INDMEARSURING UNDERWORLD OF LOGIC
In "INDMEASURING 'PEIRCE PARADOX'", this Website, the "paradox" isexplained. "Peirce's Axiom" (actually, "Theorem") cannot be axiomatically proven (proof cited ONLINE) in any system lacking negation, although negation does not appear in its statement. An inmeasuring proof is given in "INDMEASURING ...", which also lacks negation. It
is shown in this file that the conditional connective of the Theorem's statement is
indmeasurement equivalent to a disjunction involving negation. Hence, indmeasuring can explain somethiing that axiomatics cannot -- banishing the "Paradox".
But what does this mean? We now fulfill the promise (made in that previous file) to explain.
Let's indmeasure all connectives of statement logic:
INDMEASURING ALL CONNECTIVES OF STATEMENT LOGIC
| A | B | ØA | ØB | A Ù B | A Ú B | A Þ B | A Û B |
| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
Now, let's indmeasure the negations of all connectives:
NEGATIVE INDMEASUREMENT OF ALL CONNECTIVES OF STATEMENT LOGIC
| ØA | ØB | ØØA | ØØB | Ø(A Ù B) | Ø(A Ú B) | Ø(A Þ B) | Ø(A Û B) |
| 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
The operation or connective of negation exists in at least two forms:
- EXCEPTION, which is complementary to whatever it EXCEPTS, so we have coexistence which can be diagrammed in the same lattice;
- EXCLUSION, which is not complementary to whatever it EXCLUDES, hence, cannot coexist, but must be diagrammed in dual lattices such that neither can
"reach" the other.
The second connective Table above resides in an "underworld" of the first connective Table. "You can't get there from here!"
