Inconsistency of The Belief in One's Stability
If a consistent reflexive reasoner of type 4 believes that he or she is stable, then he or she will become unstable. Stated otherwise, if a stable reflexive reasoner of type 4 believes that he or she is stable, then he or she will become inconsistent. Why is this? Suppose that a stable reflexive reasoner of type 4 believes that he or she is stable. We will show that he or she will (sooner or later) believe every proposition p (and hence be inconsistent). Take any proposition p. The reasoner believes BBp→Bp, hence by Löb's theorem he or she will believe Bp (because he or she believes Br→r, where r is the proposition Bp, and so he or she will believe r, which is the proposition Bp). Being stable, he or she will then believe p.
Read more about this topic: Doxastic Logic
Famous quotes containing the words belief and/or stability:
“My belief is that no being and no society composed of human beings ever did, or ever will, come to much unless their conduct was governed and guided by the love of some ethical ideal.”
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—Stephanie Marston (20th century)