind of appropriateness rather than all of them.

I think on this issue being very specific may be crucial.

It is one thing to say - “This statement is probably true, but I don’t have a 100% proof, so I’m not going to act on it”.

It is another thing to say – “This thought is probably true, but it is inappropriate/racist/sexist, so I’m going to suppress it.”

The examples from your last comment refer to the former case, while the previous comments refer to the latter.

For the kinds of appropriateness I am focusing on it is clearly inappropriate to hold knowingly false beliefs

The problem is that, unlike mathematical theorems, our ideological views cannot generally be proven false. Instead, we need to weigh the arguments for the different sides. Thus it is often not required to hold knowingly false beliefs – simply avoiding certain thoughts as inappropriate would be sufficient to tilt the balance.
and appropriate to take reasonable measures to keep your beliefs factual.

Can you describe these measures?

I am a little fuzzy on whether I mean a more general appropriatesness but I present here an analogy in the hopes that it highlights the kind of appropriateness relevant here and how it can't be handled based on correctness alone.

You are running a village that runs agriculture. Your village does agriculture. However the surrounding birds eat on the bugs. In fact they leave only 2 species alone. A hoverfly species and bees. Hoverflys are bee mimics. Your plants need pollinators in order to grow properly and hoverflys and bees are both pollinators (and we are assuming no others are availble). However you have a problem. Your village is allergic to bees to the extent that getting stung is lethal. If you let all the bugs be as they are 100 people are going to get lethally stung. However if you kill up all the bugs the plants don't grow (fruit) and 50 people are going to die of starvation. Person A argues "having a yellow-black bug land on you undisturbed has 50% of getting you stung. This percentage is only 1% for bugs in general. Yellow-black bugs are dangerous and should be swatted on sight. Let X be the percentage of yellow-black bugs swatted on sight. We can show that the expected amount of deaths is (100*(X))+(50*(X-1)). The value of X that minimises the expect amount of deaths is 1.". Person B argues "Yellow-black bugs are only dangerous because bees are yellow-black bugs. Hoverflys are also yellow-black bugs but they are harmless. Bees have a 100% chance of being poisonous. Hoverflys have 0% chance of being poisonous. Both can equally pollinate. We should only swat the bees and not the hoverflys. This way we can have only 25 people die which is lower than 50 that get killed if we swat all.".

Person As beliefs are all correct (maximally so). In a non-reduced bug environment the swatting would be reasonable risk management as the distinction is hard to make. They are afterall mimics. Indistinguisability keeps them alive.

However the expected amount of people that are going to live is propably way more important than the effort of distinguishing. If a human is aware that hoverflys are a thing and knows what to look for the task isn't even that hard.

While it would be unreasonable to brute-force catalogue every single aspect of surrounding nature there are atleast two reasons why one ought to pay special attention to yellow-blacks. They sting people (they are a threat). And they pollinate vital plants (they are an important asset).Yellow-blacks should be on the top of your villages attention list even if you didn't know about the existence of hoverflys.

Previously the village had a belief that only one kind of coloring of bugs should be left alive. The village eventually had only one kind of bug and when a predator for it showed up there were mass starvations. This made people think that it is important to ignore bug colorings. Person A can claim that Person B is being too color-blind. The previous bug types were all of the %1 lethality type and the resultant swatting unnneccesarily reduced biodiversity. however here we have clear evidence that yellow-blacks are not %1 but 50% and it does indeed make sense to swat them for the risk they pose. And this is more based on reasoning than the previously totally made-up belief. However person A kind of has inherited a "color sensitive worldview". For him bug colorings are very fundamental categories of being. He is a colorist. He does belief that all things being equal bugs of all colorings should be allowed to live. But yellow-blacks have been identified as being poisonous independent of their colorings but they happen to be correlated.

One reason to abandon the line of logic of person A is to say that we ought to see the situation in concepts and terms that is clear and useful (without regard how it was seen in the past). The concept of hoverflies arises form the need to minimise deaths. That is not only a matter of correctness but why we are forming the beliefs.
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