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Meta-Theorem 1
Here is a theorem about our logic. It is not proven within our logic: we do not have the tools yet to do so.
Theorem:A formula that is a continued equivalence with an odd number of equivales is true if an even number of the operands are true. Likewise, a formula that is a continued equivalence with an even number of equivales is true if an odd number of the operands are true.
The proof is done in two parts.
- Odd number of equivales is true if even number of operands are true.
- If there are an odd number of equivales operators, then there are an even number of operands. If an even number of operands are true, then an even number are false. Each pair of false operands connected by an equivales is equivalent to true. Each true operand "disappears" in that true is the unit of equivales. So the entire formula is equivalent to true.
- Even number of equivales is true if odd number of operands are true.
- If there are an even number of equivales operators, then there are an odd number of operands. If an odd number of them are true, then an even number are equivalent to false. Again, pair up the false operands, since they are connected by equivales, each pair is equivalent to true. The true operands all "disappear" as units of equivales.
We can make the theorem shorter by defining parity as the "even-ness" or "odd-ness" of a number. Then the theorem becomes: The parity of the number of equivales is the opposite of the parity of the number of true operands. |