In BONUS1 I derived an "approximating" Corollary from each of Theorems 3 and 4.

COROLLARY (TH. 4): A(O(i, j)) > A(O(i + k, j)) > (2^k - 1)/2^i+j+k > 1/2^i+j, for all i,j,k >= 1.

COR. (TH. 5): A(O(i + k, j + 1)) > A(O(i, j)) > (2^k(2ⁱ - 1)/(2^i+j+k+1) > 1/2^j+1, for all i,j,k >= 1.

What are the consequences of these Corollaries?

EXAMPLE: Suppose that the compound argument involved in Theorem 5, and its Corollary, must accept 20 INDEPENDENT ASSUMPTIONS (a.k.a. HYPOTHESES) to INVOKE 3 CONFIRMED PREDICTIONS, whereas the "simpler" argument involved therein need make only 3 INDEPENDENT ASSUMPTIONS (a.k.a. HYPOTHESES) TO INVOKE these same 3 CONFIRMED EXCEPTIONS. If we substitute i = 3, j = 3, k = 19 in the COROLLARY of THEOREM 4, we find that its APPROXIMATIVE result is 1/2^i+j = 1/2¹⁺³ = 1/2⁴ = 1/16. Then we have:

A(O(1, 3)) - A(O(20, 3)) > 1/16,

which is significant.

Suppose, however, further analysis of the compounD case suggests a critical experiment derived from the hypotheses of the complex argument with its associated prediction Suppose, further, that this prediction is observed. Then the DOMINANCE seen above "changes sides":

		A(O(20, 4)) - A(O(1, 3)) > 1/16,

again significant.

But the RELATIVE change is even more significant: 1/16 + 1/16 = 1/8.

And we may learn even more if we MEASURE the ASSERBILITY change within the complex argument as it advances from 3 to 4 CONFIRMED PREDICTIONS:

A(O(20, 4)) - A(O(20, 3)) = (1 — 1/24 + 1/224) - (1 — 1/23 + 1/223 =

(1/23 — 1/24) + (1/224 — 1/223) = (2 — 1)/24 + (2 — 1)/1/224 =

1/24 + 1/224 = (220 + 1)/224 > 1/24 = 1/16.

But the BASE should be in the PREVIOUS UNIVERSE, that, a BASE of 23, so we have 2²⁰/2²³ = 1/2³ = 1/8, as before.

This ASSERBILITY change is so SIGNIFICANT that it deserves its own specification as a MEASURE-CONCEPT.

Def. 1. Given an OCKAM UNIVERSE, with OCKAM FUNCTION, O(i, j), with A(O(i, j + 1)), the ASSERBILITY MEASURE resulting when THE CONJUNCTIVE HYPOTHESIS of this universe YIELDS ONE MORE CONFIRMED PREDICTION. Let p denote the numerator of the ASSERBILITY change:

A(O(i, j + 1))  - A(O(i, j)).

Then P(O(i, j)) denotes the PREDICTION POTENTIAL of this OCKAM FUNCTION (or its universe) iff (if, and only, if) R(O(i, j)) = r + 1.

The "predictive potential" has been symbolized by "r" amd "R" (second letter of "predictive, rather than "p",which might lead to confusion with the probabiity measure. And this result leads us to another Theorem.

THEOREM 6 (WILDCAT THEOREM): A(O(i, j)) = 2ⁱ.

PROOF: R(O(i, j + 1)) - R(O(i, j)) = (1 - 1/2^{j+ 1} + 1/2^i+j+1) - (1 - 1/2^j + 1/2^i+j) = (1/2^j - 1/2^j+1) + (1/2^i+j+1 > 1/2^i+j) =
(2 - 1)/2^j+1 + (1 - 2)/2^i+j+1 = 1/2^j+1 - 1/2^i+j+1 = (2ⁱ - 1)/2^i+j+1.

Then we have r = (2ⁱ - 1. And D(O(i,j)) = 2ⁱ - 1 + 1 = 2ⁱ , as stated. (The "+1" in the definition of the prediction potential is to remove the unsightly "- 1" in the Theorem. But what does this mean?

ANALYSIS SHOWS THAT A CONJUNCTIVE HYPOTHESIS WHICH CAN YIELD ONE MORE CONFIRMED PREDICTION THAN PREVIOUSLY DOES THEREBY INCREASES THE ASSERBILITY MEASURE. But WHAT PRODUCES THE PREDICTIVE POTENTIAL? You might expect the DIFFERENCE BETWEEN "BEFORE" AND "AFTER" TO CARRY A TERM RELATED TO THE PREDICTION PARAMETER, j. But it DOES NOT. It CARRIES A TERM RELATED TO THE HYPOTHESIS PARAMETER, i. THE PREDICTION POTENTIAL CAME FROM WHAT THE HYPOTHESIS CLAIMED -- which is THE RISK TAKEN BY THE INVESTER.

Then, this result is A CLARION CALL TO WILDCATTING, professional or amateur! If one is "willing to take the risk" of "loading" the HYPOTHESES -- a procedure which scientists and philosophers peoratively label as "ad hoc" -- then the POTENTIAL "waiting to be released" -- in the event "the risky hypothesis pays off" -- is AN EXPONENTIAL OF THE TOTAL NUMBER OF "RISKS TAKEN"!

THEOREM 7 (NEGENTROPY THEOREM): GIVEN "WILDCAT" THEOREM, THE INCREASED ASSERBILITY HAS THE FORM OF THE NEGENTROPY MEASURE IN THE SZILARD-SHANNON THEORY OF INFORMATION.

PROOF: From WILDCAT THEOREM, R(O(i, j)) = 2ⁱ. In the SZILARD-SHANNON THEORY OF INFORMATION, the INFORMATION MEASURE is log₂c, where c is THE NUMBER OF CHOICES in a given situation. Here we have log₂2ⁱ = i BITS.

THEOREM 8 (NONMONOTONICITY THEOREM): A FAILED PREDICTION MUST BECOME A NEGATED PREMISE, DECREASING THE ASSERBILITY MEASURE, HENCE, NONMONOTONICITY IN LOGIC. (PROOF, obvious.)