Can integrity be trained into people?
Yes, but it most probably has to start with childhood.
People, as has been pointed out here above, will make choices based on the expected or perceived value attached to each choice. If integrity has been instilled as something of value in one's mind, that will factor into the decision.
Of course, we all learned that if we steal that gum, we will get into trouble. Or, did we? Perhaps we learned that if we get
caught stealing that gum we will be in trouble. This leads into the value calculation:
Option 1 -- If I steal that gum, I'll have gum and not have to pay for it.
Option 2 -- If I get caught, I'll have to pay for it, and I'll be punished.
I handicap the odds of getting caught and decide whether it is a better value for me to take the gum or not.
Somewhere along the growing up process, we either learned on our own or were taught that there is another, intangible value to attach to one of the options. So, the calculation evolves:
Option 1 -- If I steal that gum, I'll feel bad.
Option 2 -- If I don't steal that gum, I'll feel good.
Eventually, even this choice becomes emotionally stressful:
Option 1 -- If I think about stealing gum, I'll feel bad.
Option 2 -- If I avoid the moral dilemma over stealing, I will never have the anxiety that comes with that kind of choice.
In the end, integrity is something that you teach yourself, but your parents probably planted the seed. If we saw Dad stealing cable TV service or submitting false insurance claims and feeling pretty good about it, we could be led to believe that getting over on the cable company or "getting your money's worth" from that big insurance company are repugnant, guilt-ridden activities that Dad simply rationalized away or that they are pretty cool ways to get what you want. But if Dad never tried to pull of a "harmless" little caper like those, we are never tempted to follow the same path of rationalization.
The moral of the story: Your kids may or may not copy what you do, but they are watching you for clues about how to live. If you show no value for integrity in your own life you cannot expect them to embrace it in theirs.