Saturday, June 2, 2012

All Perfections are Possibly Instantiated

My logic teacher and I were reflecting on the ontological argument and we came up with this. Forgive the cheap operators. A perfection is a property which it is better to have than to lack. Let A be any perfection (say, being perfect) and let B be any non-perfection (like being imperfect) and let 'PA' mean 'A is a perfection' and 'PB' mean 'B is a perfection'. Then let PP be the principle that perfections only imply perfections:

(PP) For any property J and any property K, [PJ & [](x)(Jx --> Kx)] --> PK

Since A is a perfection and B is a non perfection:

[Prem] PA & ~PB

Now, assume for reductio that ~<>(Ex)(Ax). This is equivalent to []~(Ex)(Ax) which is equivalent to [](x)~(Ax). Since necessarily A is not exemplified by anything, then trivially [](x)(Ax --> Bx). So by our premise, [PA & [](x)(Ax --> Bx)] But by (PP) [PA & [](x)(Ax --> Bx)] --> PB. It follows that PB. But by our premise, ~PB. This is a contradiction. So we must reject our assumption. So ~~<>(Ex)(Ax). So <>(Ex)(Ax).