Doxastic Logic - Inconsistency of The Belief in One's Stability

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:

    Where belief is painful we are slow to believe.
    Ovid (Publius Ovidius Naso)

    ...I feel anxious for the fate of our monarchy, or democracy, or whatever is to take place. I soon get lost in a labyrinth of perplexities; but, whatever occurs, may justice and righteousness be the stability of our times, and order arise out of confusion. Great difficulties may be surmounted by patience and perseverance.
    Abigail Adams (1744–1818)