Dishonesty and other vices are often seen as corruptions, or shortcomings in a person's character or composition.
An integral person, on the other hand, is whole.
Such a person is not lacking in moral virtues.
english.stackexchange.com/users/3313/Jay
Etymology online has the following entry for integrity : mid-15c.,
"wholeness, perfect condition," from O.Fr.
integrité, from L. integritatem (nom.
integritas)
"soundness, wholeness," from integer "whole" (see integer).
Sense of "uncorrupted virtue" is from 1540s.
That doesn't answer why integrity also means honesty, but you can see that circa 1540 the word began being used to express "uncorrupted virtue," which implies honesty, as a virtuous person does not lie.
english.stackexchange.com/users/1631/Scott Mitchell
I'm not sure that integrity and honesty are entirely synonymous.
I think integrity is closer to the word consistent than the word honest.
Of course, you couldn't really be dishonest and have integrity because dishonesty requires inconsistency between your words and your thoughts.
english.stackexchange.com/users/181/John Weldon
I agree with @John
Weldon that integrity and honesty aren't synonymous.
So defining integrity becomes important, and I'm inclined to go with the crowd on that--integrity in a human being is a state of having all the important virtues.
Being "whole" or "complete" as a being of virtue.
There would probably be some disagreement about which virtues are required for integrity, of course.
In this view, honesty can be seen as a necessary condition of integrity--you can't be "integrated" without it--but you can be honest and still lack integrity if you also lack other crucial virtues.
I think people sometimes think of integrity as synonymous with honesty because it's one of the most obvious virtues, and one that most people can agree is necessary for integrity.
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In the usage where integrity means honesty, it comes from a meaning that what you communicate and your actions/intentions are coherent with each other, forming a whole that has the property of integrity.
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This question -> 1 <- was closed, but I don't see it as a duplicate of this one at all.
In the Mental Health professions, one speaks of "being Integrated" or "well Integrated", but the term "integrity" itself almost invariable has no special meaning.
Rather, it has the common usage that associates it with ethical or moral values.
A person could be "well integrated" i.e., in no way suffering from Associative Identity Disorder (the current term, not M.P.D.), and still have ethics or morals that their society would judge to be poor.
So, the answer to the closed question is No, 'Integrity' is not an antonym to 'Honesty', but 'Integrated' could certainly be considered one.
The answer to this question is that strictly speaking Integrity and Honesty are not identical, but in common usage what most other members of a society consider to be a whole, complete "well adjusted" person is usually one that they also consider honest.
english.stackexchange.com/users/597/MickeyfAgain_BeforeExitOfSO
The word  integrity  meaning  wholeness  seems to come from the word  integer  which, roughly speaking, refers to whole numbers. Why does  integrity  also mean  honesty ?
Why "integrity" means "honesty"
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<meaning>
english.stackexchange.com/questions/8131
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It is not clear whether or not "integrity" and "honesty" are actually synonymous. In fact, integrity might be closer to the word "consistent". Integrity could also be better explained as "having all the important values", of which one may be honesty, although it is possible to be honest whilst lacking integrity. Dishonesty and other vices are often seen as corruptions, or shortcomings in a person's character.An integral person, on the other hand, is whole However, integrity and honesty may not be entirely synonymous. it would be impossible to be dishonest and have integrity, because dishonesty requires inconsistency between your words and your thoughts.People sometimes think of integrity as synonymous with honesty because it's one of the most obvious virtues, and one that most people can agree is necessary for integrity.Integrity in a human being is a state of having all the important virtues.
It is not clear whether or not "integrity" and "honesty" are actually synonymous. In fact, integrity might be closer to the word "consistent". Integrity could also be better explained as "having all the important values", of which one may be honesty, although it is possible to be honest whilst lacking integrity. Integrity means wholeness, soundness and uncorrupted virtue, which implies honesty.
Integrity and honesty are not entirely synonymous.  People sometimes think of integrity as synonymous with honesty because it is one of the most obvious virtues, and one that most people can agree is necessary for integrity. It is not clear whether "integrity" and "honesty" are perfect synonyms or not. However, "integrity" may often be used to refer to "honesty" because honesty is seen is a necessary virtue in order for one to have "integrity".
A person could be "well integrated" i.e., in no way suffering from Associative Identity Disorder (the current term, not M.P.D.), and still have ethics or morals that their society would judge to be poor.
A person could be "well integrated" i.e., in no way suffering from Associative Identity Disorder (the current term, not M.P.D.), and still have ethics or morals that their society would judge to be poor.
Dishonesty and other vices are often seen as corruptions, or shortcomings in a person's character.An integral person, on the other hand, is whole. However, integrity and honesty may not be entirely synonymous. Some think "integrity" is closer to the word "consistent" than the word honest. Of course, it would be impossible to be dishonest and have integrity, because dishonesty requires inconsistency between your words and your thoughts. Being "whole" or "complete" is a virtue.Honesty can be seen as a necessary condition of integrity--you can't be "integrated" without it--but you can be honest and still lack integrity if you also lack other crucial virtues.  IThe answer to this question is that strictly speaking Integrity and Honesty are not identical, but in common usage what most other members of a society consider to be a whole, complete "well adjusted" person is usually one that they also consider honest. Dishonesty and other vices are often seen as corruptions, or shortcomings in a person's character or composition. An integral person, on the other hand, is whole. Definition of integrity and honesty Integrity and honesty may not be entirely synonymous.  I think integrity is closer to the word consistent than the word honest.  it would be impossible to be dishonest and have integrity, because dishonesty requires inconsistency between your words and your thoughts. People sometimes think of integrity as synonymous with honesty because it's one of the most obvious virtues, and one that most people can agree is necessary for integrity. The defintion of integrity integrity in a human being is a state of having all the important virtues. "whole" or "complete" as a being of virtue.  There would probably be some disagreement about which virtues are required for integrity, of course.   Honesty can be seen as a necessary condition of integrity--you can't be "integrated" without it--but you can be honest and still lack integrity if you also lack other crucial virtues.   People sometimes think of integrity as synonymous with honesty because it's one of the most obvious virtues, and one that most people can agree is necessary for integrity. but the term "integrity" itself almost invariable has no special meaning. A person could be "well integrated" i.e., in no way suffering from Associative Identity Disorder (the current term, not M.P.D.), and still have ethics or morals that their society would judge to be poor. The answer to this question is that strictly speaking Integrity and Honesty are not identical, but in common usage what most other members of a society consider to be a whole, complete "well adjusted" person is usually one that they also consider honest.
Integrity means wholeness, soundness, perfect condition.  Integrity means uncorrupted virtue, which implies honesty. 
It is unclear whether "integrity" and "honesty" truly are synonyms. It is possible that "integrity" has become synonymous with "honesty" because "honesty" is seen as necessary for one to have "integrity".