Logicism - The Unit Class, Impredicativity and The Vicious Circle Principle

The Unit Class, Impredicativity and The Vicious Circle Principle

A benign impredicative definition: Suppose the local librarian wants to catalog (index) her collection into a single book (call it Ι for "index"). Her index must list ALL the books and their locations in the library. As it turns out, there are only three books, and these have titles Ά, β, and Γ. To form her index-book I, she goes out and buys a book of 200 blank pages and labels it "I". Now she has four books: I, Ά, β, and Γ. Her task is not difficult. When completed, the contents of her index I is 4 pages, each with a unique title and unique location (each entry abbreviated as Title.Location_T):

I ← { I.L_I, Ά.L_Ά, β.L_β, Γ.L_Γ}.

This sort of definition of I was deemed by Poincaré to be "impredicative". He opined that only predicative definitions can be allowed in mathematics:

"a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it".

By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In the teeth of these, Russell would enunciate a strict prohibition—his "vicious circle principle":

"No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Gödel 1944 appearing in Collected Works Vol. II 1990:125).

A pernicious impredicativity: α = NOT-α: To create a pernicious paradox, apply input α to the simple function box F(x) with output ω = 1 - α. This is the algebraic-logic equivalent of the symbolic-logical ω = NOT-α for truth values 1 and 0 rather than "true" and "false". In either case, when input α = 0, output ω = 1; when input α = 1, output ω = 0.

To make the function "impredicative", wrap around output ω to input α, i.e. identify (equate) the input with (to) the output (at either the output or input, it does not matter):

α = 1-α

Algebraically the equation is satisfied only when α = 0.5. But logically, when only "truth values" 0 and 1 are permitted, then the equality cannot be satisfied. To see what is happening, employ an illustrative crutch: assume (i) the starting value of α = α₀ and (ii) observe the input-output propagation in discrete time-instants that proceed left to right in sequence across the page:

α₀ → F(x) → 1-α₀ → F(x) → (1 -(1-α_o)) → F(x) → (1-(1-(1-α_o))) → F(x) → ad nauseam

Start with α₀ = 0:

α₀ = 0 → F(x) → 1 → F(x) → 0 → F(x) → 1 → F(x) → ad nauseam

Observe that output ω oscillates between 0 and 1. If the "discrete time-instant" crutch (ii) is dropped, the function-box's output (and input) is both 1 and 0 simultaneously.

Fatal impredicativity in the definition of the unit class: The problem that bedeviled the logicists (and set theorists too, but with a different resolution) derives from the α = NOT-α paradox Russell discovered in Frege's 1879 Begriffsschrift that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but from the function's own output as well.

As described above, Both Frege's and Russell's construction of natural numbers begins with the formation of equinumerous classes-of-classes (bundles), then with an assignment of a unique "numeral" to each bundle, and then placing the bundles into an order via a relation S that is asymmetric: x S y ≠ y S x. But Frege, unlike Russell, allowed the class of unit classes (in the example above, ]) to be identified as a unit itself:

, ] ≡ ■ ≡ 1

But, since the class ■ or 1 is a single object (unit) in its own right, it too must be included in the class-of-unit-classes as an additional class . And this inclusion results in an "infinite regress" (as Godel called it) of increasing "type" and increasing content:

, ] ≡ ■

, ]]] ≡ ■

, ]]]]]]] ≡ ■, ad nauseam

Russell would make this problem go away by declaring a class to be a "fiction" (more or less). By this he meant that the class would designate only the elements that satisfied the propositional function (e.g. d and s) and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". It is an assemblage e.g. d,s but it is not (in Russell's view) worthy of thing-hood:

"The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus x ε u will mean " x is one of the u 's." This must not be taken as a relation of two terms, x and u, because u as the numerical conjunction is not a single term . . . Thus a class of classes will be many many's; its constituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents." (1903:516).

This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchy of "types" of classes.