subsumes(R\2,R'\2) := all x,y, R'(x,y) -> R(x,y)
transitive(R\2) := all x,y,z, R(x,y) -> R(y,z) -> R(x,z)

In second-order logic:
 tc(R\2,R'\2) :=
  subsumes(R',R) ^
  transitive(R') ^
  all R", (subsumes(R",R) ^ transitive(R")) -> subsumes(R",R')

In monadic second-order logic:
 closed(X\1,R\2) := all u,v, (X(u) ^ R(u,v) -> X(v))
 tc'(R\2,x,y) := all X, (closed(X,R) ^ X(x) -> X(y))
 tc(R\2,R'\2) := R'(x,y) <-> TC'(R\2,x,y)

In first-order logic:
 ???