Interpretation of a valuation function when we have a known truth and an assumed truth in a proof

Question

To provide context, I am currently learning about Conditional Introduction. This is the first time in this propositional calculus unit that I have encountered an assumed truth ; previously, all premises have simply been evaluated as known truths.

Here is a picture to provide further clarification:

The conditional introduction itself is not confusing. However, what IS confusing is the idea that a premise and an assumption are being classified as different entities. My confusion is as follows:

When it comes to a valuation function, what is the difference between "Assuming" a truth value and "Knowing" a truth value. Using the above picture, we "know" that the premise $R$ is true....under some value function $v_i$. However, when we say that we "Assume" $Z$ is true...this is presumably the truth value being assigned under the same value function $v_i$.

If these statements are being evaluated using the same value function, why is it that one statement's truth value ($R$) is "known" but the other statement's truth value ($Z$) is "assumed"? Is the idea that we do not know the full extent of $v_i$'s mapping strategy?

Answer 1

One can look at the assumption on line 2 as an "additional assumption". It is temporary unlike the premises. It will be discharged on 5 when we close the subproof from lines 2 to 4 replacing it with that conditional proposition.

Here is how the authors of the forallx logic textbook describe what is going on. (page 107)

The general pattern at work here is the following. We first make an additional assumption, A; and from that additional assumption, we prove B. In that case, we know the following: If A,then B.

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P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf