A Proof of Self-Proof; 5 of 15

        Dualities

          What is the negation of self-doubt? It would seem to be self-trust, for self-doubt asserts that ‘this’ sentence is unprovable, while self-trust asserts that ‘this’ sentence is provable; but it’s a different ‘this’ for each one. Self-reference implies a twist in logic. Paradox pervades this study.

          In fact the negation of self-doubt is not self-trust but self-shame; and the negation of self-trust is not self-doubt but self-pride.

          Proof:

          (not D)   =   not not prv D   =   prv D  =  prv not (not D)

          Therefore (not D) fits the self-shame equation   S = prv not S.

          (not T)   =   not  prv T   =   not prv not (not T)

          Therefore (not T) fits the self-pride equation   P = not prv not P.

         

A Proof of Self-Proof; 4 of 15

        Four Logical Quanta

          Four quanta are of particular interest to logicians. Here I nickname them the quanta of Self-Doubt, Self-Shame, Self-Pride and Self-Trust.

          The quantum of Self-Doubt says

          “ ‘Is unprovable when quined’, is unprovable when quined.”

          Or in other words: “This sentence is not provable.”

          Or in other words: “Doubt me.”

          It is the mathematical quantum of uncertainty. Its equation is:

          D       =        not prv D

          The quantum of Self-Shame says

          “ ‘Is refutable when quined’, is refutable when quined.”

          Or in other words: “This sentence is provably false.”

          Or in other words: “Refute me.”

          It is the mathematical quantum of error. Its equation is:

          S        =        prv not S

          The quantum of Self-Pride says

          “ ‘Is irrefutable when quined’, is irrefutable when quined.”

          Or in other words: “This sentence is possible.”

          Or in other words: “Tolerate me.”

          It is the mathematical quantum of power. Its equation is:

          P        =        not prv not P

          The quantum of Self-Trust says

          “ ‘Is provable when quined’, is provable when quined.”

          Or in other words: “This sentence is provable.”

          Or in other words: “Trust me.”

          It is the mathematical quantum of certainty. Its equation is:

          T        =        prv T

          The quantum of self-doubt is also known as a Gödelian sentence. It is literally a paradox, for it calls itself beyond belief. The quantum of self-trust is also known as a Henkin sentence (after the man who asked if it is true) or a Löbian sentence (after the man who proved that it is). The quanta of self-shame and self-pride are unclaimed, for good reason.

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.

A Proof of Self-Proof; 2 of 15

          Notation

          Every proof uses its own forms of notation, which has its own conveniences and hidden assumptions. I set forth this paper’s notation here in order to simplify critique. To quote the famous logician Humpty Dumpty, “when I use a word, it means just what I choose it to mean – neither more nor less.”

          This paper discusses mathematical propositions, called ‘sentences’. They are defined in terms of each other, using logical connectives such as ‘and’, ‘or’ and ‘no’.  Call logically equivalent sentences ‘equal’; denote equality by ‘=’. Let T denote anything obviously true, and F denote anything obviously false. Therefore T = not F; the former can mean ‘x=x’ or ‘1+1=2’ or ‘0 does not equal 1’; and F denotes the denial of those statements.

          Some sentences are provable. The assertion that the sentence ‘S’ is provable is another sentence; call it “prv S”. To assert prv S is to assert that S is necessarily true, therefore true in all mathematical models.

          Some sentences are true on occasion. Such sentences are ‘possible’. The assertion that the sentence ‘S’ is possible is another sentence; call it “poss S”. To assert poss S is to assert that S is true in some mathematical model; and therefore it cannot be proven false. Therefore poss and prv are conjugate to each other:

          poss S          =        not prv not S

          prv S            =        not poss not S

          The possible is what you can’t prove false; and the provable is what you can’t possibly deny. We can form combinations of these:

          Poss T          =        ‘truth is possible’

          Prv F           =        ‘falsehood is provable’

          Poss F          =        ‘falsehood is possible’

          Prv T           =        ‘truth is provable’

          The last two are easy to simplify. T is provable in all mathematical models, in fact it is the standard of proof; therefore prv T = T. Similarly poss F is false in all mathematical models, therefore poss F = F.

          The statements poss T and prv F involve a deep conundrum; namely, is our proof system itself valid? Perhaps our reasoning methods contain fatal flaws. Is our logic in fact consistent? Does it have a mathematical model? If so, then truth is possible; if not, then falsehood is provable.

A Proof of Self-Proof; 1 of 15

          A Proof of Self-Proof

          On The Metamathematics of Belief

          Introduction

          Notation

          Quining Quanta

          Four Logical Quanta

          Dualities

          Evaluating Four Quanta

          Some Tables

          The Miracle of Doubt

          The Failure of Shame

          The Fall of Pride

          The Vanity of Faith

          How heavy is your theory?

          Paradoxes of Orthodoxy

          Do I exist?

          Cosmos from Chaos

          Summary

          Appendix 1: A Dialectical Game

          Appendix 2: A Proof of Self-Proof

          Introduction

          This paper is about the mathematical logic of belief systems. I  discuss four forms of self-reference, and their paradoxical properties. These imply Gödel’s Incompleteness Theorems, which in turn imply Löb’s Theorem.

          Löb’s Theorem says that any statement that asserts just its own provability is, in fact, provable. It is a logical bootstrap; by declaring itself necessary, it makes itself necessary. A Löbian statement, by its perfect faith in itself, attains truth.

          But why? How can anything so vain as self-belief attain absolute certainty? Read on, and I shall prove it to you.