How q is reached as inference, even when ~q is in the premise?

Question

This screen grab from the book, Elements of Discrete Mathematics by C L Liu. On the first chapter about Sets and Propositions, under the heading THEORY OF INFERENCES there is an example problem.

The problem states,

Example 1.26 Show that q is a valid inference from the premises p -> q, p v q and ~q

Though the answer is derived using the laws (as it is evident in the screen grab), the confusion I have is, how can possibly q be inferred when we know ~q is infact true? Is it due to the combination of illegitimate premises p -> q and p v q?

Answer 1

We don't know ~q true. We can assume ~q as true. Then we can derive q as true also (given the other assumptions and valid rules of inferences). Thus, we can determine that the premise set is not consistent.