TITLE: What does it mean to 'discharge assumptions or premises'?
QUESTION [21 upvotes]: When constructing proofs using natural deduction what does it mean to say that an assumption or premise is discharged? In what circumstances would I want to, or need to, use such a mechanism?
The reason I'm asking this question is that many texts on logic use this term as understood by the reader and don't take the time to adequately explain the technical sense in which they are using it.

REPLY [5 votes]: In the spirit of Kenny's observation, note also that we can formulate classical logic using a Peircian inference rule (equivalent to the usual theory in the presence of ex falso quodlibet) which clearly modifies the inferential properties of implication:

${{{A \rightarrow B} \atop \vdots} \atop A } \over A$

but in an odd way: is it an introduction rule?  But there is no logical structure in the conclusion.  Is it an elimination rule?  But the implication above the rule appears among the assumptions to the subderivation, not the conclusion.  It seems to be something like an implication elimination-eliminating rule, a kind of structurally double-negative introduction rule, where you can introduce logical structure by discharging it in the assumptions.
All the rules for adding classical-strength inference to the usual, well-behaved intuitionistic natural deduction involve inference rules that are eccentric in some way or another.  Parigot's lambda-mu calculus shows how the fundamental structural glue of (through a classical Curry-Howard correspondence) natural deduction can be tweaked to make it as good a fit for classical logic as the sequent calculus.
In chapter 3 of my DPhil dissertation (Stewart 2000), I give what I think is a successful reconstruction of Prawitz's inversion principle for the lambda-mu-based natural deduction, and show how this provides the basis for something we might call "classical constructivism", where the principle of the excluded middle is admitted as having constructive content by being a principle that provides a constructive, dialectical mediation between proofs and refutations.
Ref. Stewart (2000) On the formulae-as-types correspondence for classical logic.