Vicious circle
Vicious circle
A vicious circle in proof (circulus in probando) occurs if p1 is used to prove p2, p2 to prove p3, . . . , pn-1 to prove pn, and finally pn to prove p1 — p1,p2, . . . , pn being then taken as all proved. This is a form of the fallacy of petitio principii (q.v.).
A vicious circle in definition (circulus in definiendo) occurs if A1 is used in defining A2, A2 in defining A3, . . . , An-1 in defining An, and finally An in defining A1. (The simplest case is that in which n = l, A1 being defined in terms of itself.) There is, of course, a fallacy if A1, A2, . . . , An are then used as defined absolutely. Apparent exceptions, such as definition by recursion (q.v.), require special justification, e.g., by finding an equivalent form of definition which is not circular.
The term vicious circle fallacy is used by Whitehead and Russell (1910) for arguments violating their ramified theory of types (q.v.). Similarly, the name circulus vitiosus is applied by Hermann Weyl (1918) to an argument involving impredicative definition (q.v). — A.C.