(Philosophy novice here and not sure how best to phrase this question; if it's unclear please point out the problems).
Imagine this conversation:
Alice: I believe that X.
Bob: Do you also believe Y? (Alice says yes) But that means you believe in Z.
Alice: That's true, but I still don't believe in Z.
Alice's position is clearly inconsistent. The question is, so what? From what I have seen in philosophy, people simply assume that things must be consistent. I'm wondering what happens if we accept inconsistency.
Based on what I've seen, if we accept inconsistency then everything becomes possible. However this seems to be a mathematical proof that isn't directly translate-able to real life. For example we know quantum mechanics is inconsistent with general relativity, yet the world goes on as normal (and things don't start falling upwards).
The other possibility is that, if we accept inconsistency, then logic and rational discourse becomes impossible. By accepting both X and ~Z, Alice is saying she cannot be reasoned with. But this doesn't seem to work either. Most people don't make it past PhilosophyExperiment's Battleground God unscathed, yet we are still able as a society to agree on certain things (such as "murder is morally wrong"). Descartes famously claimed that God can make 2x4 not equal 8, which most people would call inconsistent, and yet most people would also not say Descartes is impossible to reason with either.
Why is it common to assume that things must be consistent (indeed, proof by contradiction is commonly used in mathematics)? Does inconsistency lead to disaster? If so, why are we able to hold inconsistent beliefs and still go about our lives without problems?