When the costs of production increase, businesses increase the prices of products to recover the costs. This causes rising inflation.
Higher minimum wages do not create higher standards of living and a reduction in poverty. As the cost to provide services and produce products increases, businesses increase prices, leading to increased inflation, and reduced purchasing power of low-income earners. This increase in inflation offsets any increase in living standards, leaving the same low-income families the minimum wage was designed to help to languish in poverty.
While this was once the case, in the age of automation and technological solutions, higher minimum wages do not equate to higher inflation. Productivity has been rising, which has been lowering the costs of production. Implementing or raising a minimum wage would put more money in the hands of low-income workers. The increased wage costs on businesses could be offset by the increased productivity unlocked through more automation. This could keep prices, and inflation, low, while increasing standards of living and the purchasing power of the lowest earners. 
[P1] The minimum wage increases the cost of production. [P2] Businesses pass this cost onto the consumer by increasing prices. [P3] Increased prices, cause a rise in inflation. [P4] This rise in inflation keep low-earners purchasing power down. [P5] Therefore, the minimum wage does not reduce poverty.
Rejecting the premises
[Rejecting P1] Improved technology and automation have improved productivity. This productivity improvement has reduced the costs of production. These reduced costs more than offset the cost increases caused by a minimum wage. Therefore, employers would not need to raise prices and inflation would not increase.