Wednesday, February 26, 2014

A Proof of Self-Proof; 3 of 15

        Quining Quanta

          Sentences can be defined in terms of each other; can a sentence be defined in terms of itself? Yes! Self-reference is possible, even in a rigorously hierarchical logic system, due to a technical trick called ‘quining’. To ‘quine’ a predicate means to apply it to its own quotation, For instance:
          ‘Is a predicate’ is a predicate.
          ‘Is not a predicate’ is not a predicate.
          ‘Is a statement when quined’ is a statement when quined.

          The first and the third are true, the second is false. The third is self-referential; when the quoted phrase is quined, the result is the original sentence. In general, the statement
                    “Has property F when quined’ has property F when quined.
          is self-referential; it says that it has property F:
                    “This statement has property F.”
                      S      =        F(S)
          Statement S generates itself out of itself. It is a self-propagating process; an organic structure. I call it a ‘logical quantum’.
          Quanta bootstrap themselves into definition. Like the Earth that we stand on, they rest upon themselves. Quanta need no prior ‘foundation’, any more than our round planet, afloat in the void, needs to lie on the back of a space turtle.

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