One of the most common ways to address that worry is to appeal to something not reducible to structure, for instance, causality—the symbol is connected to the object because, by whatever means, it is caused by it. But I don’t think that’s a good option, because it gets into problems with misattribution—if, for instance, you mistake a cat for a dog, then in some ways, the ‘dog’ symbol was caused by a cat. But if meanings are due to causal influences, what’s the difference between the ‘right’ and ‘wrong’ instances? Shouldn’t the symbol then simply mean ‘dog or cat’?
Another one is the ‘biosemantics’ approach due to, mainly, Ruth Millikan. This identifies the meanings of symbols by an evolutionary account: the symbol X means Y if it’s evolves function is to indicate the presence of Y.
But you’re completely right to note that it’s an open problem, generally referred to as the question of ‘intentionality’ in the philosophy of mind. (It was the main motivation for coming up with my ‘von Neumann mind’-model.)
Again, these will just be more relation—‘just more theory’. It will do nothing to address the problem (in fact, it will only make it harder).
Exactly, and that’s all there is to the objects in the argument, and all that’s needed. The just form a set of n things, and that’s it—the rest, after all, is supposed to be supplied by means of relations.
If they show up in some relation, they’ll be ‘discoverable’ by those means. If they don’t, they’re also not relevant for anything, and we can just excise them from the domain.
If what you’re talking about includes a structure that is defined over those objects—i.e. if there’s some relation fulfilled by them—then they’ll be just as much part of the domain. If there isn’t, then, again, we can just do without, since they relate to nothing. It’s like saying there’s an electron that doesn’t influence anything else—if we talk about the universe, we can just forget about it.