There are two laws that may have been broken here -- the
Logan Act and the crime of making false statements.
The Logan Act makes it a crime for a private citizen to communicate with a foreign government without proper authority in an attempt to influence the actions of the foreign government. President John Adams signed the Logan Act in 1799 in response to the
actions of a state legislator (George Logan) who apparently went behind Adams' back and traveled to France to try and negotiate peace with that country during the
undeclared Quasi War.
Flynn is guilty of violating the Logan Act if he (1) had communication with a foreign government; (2) with the intent to influence that foreign government, (3) while being a U.S. citizen without the authority to engage in diplomatic discussions on behalf of the United States. If Flynn violated this law, he would be guilty of a felony and face up to three years in prison.
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The Logan Act is not the only law that Flynn may have broken. Federal law also prohibits someone from
making a false statement when discussing a matter within the jurisdiction of the federal government if there is an intent to deceive a government agency about an important matter. Making a false statement is a felony that carries up to five years in prison. Unlike the Logan Act, prosecutions under the false statements statute
occur all the time.
A false statement can be a lie, or an attempt to cover something up. The critical issue is whether the person intended to mislead a government official about an important matter. Did Flynn make a false statement? Based on what's been reported, it appears that he did. At some point, it appears that Flynn told Vice President Pence that his discussions with the Russian ambassador did not involve the recently imposed sanctions.
That statement was made to a government official, concerned a matter within the jurisdiction of the federal government, and now appears to be false. The only questions are whether Flynn intended to lie or just made an honest mistake, and whether any false statement he may have made was intended to impair the government's investigation of his discussions with the Russians.