TITLE: Weakening and Contraction
QUESTION [6 upvotes]: I saw this site saying weakening is a structural rule where the hypotheses or conclusion of a sequent may be extended with additional members and that contraction is a rule where two equal (or unifiable) members on the same side of a sequent may be replaced by a single member (or common instance). However, I can't seem to get the terms yet. There are even these symbols that looks like a T turned counterclockwise.
In simple terms, what are weakening and contraction? Can they even be described without the symbol that looks like a T turned counterclockwise?

REPLY [2 votes]: Given statements p and q, where p proves q, weakening means we can add any number of statements r to the left using conjunction (and, &), yet "p & r proves q" is still valid. This is the case because we know that we can prove q if p is true, and we can prove p if "p & r" is true (using conjunction elimination, or subtraction). Therefore, we can prove q if "p & r" is true, no matter what r is. 
Contraction means we can add any number of statements r to the left using disjunction (or, v), yet "p proves q v r" is still valid. In this case, we know we can prove q if p is true, and "q v r" can be proven if q is true (using disjunction introduction or addition). (Don't forget that "p v true" is a tautology, meaning it's always true). Therefore, we can prove "q v r" if p is true, no matter what r is.