For example, were all politicians to fully disclose every personal foibles, character flaw, and detail concerning personal life, few honest politicians would ever by elected.
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The sentence could be rewritten as
"If all politicians were to fully disclose every personal foible
s, character flaw, and detail concerning personal life, (then) few honest politicians would ever be elected"
It's a way to create an if-then sentence without actually saying either of those words.
Another example is "Had I known before that it would be this cold, I would have brought a jacket."
That sentence means the exact same thing as "If I had known before that it would be this cold, (then) I would have brought a jacket."
It's important to note that you can only start this sentence structure with an auxiliary verb (i.e. you can't say "Knew I before that it would be this cold")