No Mathematics

The Correct Definition of if in Binary

0 implies 0 and 1 implies 1. Finite state machines replace plans and linear time suits some situations.


The biosphere exists. Our neurons exists. Snails whose behaviour Joseph LeDoux observes and describes in Synaptic Self exist, and behave in agreement with the correct definition of if. Our neurons are active in ways which agree with the correct definition of if. Our biosphere agrees with the correct definition of if. Therefore I also call this Nature's if. It seemed money was invented from nothing due to ambiguities in language inviting cognitive errors. Removing ambiguities and cognitive errors lets us earn real money in this real world, and keep our real lives in this biosphere.



Corrections in Basic Logic


With the incorrect definition of if, A ior A is in agreement with Not-A => A. However, with the correct definition of if, A => A is in agreement with A and A.

Also due to the correct definition of if, the following Axioms of Replacement are invalid due to they are mistaken consequences of the incorrect definition of if.

Transposition is invalid. Instead the correct statement is if A => B then B => A and iff is redundant.

Material Implication is invalid. Instead A => B is similar to A AND B with the only difference being a factor of time.

Material Equivalence is invalid.

The Tautology A = A AND A is valid.

The Tautology A = A IOR A is valid.

The Tautology A = A XOR A is invalid since the statement A XOR A is false when A is true, and is false when A is false.

DeMorgan’s Rule is valid for the inclusive-or (IOR) and invalid for the exclusive-or (XOR).

Commutativity, Associativity and Distribution are valid.

Cause is normally thought of (in Basic Logic) in terms of sufficient conditions and necessary conditions.

However, visible in human trade relations and in Nature, the actual conditions for cause are threefold: sufficiency, necessity and something extra. The something extra prerequisite to cause is quantified as a function of and in proportion to the consequent of cause, and this may be new work not previously considered.

I prefer replacing Syllogisms with Predicate Logic since the latter is able to state all of the former without confusion.

I prefer teaching everyone to reject fallacies as their initial cognitive development training. Yes, advanced logicians do see the practical value of the fallacies, however letting beginners see and reject fallacies could strengthen their cognitive self-defense.

Reference: Introduction to Basic Logic by Patrick J Hurley Return to Persuasive Logic.

Probability of B Given A


With conditional probability we show the unviability of the old definition of if.

The conditional probability of event A occurring if condition B is met is defined by the formula:
P(A|B) = P( A intersect B) / P(B).

Paraphrased from source:Encyclopaedia of Mathematics, Volume 7, Kluwer Academic Publishers, Managing Editor M. Hazewinkel

I've reversed A and B from their traditional roles to deliberately test the old 'only if'. Thus suppose we have A => B with the old definition of if:

=> 0 1
0 1 1
1 0 1

The conditional probability that is worth testing is in the reverse direction: A only if B, and be clear to use the same truth table without flipping it over for convenient escape of the question. So A is on the left-hand side and B is on the top of the truth table, same as for A => B, and what we are testing is B => A with conditional probability and both definitions of if.

Observe that P(B) = 1 in any situation in the old definition of if. Therefore, the statement that B is given is meaningless. Therefore, in the old definition of if, P(A|B) = P(A) in all four cases: A and B have empty intersection; all three of (A intersect B), A\B and B\A have non-empty intersection; or either is a proper subset of the other.

Observe the new definition of if has in some cases the only if direction being identical to the if direction, preserving the formula from the Soviet Encyclopaedia of Mathematics.

Infinity


I prefer clearly disambiguating finite from countably infinite from more than countably infinite.

Prove we have more than countably infinite real numbers without reference to Cantor's Diagonal Argument:

Let N be the set of all natural numbers, 0, 1, 2... and let R be the set of all real numbers. Both N and R are infinitely large. The size of N is called countable infinity and the size of R was traditionally called uncountable infinity, now known as more than countably infinite. The set Q of rational numbers is countably infinite.

Proof of there being something more in R than in Q:

Let Q be the set of rational numbers. Write Q as the union over each n in N of {m/n | m is each integer}. Proof that the set of all real numbers, R, is beyond countably infinite may be done by showing that the proper subset of irrational numbers is uncountable. For this proof, hold N as N without 0. Suppose that the set of irrational numbers, R\Q, is countable. Then there is a one-to-one correspondence between Q x Q and Q x (R\Q). List Q x Q with the m values ordered 0, +1, -1, +2, -2, ... and with the n values 1, 2, 3, ... such that our matrix has 0 in the upper left corner the top row is 0, +1, -1, +2, -2,..., and the second row is 0, +1/2, -1/2, +2/2, -2/2,..., and so on.) According to our supposition, all real numbers are now in our matrix. Remember Pi. We know Pi is transcendental and therefore is without a rational representation. Therefore R\Q is beyond enumerable and therefore R is beyond countably infinite.

I confess the proof contains a contradiction.

Multiply-Interpretable Present


Now exists, and is always happening. Multiply-interpretable present situations have different optional futures.

General Comment about malevolence, how to see and remove that: 

Due to the correct definition of if, some mathematicians shall re-write some mathematics proofs which were proofs by contradiction, and I think it makes sense to rewrite also proofs which relied on Modus Tollens and other now inoperable Axioms of Replacement. I accept arguments which clearly prove existence of some things while holding open constructive options per positive motivation.

Friedman's operating assumption is the universe appears the same as interpreted via any direction from any point inside the universe. However, insofar as human information systems may be analogous to space manifolds, clearly the reported local area data stream differs rather than as intrinsic fluctuation, instead as an expression of differences of the quality of measuring and reporting devices. Thus, does anyone have any right to say anything is positive and constructive, or that could be relative? Instead, I prefer retaining reference to the biosphere and to ancient financial and trade systems which work together with ternary logic and the biosphere since I think this might suffice to remove malevolence from anyone pretending to do good while merely mugging the ones who construct. To what extent does the analogy of human minds in societal data generalize to particles in space? Parasites?