Derivable and Admissible rules

• derivable from a set R of other rules if there exists a proof which uses only rules in R, has C as conclusion and has P₁ .... P_n as assumptions. That is there exists

  
            P1  ....   Pn
            \           /
             \         /
              \       /
               \     /
                \   /
                  C
  

  
  
• admissible in a set R of other rules if, given n proofs of P₁ .... P_n which use only rules of R, it is possible to find a proof of C which uses only rules of R. That is, given

        \ Gamma1 /             \ Gamman /
         \      /               \      /
          \    /        ....     \    / 
           \  /                   \  /
            P1                     Pn
  

  where the Gammai's are sets of assumptions, there exists

            Gamma1,...,Gamman
            \               /
             \             /
              \           /
               \         /
                \       /
                 \     /
                  \   /
                    C