The idea that any claim that contradicts reality is not arbitrary but false essentially means that any arbitrary claim is not arbitrary but false. In essence, what you are saying is that there is no such thing as an arbitrary claim.
Interestingly, I was going to ask you if you think there is such a thing as a false claim. Here is how I understand true, false and arbitrary – I want to know if you disagree with any part of this:
Truth is a relationship between a proposition and fact, i.e. the recognition of reality. If the product of your cognition (as a proposition) corresponds to reality, it is true. If the mental content does not correspond and contradicts reality – it inversely corresponds – it is false. An arbitrary claim has no relationship to evidence and thus to man’s means of acquiring knowledge.
I surmise that you reject my position that a claim contradicting reality is false (that is how I understand “The idea that any claim that contradicts reality is not arbitrary but false essentially means that any arbitrary claim is not arbitrary but false”, followed by the conclusion that this would mean that there is no such thing as an arbitrary claim).
If there is no evidence, the claim is arbitrary and the claim cannot be processed. If there is evidence, the claim can be cognitively processed, and it is either true or false: if the proposition describes a fact, it is true; if it anti-describes a fact (contradicts reality) it is false.
If I claim that there are green gremlins on Mars, then there is certainly ample evidence pointing away from this claim: Mars has no atmosphere to support life, has no water to support life, etc.
I will admit that I’m not sure about why Peikoff used the Hegel-studying Venusian gremlin convention example, and maybe I should stop using that one. If you’d like to substitute Mars, I don’t think it matters. An arbitrary claim can be made non-arbitrary by adding evidence: being “arbitrary” is not a permanent state, where once someone advances an arbitrary claim, the claim – even if it actually described a fact – is forever doomed to be arbitrary. You’ve contributed evidence, and that evidence shows that the claim is in fact false. That was a nice contribution, but by no means required since nobody here is seriously giving a second’s worth of consideration to the gremlin claim. I don’t think that Peikoff is saying that it is wrong to contribute evidence bearing on an arbitrary claim, just that one has no intellectual obligation to even consider such a claim that has no evidence.