If I am in a certain intentional state - say believing something particular to be true (that today is Friday), then I will in general behave and speak in a ways that is consistent with this. The ways that might be consistent cannot be spelled out in advance, and not only because they are complex - also because unexpected situations will arise.
Who will adjudicate? I and my interlocutors, playing the shared language game within which the statement of my intentional state is held to be true. The intelligibility, the playability, of these adjudications will be one part of what makes the whole game possible; and if the become idiosyncratically unintelligible then so does the original statement of my intentional state - we will find that we did not know, after all, what we meant by attributing this state to me.
This process of adjudication and exploration is what the history of mathematics reveals to us about addition. Someone who has been quadding so far, but thinks they were adding, is not a competent participant in this conversation.
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