(A little calendar wisdom riffed on from Stephen R. Covey’s 2011 “The 7 Habits of Highly Effective People” calendar, months September and December)
Integrity includes but goes beyond honesty.
Honesty is telling the truth—in other words, conforming our words to reality.
But integrity is conforming reality to our words—in other words, keeping promises and commitments, and fulfilling expectations. And to do this requires character—an integrated identity that is in alignment with fundamentally correct life principles, instead of identity diffusion (several contradictory and competing and revolving part-identities as opposed to a coherent unitary self). Integrity requires a oneness primarily within oneself but also with life, truth, and correct fundamental life principles.
Integrity in an interdependent relationship or nexus of relationships is simply this: you treat everyone according to the same set of principles.
And as you do, people will come to trust you.
They may not at first appreciate the honest confrontational experiences that such integrity might generate. Many people would prefer to take the course of least resistance and belittle and blame and betray confidences, be dishonest, overpromise and or go back on their word, or gossip about others behind their back. Confrontation generated by integrity takes courage, but in the long run people will trust and respect you if you are open and honest and transparent and understanding with them.