Minimum wage laws cause unemployment, these opponents say, because they price those workers whose skills don't justify the minimum wage out of the market completely. If a worker only has the skills to produce $14/hour worth of benefits to an employer, the employer is better off not employing that person rather than losing $1 dollar/hour doing so, if the minimum wage is $15/hour. And regardless of where the minimum wage is presently, any increase in the price of labor will result in less demand for labor, all other things being equal.
That's basic economic reasoning and wasn't even controversial until recently when, for political reasons, economists like Paul Krugman began contradictingtheir own earlier writing on the same subject. But as economically sound as the unemployment argument against minimum wages may be, it ignores a previous and much more important one: you don't own other people.
We think of the basis of what used to be called "the liberal tradition" as being the fundamental rights to life, liberty, and the pursuit of happiness. Governments are ostensibly instituted among men "to secure these rights." But these rights are pillars, not the foundation of a free society, according to the essay Jefferson himself said established the "general principles of liberty and the rights of man, in nature, and in society," as Americans of his time understood them.