When we speak of a property or trait instantiated by an object, we take two assumptions into account: the object in question has a property Bo which causes it to interact with the surrounding world in a particular way, and the perceiver has the property Bp which causes them to perceive those interactions in a particular way. This is an asymmetric relationship between perceiver and perceived.
The colloquial way of defining what it means for a statement to be true is that it corresponds to reality: if I say “it is raining” and it’s also the case that it’s raining, then what I said is true; if I say “it is raining” and it’s not the case that it’s raining, then what I said is false. This is an extensional truth condition – the extension of the proposition must be the case in reality for the statement to be true. But is this really how truth works? In what follows, I am riffing on some ideas floating around in my head, so feel free to point out any problems so as to help me clarify my thoughts.