A minimum wage reduces poverty
A minimum wage guarantees that all workers can earn a livable wage and support their families.
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A legally guaranteed minimum wage ensures that all workers can earn enough to put them above the poverty line.
Guaranteeing workers an annual income above the national poverty line can ensure no workers are underpaid and exploited. Left to its own devices, the free market creates a race to the bottom where workers are forced to accept lower salaries to compete with others for work. The minimum wage puts a platform at the bottom to prevent anybody from sliding into conditions of abstract poverty.
A minimum wage does not reduce poverty. It benefits workers who have jobs, and keeps them from becoming impoverished, however, it reduces the volume of jobs available, leading to layoffs among less-experienced workers. For example, when Seattle increased its minimum wage, less experienced workers lost their jobs, forcing more onto government welfare programs.  The establishment or raising of a minimum wage doesn't reduce poverty, it increases through elevated unemployment levels and increased reliance on social security.
[P1] All workers earning a minimum wage can live above the poverty line. [P2] Therefore, the establishment of a minimum wage reduces poverty. [P3] Therefore, a minimum wage benefits workers.
Rejecting the premises
[Rejecting P2] Minimum wages do not reduce poverty because they increase unemployment.