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 late 19th / early 20th century philosopher and mathematician Gottlob Frege famously came up with the Sense and Reference distinction in order to clarify issues in his logical system. Briefly put, the reference of a word (I’ll use my favorite example of “table”) is the actual object that the word signifies: the referent of the word “table” is the actual, physical table existing out there in the real world. The sense of a word is the way in which it exists as a psychological representation: the sense of “table” is how I conceive the table in my mind. This was important to Frege because examples where we have two (or more) words that signify a single referent can have a different sense, and therefore lead to different inferences, even if the referent is the same. How, though, might we relate these ideas?
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.