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

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

COR. (TH. 5): S(O(i + k, j + 1)) > S(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. TIPOFF's) to INVOKE 3 OBSERVED ANOMALIES, whereas the "simpler" argument involved therein need make only 3 INDEPENDENT ASSUMPTIONS (a.k.a. TIPOFF's) TO INVOKE these same 3 OBSERVED ANOMALIES. 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:

S(O(1, 3)) - S(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 ANOMALY. Suppose, further, that this Anomaly is observed. Then the DOMINANCE seen above "changes sides":

		S(O(20, 4)) - S(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 SECURABILITY change within the complex argument as it advances from 3 to 4 OBSERVED ANOMALIES:

S(O(20, 4)) - S(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 SECURABILITY 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 S(O(i, j + 1)), the securaBILITY MEASURE resulting when THE CONJUNCTIVE TIPOFF of this universe YIELDS ONE MORE OBSERVED ANOMALY. Let q denote the numerator of the SECURABILITY change:

S(O(i, j + 1))  - S(O(i, j))

. Then Q(O(i, j)) denotes the ANOMALY POTENTIAL of this OCKAM FUNCTION (or its universe) iff (if, and only, if) Q(O(i, j)) = q + 1.

The "anomaly potential" has been symbolized by "q" amd "Q" (letters following "p", which might lead to confusion with the probabiity measure. And this result leads us to another Theorem.

THEOREM 6 (INTELLIGENCE THEOREM): Q(O(i, j)) = 2ⁱ.

PROOF: Q(O(i, j + 1)) - Q(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 q = (2ⁱ - 1. And D(O(i,j)) = 2ⁱ - 1 + 1 = 2ⁱ , as stated. (The "+1" in the definition of the anomaly potential is to remove the unsightly "- 1" in the Theorem. But what does this mean?

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

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

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

PROOF: From INTELLIGENCE THEOREM, Q(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 UNOBSERVED ANOMALY MUST BECOME A NEGATED PREMISE, DECREASING THE SECURABILITY MEASURE, HENCE, NONMONOTONICITY IN LOGIC. (PROOF, obvious.)