INDMEASURING "THE PEIRCE PARADOX"

"Peirce's Axiom" (actually a Theorem) is considered by some to be "the difference" between an intuitionist system of propositional calculus: accepted in a classical system, rejected by intuitionists.

"Peirce Pardox": Although the "axiom" contains no negation connective, yet it can only be axiomatically proven by the Contapositive Axiom, hence, cannot be proven in any system lacking negation. (Proof at http://metamath.planetmirror.com/mmexplorer/peirce.html )

Modern logic shows that separability operates by setting forth an axiomatic system which lacks one or more of axioms of a complete ssystem. Hence, "Peirce Paradox".

However, Peirce's Axiom can readily be proven by Indmeasuring ("truth table" method):

PROOF OF (((a Þ b) Þ a) Þ a)
a
b
(a Þ b)
((a Þ b) Þ a)
(((a Þ b) Þ a) Þ a)
0
0
1
0
1
0
1
1
0
1
1
0
0
1
1
1
1
1
1
1

The final column of all ones measures the validity.

And this result also seems "paradoxical", because it, too, lacks negation.

Never fear! Indmeasuring is here! Inmeasuring will show that negation is implicit or hidden in the above indmeasurement -- hiding in the conditional operator Þ.

COMPARING A ÞB AND ØA Ú B
A
B
A Þ B
ØA
ØA Ú B
0
0
1
1
1
0
1
1
1
1
1
0
0
0
0
1
1
1
0
1

The third and fifth columns of INDMEASURES are THE SAME! This INMEASUREMENT PROVES EQUIVALENCE OF A ÞB AND ØA Ú B! So, NEGATION IS HIDDEN IN THE INDMEASUREMENT OF "PEIRCE'S AXIOM". The "paradox" cannot be explained axiomatically, but it is banished by indmeasurement.

But what does that mean? The next file, "INMEASURING UNDERWORLD OF LOGIC", answers this question.