Embedded in the "Not My President" statement is a complex philosophical concept and a major contradiction. As an anarchist, I agree with this statement, but I find fault with the sentiment it usually represents. Is it really a principled protest against an encroachment of liberty, or a whining complaint of a sore loser? Well, it's actually both...

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Argument from precedent

Appeals to tradition are irrelevant in rational discourse, but history can reasonably be used to cite precedent while building a case for why a right is recognized.

According to the Declaration of Independence, it is the right of the people to withdraw their consent when they believe government fails to represent them. This argument is built on he principle of self-ownership and the argument that a government's claim to authority requires consent to avoid being a trespass against natural rights.

While the US Civil War is sometimes presented as a counterargument, serious analysis shows that neither side in that conflict was interested in the rights of the people, and both sides were pursuing the interests of the political classes in the North and South while hiding their intent behind propaganda. History is filled with examples of secessions and independence movements.

Argument from individual rights

This brief post cannot possibly provide a comprehensive coverage of natural rights theory and all its attendant distinctions and arguments. In a broad sense, natural rights theory typically defines rights as "negative," meaning rather than creating an obligation to act, they define the sphere where another individual's action is a trespass. They are also reciprocal and universal, meaning they must apply equally to any two parties at any time and place. In short, if I do not have the authority to govern other people, and other people do not have the authority to govern me, I cannot rationally delegate such authority to anyone else. Anyone claiming such delegated authority bears the burden of proof to show it.

There is no evidence for representation. There is no agent/principal relationship between the government and the governed. Any such claims to such delegated authority are thus inherently usurpation, and cannot be supported by reason and evidence. As such, political authority is illegitimate. It is entirely rational for people to argue that Trump is not their president. It is also rational and consistent to say that Obama was not their president either. George W. Bush, Bill Clinton, George Bush Sr, Ronald Reagan, etc. can all be disavowed as legitimate presidents in a consistent application of this line of reasoning if the foundation principles are human action and self-ownership.

The Fault in the Protester's Claim

The typical protester's argument is not based in historical precedent or individual rights. Instead, it is a hypocritical complaint rooted in the desire to impose a different unwanted ruler upon others. It is absurd to argue that opposition to Obama was treason while simultaneously rejecting Trump.

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If you're concerned that the "wrong guy" could get power, maybe the root problem is the presumption that anyone has the authority to wield that kind of power?