Day: June 14, 2007

Meta-theorem 22:
The morphology of our world implies that if there is a unicorn, life would be more beautiful, iff the union of our morphology and the statement ‘there is a unicorn’ also implies that ‘life would be more beautiful’.

Proof:

1. Assume ? ? ‘there is a unicorn’ ? ‘life is more beautiful’.
2. Then we know that every valuation that simultaneously satisfies ?, also satisfies ‘there is a unicorn’ ? ‘life is more beautiful’.
3. For every valuation which simultaneously satisfies ?, it must be the case that V(‘there is a unicorn’)=F or V(‘life is more beautiful’)=T.
4. Assume ? ? {‘there is a unicorn’} ? ‘life is more beautiful’
5. Then there must be a V such that ? ? {‘there is a unicorn’} is simultaneously satisfied, but V'(‘life is more beautiful’)=F and V'(‘there is a unicorn’)=T.
6. However, this is a contradiction to line 3, hence:
7. ? ? {‘there is a unicorn’} ? ‘life is more beautiful’

 

8. Now assume ? ? {‘there is a unicorn’} ? ‘life is more beautiful’.
9. Then we know that every valuation which simultaneously satisfies ? ? {‘there is a unicorn’}, also satisfies ‘life is more beautiful’.
10. It must be the case that V(‘life is more beautiful’)=T or ? ? {‘there is a unicorn’} is not simultaneously satisfiable, such that V(‘there is a unicorn’)=F.
11. Assume that ? ? ‘there is a unicorn’ ? ‘life is more beautiful’.
12. Then there is a valuation V’ that simultaneously satisfies Γ, such that V'(‘there is a unicorn’ ? ‘life is more beautiful’)=F.
13. Then it must be the case that V'(‘there is a unicorn’)=T or V'(‘life is more beautiful’)=F.
14. However, this is a contradiction to line 10, hence:
15. ? ? ‘there is a unicorn’ ? ‘life is more beautiful’