[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. Please forgive my typos, as proofreading is incomplete. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.5: Propositional Logic Derivations
5.2 Premises and the Goal Proposition
In a proof, we can for convenience write the goal proposition, which is also the conclusion, next to the ‘P’ for the final premise, in the justification column.
In our proof, we have propositions that serve as the premises. They are noted as such in the justification column with the letter P. Suppose we wanted to make a proof for the following argument:
A→B, B→C ⊢ C→D
[To make the argument more obvious, I will change the conclusion to A→C in the diagrams. So instead, the argument would be:
A→B, B→C ⊢ A→C
] We would set up the premises like this:
Our proof would like to arrive upon the conclusion, which in this case is A→C. We call this the goal proposition, and for convenience we can write it beside the ‘P’ of the last premise.
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.