Weakening Glossary
Definition
Weakening is the logical principle that whatever follows from just some of the assumptions of a sequent thereby follows from all of them. In other words, in the course of a proof, extra assumptions may be added to the left of any line, even if those extra assumptions play no part in the derivation. This principle underlies vacuous discharge, which allows such phantom premises to be blamed for a contradictions in the case of RAA, or used as antecedents of conditionals in the case of IMPI.
Comments
Weakening is a feature of classical logic and similar logics (including fuzzy and constructive logics). In the case of relevant logic, weakening is restricted to cases in which it does no harm, by being disallowed for the kind of premise combination which permits discharge.
Given that the premises of a sequent are collected into a set rather than a more complex bunch, weakening is simply the monotonicity property of the consequence relation.
Example
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SP {p TS (q IMP p)}
PL {1} {1} {p} {A}
PL {2} {2} {q} {A}
PL {1,2} {3} {p} {1 weakening}
PL {1} {4} {q IMP p} {3 [2] IMPI}
EP
The step introducing the conditional at line 4 discharges assumption 2 from line 3, although assumption 2 was in no sense used to obtain line 3. It occurs among the assumptions of line 3 simply as a result of being inserted there by weakening the sequent on line 1.