Since the truth of a proposition relates to a state of the world (reality), the notion of a contradiction relates to this as well. Hence, we have to shift our attention to the very nature of this world/reality and its representation, when facing a contradiction. A contradiction just refers to an ambiguous state, expressed as a non-exclusive duality of logical values. Now, we may ask, is it the world, which is ambiguous, (we may never know), or is it our representation, or, is there possibly more than a single representation/context at once?
A (very simple) example: If we subscribe to the notion of color as a property of an object, a solid colored object _is_ of a color, meaning, it may be either blue or red, but not both at once. But, if our description of the world includes the notion of a spectrum and mixed colors, it may be both blue and red at once, commonly related to as a shade of violet or purple. The very context may include the spectral description or not, it may be even ambiguous about this.
Suspension of disbelief: I'm collaboratively participating in a narrative and asserting certain truths is a premise to this collaboration, which is the very act of making meaning. At the same time, I do not believe in the reality of what is actually my psychological reality. This is a certainly a matter of context switching, which is not reserved to art, but may also be observed in real life.
In certain cases, the narrative which is the description of the real world, which we wholeheartedly assert, becomes complex to an extent, where, while we still acknowledge the correctness of this description, we find it increasingly difficult to believe in, even on the basis of a working model, while still working in and with it. Compare quantum physics, certain aspects of cosmology, etc.
Parallel acknowledgement of contesting messages regarding the state of the world or facts therein, especially in politics or systems of belief. I do know that the regime I was asserting did this and this, but still I do not believe that the leaders would have done so. (Compare German citizens guided through concentration camps after WW II; I see it, I do know it to be true, but I do not believe it.) Or, by the example of current SCOTUS hearings, I may believe a certain witness (i.e., assert the content to be true), but still do not believe that the other person would have committed the incriminated act, still, I assert the testament of the witness as truth, which implies that the act has been committed by this person, etc. (All kind of distortions arise from this, some are even part of what we regard as objective history.)