Note that parentheses and commas are not counted. The latter is the same as saying that the universal closures of phi and of psi are logically equivalent. We introduce two notions of efficiency for finite sets of formulas. Investigate efficiency, alpha efficiency, and beta efficiency in predicate calculus. Take standard theories in predicate calculus and give alpha and beta efficient axiomatizations.
R holds everywhere.
R holds somewhere. R is reflexive. R is irreflexive. R is symmetric. R is antisymmetric.