I wonder if there is a tendency for organisations to be quite 'shallow' in their approach to honesty and integrity because they need to protect themselves from negativity. Whereas if a different lens is used then the focus can be on strengths which means that leaders and staff can push themselves to constantly improve but avoid the defensiveness.

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I suppose that the depths of organisations head off in different directions - towards preservation and reputation rather than towards openness and honesty? Which of course destroys opportunities to learn and improve...
The failure-tolerant leader paper by Farson and Keyes comes to mind with this. They separate mistakes from failures and describe approaches to encourage learning and improvement.
I would say that preserving professional reputation is important and a high ethical principle. The concept of a 'professional' as someone who is to be trusted is needed in society. I do though think that this does not necessarily need to clash with openness and honesty.
I would like to cite my interpretation of integrity. I have moved away from the idea that it is always doing what you say you will do to a more nuanced understanding. I see integrity as trying to always do what you say you will do, accepting that this is not always possible, and proactively trying to close the gap.