On the Danang Operator Within Standard Formal Systems

Filed under: , by: Kevin

During the foundation of the Waltham Circle, it was agreed upon by various of the core members that it would be one of the projects of the Circle to formalize a robust concept that had formed within the Circle: that of 'danangness,' signifying the ultimate degree of interestingness. The project has since withered, and many of my colleagues have reverted to using the predicate in its unformalized, primitive form. Perhaps this is best. Nevertheless, I present the rough, first draft of an unpublished manuscript that provides the first operational definitions of the danang predicate. It would have been the task of the Circle to further refine the concepts, distinguishing them from other concepts of formal systems (satisfaction, Tarskian conceptions of truth, etc.). Likely, the product is simply non-sense (not nonsense, but non-sense).


D1: A sentence S is 'danang' if and only if it demonstrates an astonishing degree of awesomeness.
H1: For any proposition p of which we can predicate danangness, the formal result will be a conversion of truth values (assuming, for the moment, a truth functional sentential logic) into an absolute, language-transcendent form of 'Truth' and 'Wrong.'
D2: The danang operator (δ) can be reconstrued as an imposition of a factorial form on truth values in a bivalent canonical system.

The 'danang' symbol (δ) serves as a logical operator like the negation: unlike the other logical operators, it is not a connective between propositions, but functions to convert single propositions (or strings of propositions encased in brackets following the δ-operator). The δ-operator translates the truth value of any given proposition into its factorial.
D3: An arithmetical factorial ('!') can be recursively defined for n > 0 as n(n-1)!.

Because classic sentential logic is bivalent, we might be inclined to apply the factorial to 0 for false, and 1 for true. This is incoherent, for in such a case F! = T! (0! = 1! = 1). The δ-operator must preserve the initial form of a statements truth conditions, or else we will end up affirming such impossible propositions as p & ~p.

The point is to convert the arithmetical definition of the factorial into terms that can be operated on in a bivalent sentential logic. In other words, we should recognize a homologous role of factorials in, first, truth-functional formal systems and, second, quantified predicate systems. In a truth-functional system, set theoretically defined, we can define the factorial of 'True,' or 'T!' as:
D4: T! is the product of all the elements of a set A which, when operated on by a function S(x), combine to specify the class of sentences which are, for that set, true.

Similarly, F! can be defined as the product of all the elements of a set A which together, under S(x), specify the class of false sentences. For our purposes, S(x) is any truth-apt proposition within a given language L1. Its range is restricted to the set A, and likewise any S'(x) within language L2 cannot extend over the class of sentences designated by S(x).

When we consider the elements of a set A that, on S(x), altogether specify the class of true sentences in A, however, we should note that these elements are, in terms of informal language, the foundation for the possibility of S(x)'s truth-aptness. They therefore precede the sentence S(x), as well as any sentence Sn(x) in language Ln. Any truth functional factorial that functions through S(x) therefore eliminates the language restrictions of that sentence and allow us to generate a subset of a given class, the members of which serve to make any sentence within that set true or false. The factorial therefore eliminates the linguistic constraints on our truth values.

It is our position that while the fundamental unit of meaning is the sentence, the sentence's meaning is conditioned by its position within a holistic system of beliefs, attitudes, and experiential conditionings. The truth value of any sentence is therefore only intelligible within the language of that sentence. This is a point made by Quine. Blackburn similarly points out that if an entire language (or theory, in his nomenclature) is rejected, there is no way to make “uncontaminated” attributions of truth or falsity to a sentence formulated within that system. Because every semantic system that is not complete by virtue of its syntax can be intelligibly rejected, every truth functional proposition can, in principle, be subject to what we will here call 'contamination,' whereby operations on the system will contaminate our ability to make sharp attributions of truth or falsity to any statement made within that system.

Applying a sentential factorial will serve to specify the members of a set that make a sentence true or false, outside of the contaminable structure of an incomplete formal or informal system.

The significance of the danang operator is to syntactically specify those elements of a set that, combined, make a sentence true or false. The truth table of any proposition to which the danang operator is applied will look like:
p δ(p)
T T!
F F!

We can colloquially signify the canon by stating that the δ-operator translates any attribution of “True” or “False” to a proposition p in language Ln to a specification of “Truth” or “Bullshit,” respectively.

Even though within a truth functional sentential logic, the δ-operator works only on truth values, in a quantified predicate system it can be treated as a predicate (in the same way that we might canonize “x is true” as “Tx”). In the end, both canonical treatments can be given the same analysis.
D5: Any successful application of the δ-operator to a truth-apt proposition that yields a value of T! can also be said to yield a 'Truth-nugget.' In other words, the value of any true statement to which we can appropriately apply the δ-operator is coextensive with a 'Truth-nugget.'

Conjecture: For any true, danang propositions in a finite set, there will be a coextensive truth nugget.

Theorem: There exists at least one thing that is danang.

Or, we might put it,

This is demonstrated by Mr. Brantner's ingenious proof that, assuming the negation of this theorem, the possible world containing no elements of which 'danang' can be predicated would be so astonishingly impoverished that it would, itself, entail danangness.