How is a proposition “this table is primarily treated as firewood” to be formalized in Objectivists’ terms? Is it something like: “the concrete object x as conceptualized as an integrated identity A is employed in reality to the extent that its expressing his integrated identity B and thus negating the A-identity”?
At this point I think a more realistic account is this one:
1] The needs and wants of people are met when an object x is created with the capacity to perform a desired function F.
(e.g. to place stuff on, preferable foods or decorative materials).
2] The capacity of a concrete object x to perform function F is equivalent to having an identity A1.
(e.g. all table-like objects have the capacity to perform table-functions.)
3] In order for objects to exist they must be made of matter. The chosen materials for object x have identity M.
(e.g. this table is made of wood.)
4] A concrete object x can be utilized for any desired function F2, F3, F4,….Fn, as long as it doesn’t contradict the physical properties of x, i.e. the only constrain is its identity M.
(e.g. this table can be used as a raft, firewood, shelter, .)
5] Reiterating 2]: having the capacity to perform functions F2, F3, F4,….Fn is equivalent to having identities A2, A3,A4, An.
6] Thus object x can be identified as: (A1,A2,A3, and/or An) and M.
7] As 1] and 4] are predicated on the needs and wants of people, the identification is subjective in functional terms and only objective in reference to in its physical properties.
(Although this specific object is meant as a table, commonly used as table, his material M fully enabling the usage as a table, we have to conclude that x is not uniquely a table to all people at all times).
8] Conclusion: identification cannot be a purely objective act.
At this moment I personally think this last ‘deconstruction’ of identity is plausible and it seems to have some parallels with the accounts some Objectivists has given thus far. At the same time this conception of identification seems to differ substantial from what Ms Rand wants her readers to accept.
The standard you wish to use to define objectivity is in principle making objectivity impossible for all actual persons. Nothing is ever “uniquely X to all people at all times.”
A statement can be true on a desert island. As truth depends on the coherence with reality. When a second person would arrive he could verify the statement. The verifiably of a statement is a precondition for it to be objective, the actual verification is the explicit claim of a second person that this statement indeed is true. Without an actual verification a statement can be true and (maybe) objective but an observer stranded on an island has no absolute mean to be sure of that.
I would dispute that “The verifiably of a statement is a precondition for it to be objective”, since by your own proposed standard of “uniquely X to all people at all times” simply getting one other person to agree with you is entirely inadequate. Surely a poll must be taken of all people across all time and all space and nothing less than unanimous agreement will suffice to prove objectivity.
Your position has internal logical consistency problems.