DEFINITION: The BASE of an INDICATOR TABLE equals the NUMBER of SIGNS it contains.
(As some of you may know, an INDICATOR TABLE BASE always has the FORM 2a, where a is THE NUMBER OF ANOMALIES involved -- since each is BIVALENT -- "OBSERVED" or "NONOBSERVED" -- "1" or "0". This follows as a Theorem from the BASE DEFINITION and the TABLE DEFINITION.)
DEFINITION: The TALLY of a TABLE equals the NUMBER of ONES it contains.
DEFINITION: The SECURABILITY of a LOGICAL FORM given by S(OBSERVED) is the TALLY (in its FINAL SUBTABLE) in RATIO to the BASE: S(OBSERVED) = TALLY/BASE.
We need a FORMULA for the GENERAL CASE, namely,
(((&iTi) -> (&jAj))&(&jAj)) -> (&iTi), i=1,2,...,n; j=1,2,...,n
That is, a number (i) of TIPOFF's IMPLY a number (j) of ANOMALIES, which are OBSERVED; so we CLAIM "ACCURACY" OF THE TIPOFF SUM.
We need a FORMULA on the PARAMETERS i, j in this GENERAL FORM. For reason given below, we'll call this GENERAL FORM, "The Ockam Function", labeling it as O(i, j), function of i, j PARAMETERS. This is convenient since, as shown in the PROOF, it can be represented by two logical forms, namely, FAC ("fallacy of asserting the consequent") and FC ("fallacy of the contrary").
(Mote: This is a LOGICAL procedure adapted to SECURITY purpose. Certainly, William of Ockam (x-y) contributed to logical developments. It may interest the reader that monk (played by Sean Connery) who solved the murder in the 1986 film, "The Name of the Rose", was modeled on William of Ockam.)
THEOREM 1 (FORMULA-THEOREM: S(O(i, j)) = 1 - 1/2 j + 1/2i+j.
(Proof is hyperlinked from FRONTPAGE.)
Many BONUSES (THEOREMS) follow from this FORMULA -- some MATHEMATICALLY DERIVING INTUITIONS of long standing literature -- SOME VERY SURPRISING. The first one FORMALIZES what we found in another file as "success increases".
THEOREM 2 (PROMISEDLAND THEOREM): GIVEN S
(O(1, j)) = 1 - 1/2j + 1/21+j, THEN lim(j -> 
)
S(O(1, j)) = 1.
That is, "IN THE LIMIT", S(O(1, j)) IS "AS GOOD AS A TAUTOLOGY", such as MODUS PONENS, the primary "prover" in LOGIC.
PROOF: As j -> , the terms - 1/2j
+ 1/21+j GO TO ZERO, leaving only 1 remaining.
Before the next BONUS, a little history. William of Ockam, born in Ockham, England, was a Franciscan Friar of many accomplishments.
- Long before Galileo or Newton, William expressed the essentials of the physical concept of "inertia".
- Considered many-valued logic, now a special field of LOGIC.
- Expressed ideas interpreted as "Ockam's Razor: Don't multiply entities" -- that is, IF A SIMPLE EXPLANATION OR HYPOTHESIS (or TIPOFF) DOES AS WELL AS A MORE COMPLICATED ONE, DECIDE UPON THE SIMPLE ONE.
Galileo used "Ockam's Razor" to defend "The Heliocentric Theory" against the Ptolomaic "Geocentric Theory", because it was SIMPLER.
Because the Franciscans advocated poverty for the Church and William advocated sharing power with The Vatican Council (a step adopted by Pope John VII in 1950), the Church denounced William as a "heretic", saying it would "be no sin to kill him". William of Ockam disappeared during THE BLACK PLAGUE, circa 1249-50.
THEOREM 3 (OCKAM-RAZOR-THEOREM): S(O(i, j)) > S(O(i + 1, j).
That is, REQUIRING ONE MORE HYPOTHESIS (TIPOFF), Ti+1,
TO YIELD THE SAME j PREDICTIONS (ANOMALIES), &j
Aj, DECREASES THE SECURABILITY. That is, THE SIMPLER FORM HAS GREATER
SECURBILITY MEASURE THAN THE MORE COMPLICATED ONE, FOR THE SAME PREDICTIONS (ANOMALIES).
"Ockam's Razor" has hitherto been an ANSANTZ: ASSUMED for PURPOSE. But I derive it
as a Theorem, using the Formula-Theorem.
The following LEMMA (to the next THEOREM) shows that the SECURABILITY
Measure is OPTIMALLY SENSITIVE TO PREDICTABILITY (ANOMALIES).
LEMMA: Given a "simple" argument and a "complex" argument, represented, respectively, by Ockam-Functions
O(i, j) and O(i + k, j), both argument invoking
the same j confirmed predictions. Then, S(O(i, j)) >
S(O(i + k, j)).
PROOF: This is, of course, simply another case of the OCKAM-RAZOR-THEOREM, already proven. But the
"margin of victory" will be relevant to the Theorem which follows:
For large values of k, the compound argument can be "very
far behind". But the following Theorem proves that -- no matter "how far behind" -- it can "always
catch by making only one more confirmed preduction (implying one more observed exception)" than
the simpler argument.
THEOREM 4 (TORTOISE&HARE-THEOREM): S(O(i + k,
j + 1) > S(O(i,j)).
PROOF: S(O(i + k, j + 1) - S(O(i, j) = (1 - 1/2j+1 + 1/2i+j+k+1) - (1
- 1/2j + 1/2i+j) = (1/2j - 1/2j+1) + (1/2i+j+k+1
- 1/2i+j) = ((2 - 1)/2j+1) + ((2K+1 - 1)/2i+j+k+1
) = 1/2j+1 + ((2k+1 - 1)/2i+j+k+1) = (2k(2i
- 1)/(2i+j+k+1) > 0, for all i,j,k >= 1.
Thus, no matter "how far behind" is the COMPOUND ARGUMENT WITH MORE TIPPOFF's (a.k.a.
ASSUMPTIONS, CONDITIONS, HYPOTHESES), it can ALWAYS "catch up and forge ahead" of the SIMPLER one
if it INVOKES ONE MORE CONFIRMED PREDICTION (OBSERVED ANOMALY) THAN THE SIMPLER ONE. The "Tortoise"
can always "catch up and forge ahead" of the "Hare".
And the "margin" shown above can be considerable as PARAMETER k. It was
this that suggested the next ("RISKY") THEOREM, the most suprising of all these Theorems.
To prepare for this, I shall derive an "approximating" Corollary from each of Theorems 3 and 4.
COROLLARY (TH. 4): S(O(i, j)) - S(O(i + k, j)) = (2k - 1)/2i+j+k > 1/2i+j
, for all i,j,k >= 1.
COROLLARY (TH. 5): S(O(i + k, j + 1)) - S(O(i, j)) = (2k(2i - 1))/(2i+j+k+1)
> 1/2j+1, for all i,j,k>= 1.
To study the consequences of these Corollaries, see more
bonuses.
PROOF: S(O(i, j)) - S
(O(i + 1, j))=(1-1/2j+1/2i + j) - (1-1/2j+ 1/2i+j+1)=
(1/2i+j) - (1/2i+j+1) = (2 - 1)/2i+j+1 = 1/2i+j+k+1
> 0,
for all i,j >=1 .
S(O(i, j)) -
S(O(i+k, j)) = (1-1/2j+1/2i+j) - (1-1/2j + 1/2i+j+k)
= (1/2i+j) - (1/21+j+k) = (2k-1)/2i+j+k > 0,
for all k > 1.