There is a problem with understanding the distinction between the arbitrary and the possible, especially when you’re speaking of the “theoretically possible”. The existence of element 117 is (AFAIK) both allowed and demanded by all reasonable physical theories, and these are theories for which there is a good supply of evidence (we’re not talking about some hyper-abstract 15th dimensional string-theoretic deduction). This element has never been observed, so the direct evidence that it does indeed exist is null. It would still be wrong to say that the claim for this element is arbitrary (and fortunately, nobody here has said it is), because of how compelling the underlying physical theories are which predict the element. Denying the existence of the element means denying certain general conclusions which are well-supported scientifically. But the denial of an established scientific conclusion must itself be based on the integration of some new truth – not the arbitrary denial of an existing conclusion. Existing knowledge cannot (assuming a rational epistemology) be cast aside ad libitum just because you can. Thus the grander claim that element 117 exists is not arbitrary, and it is even probable, given the weight of the underlying theory (and its experimental support).
Let us compare this to the BIV. There is no theory which is even in the least supported to the effect that we must be BIVs or that we might be BIVs. We are even aware of the cognitive difference between actual direct experiences of real events and dreams or hallucinations, and we are not aware of being BIVs. Nothing proves, or remotely suggests, that the BIV story is possible at a physical level, so saying that the BIV story is “possible” is an examplar of the arbitrary. Notice Vladimir’s statement – 'Remember that the “possible” is simply that which may be; containing all assertions which don’t contradict known facts, relations, physical laws, etc. You don’t need evidence of something for it to be considered “possible.” ’ Okay, now what is the evidence that BIV is compatible with known facts, relations, physical laws, etc? There is evidence that it contradicts known facts, relations, physical laws, and no evidence presented that it is consistent with scientific theories.
Vladmir has proposed that what the category arbitrary “seems to be trying to get at” is the utterly untestable. It’s interesting that one of the first examples of the arbitrary mentioned by Peikoff is the convention of Venusian gremlins. This is not a thoroughtly untestable claims, it’s simply difficult at present to test the claim. To quote p. 164 “An arbitrary claim is one for which there is no evidence, either perceptual or conceptual. It is a brazen assertion, based neither on direct observation nor on any attempted logical inference therefrom.” Thus given his misunderstanding of the concept “arbitrary”, I do see how he made the mistake of not seeing that BIV is an arbitrary claim. Once you understand what the concept really refers to, BIV goes into the waste-bin of arbitrary claims.
What about those squirrels and trans-Plutonic monsters? They are both arbitrary claims, but still we can distinguish between different kinds of arbitrary claims. The original squirrel claim is too much like the monster claim, and Inspector, I wonder if you really meant all of those details. An evil black squirrel living under your house plotting to overthrow the government combines the possible and the impossible. The possible is “black squirrel living under your house”, but it is not possible for a squirrel to be evil or to plot, or even grasp the concept “overthrow”. You can check whether there is a squirrel, but you can’t check his morals.
We can compare three things: the trans-Plutonic monster claim, and the Venusian-gremlin claim, and the ordinary squirrel under the house claim. The squirrel claim is an innocent arbitrary claim – it reflects a low-level of epistemological slackness in asserting something that could easily be true (is consistent with what we know about squirrels and houses), without basing that assertion on appropriate evidence. The Venusian-gremlin claim is a stupid arbitrary claim, because it could, with enormous effort, be verified, but there is virtually no evidence to suggest that it really is true, and you would have to have a really stupid epistemology to say that this state of affairs is “possible”. The trans-Plutonic monster claim is an evil arbitrary claim: it cannot ever, in principle, be verified, and only an evil epistemology would allow such a claim to be considered for a nanosecond. The BIV claim is of this third, evil-epistemology type.