What does this phrase mean:

'Logic doesn't pass muster'

It doesn't make sense as written, because no logical person would say that logic doesn't pass muster.

But, one might say, "The politician's logic does not pass muster."

That means that the politicians logic is not logical, or does not pass the scrutiny of someone who knows the fact of the issue.

The expression "pass muster" comes from the military. Soldiers have to line up at certain times of inspection that are called muster (which can also mean to assemble for battle, or orders, or other things, as well).

If a soldier doesn't "pass muster," it means he doesn't measure up some way to the inspection, e.g., his uniform isn't properly pressed, his shoes aren't cleaned and shined to a high enough sheen, etc.

That's where the expression originated and now can mean to not measure up about anything.

For example, you could say, "John's table manners do not pass muster."