There is a shorthand and implied understanding in mathematical word problems (else they would be formulas!). As soon as you ask "what is the probability" you are implying either a sampling/generative process, or a subjective (e.g. Bayesian) probability framework. I think it's clear this is the former case.
The generative model implied for each child is: pick uniformly from boy/girl, and pick uniformly from Mon-Sun. The generative model for the father is: generate two children.
This generative process has a well-defined outcome distribution and it is completely reasonable to ask "what is the probability of generating an outcome with two boys, given that you generated an outcome with a Tuesday boy?"
The generative model implied for each child is: pick uniformly from boy/girl, and pick uniformly from Mon-Sun. The generative model for the father is: generate two children.
This generative process has a well-defined outcome distribution and it is completely reasonable to ask "what is the probability of generating an outcome with two boys, given that you generated an outcome with a Tuesday boy?"