Vicious circle principle
Encyclopedia
The vicious circle principle is a principle that was endorsed by many predicativist mathematicians in the early 20th century to prevent contradictions. The principle states that no object or property may be introduced by a definition that depends on that object or property itself. In addition to ruling out definitions that are explicitly circular (like "an object has property P iff
it is not next to anything that has property P"), this principle rules out definitions that quantify over domains which include the entity being defined. Thus, it blocks Russell's paradox
, which defines a set S that contains all sets that don't contain themselves. This definition is blocked because it defines a new set in terms of the totality of all sets, of which this new set would itself be a member.
However, it also blocks one standard definition of the natural numbers. First, we define a property as being "hereditary" if, whenever a number n has the property, so does n + 1. Then we say that x has the property of being a natural number iff
it has every hereditary property that 0 has. This definition is blocked, because it defines "natural number" in terms of the totality of all hereditary properties, but "natural number" itself would be such a hereditary property, so the definition is circular in this sense.
Most modern mathematicians and philosophers of mathematics think that this particular definition is not circular in any problematic sense, and thus they reject the vicious circle principle. But it was endorsed by many early 20th century researchers including Bertrand Russell
and Henri Poincaré
. On the other hand Frank P. Ramsey
and Rudolf Carnap
accepted the ban on explicit circularity, but argued against the ban on circular quantification. After all, the definition "let T be the tallest man in the room" defines T by means of quantification over a domain (men in the room) of which T is a member. But this is not problematic, they suggest, because the definition doesn't actually create the person, but merely shows how to pick him out of the totality. Similarly, they suggest, definitions don't actually create sets or properties or objects, but rather just give one way of picking out the already existing entity from the collection of which it is a part. Thus, this sort of circularity in terms of quantification can't cause any problems.
This principle was the reason for Russell's development of the ramified theory of types rather than the simple theory of types.
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
it is not next to anything that has property P"), this principle rules out definitions that quantify over domains which include the entity being defined. Thus, it blocks Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
, which defines a set S that contains all sets that don't contain themselves. This definition is blocked because it defines a new set in terms of the totality of all sets, of which this new set would itself be a member.
However, it also blocks one standard definition of the natural numbers. First, we define a property as being "hereditary" if, whenever a number n has the property, so does n + 1. Then we say that x has the property of being a natural number iff
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
it has every hereditary property that 0 has. This definition is blocked, because it defines "natural number" in terms of the totality of all hereditary properties, but "natural number" itself would be such a hereditary property, so the definition is circular in this sense.
Most modern mathematicians and philosophers of mathematics think that this particular definition is not circular in any problematic sense, and thus they reject the vicious circle principle. But it was endorsed by many early 20th century researchers including Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
and Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
. On the other hand Frank P. Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
and Rudolf Carnap
Rudolf Carnap
Rudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
accepted the ban on explicit circularity, but argued against the ban on circular quantification. After all, the definition "let T be the tallest man in the room" defines T by means of quantification over a domain (men in the room) of which T is a member. But this is not problematic, they suggest, because the definition doesn't actually create the person, but merely shows how to pick him out of the totality. Similarly, they suggest, definitions don't actually create sets or properties or objects, but rather just give one way of picking out the already existing entity from the collection of which it is a part. Thus, this sort of circularity in terms of quantification can't cause any problems.
This principle was the reason for Russell's development of the ramified theory of types rather than the simple theory of types.
An analysis of the paradoxes to be avoided shows that they all result from a kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that “all propositions are either true or false.” It would seem, however, that such a statement could not be legitimate unless “all propositions” referred to some already definite collection, which it cannot do if new propositions are created by statements about “all propositions.” We shall, therefore, have to say that statements about “all propositions” are meaningless.… The principle which enables us to avoid illegitimate totalities may be stated as follows: “Whatever involves all of a collection must not be one of the collection”; or, conversely: “If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.” We shall call this the “vicious-circle principle,” because it enables us to avoid the vicious circles involved in the assumption of illegitimate totalities. (Whitehead and Russell 1910, 37) (quoted in the Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
entry on Russell's Paradox)