1. There is a thing, x, such that it is a contingent proposition.<\/p>\n
\u2203xCx<\/p>\n
2. For any thing, x, if x is a contingent proposition then there is a possible world, w, where x is true, and there exists a possible world, w’, where x is false.<\/p>\n
\u2203xCx\u2192\u2203x\u2203w\u2203w’Pxaw\u2227Pxbw’<\/p>\n
3. Thus, there is a thing, x, and there is a possible world, w, and there is a possible world, w’, such that x has the property true in world w and x has the property false in w’.<\/p>\n
\u22a2 \u2203x\u2203w\u2203w’Pxaw\u2227Pxbw’ [1, 2, MP]<\/p>\n
4. For any thing, x, and for any possible world, w, if x has the property true or has the property false in w, then x exists in w.<\/p>\n
\u2203x\u2203wPxaw\u2192\u2203x\u2203wExw<\/p>\n
5. Thus, there is a thing, x, and there is a possible world, w, and there is a possible world, w’, such that x exists in w and x exists in w’ and<\/p>\n
\u22a2\u2203x\u2203w\u2203w’Exw\u2227Exw’\u2227Pxaw\u2227Pxbw’ [3, 4, MP]<\/p>\n
6. For any thing, x, for any possible world, w, x has the property true iff x does not hasve the property false.<\/p>\n
\u2200x\u2200w(Pxaw\u2194\u00acPxbw)<\/p>\n
7. For any thing, x, and for any thing, y, for any possible world, w, and for any possible world, w’, if x exists in w and y exists in w’, then (x and y are identical iff for any property, z, if x has it in w, then y has it in w’).<\/p>\n
\u2200x\u2200y\u2200w\u2200w'(Exw\u2227Eyw’)\u2192(x=y\u2194(\u2200zPxzw\u2194Pyzw’))<\/p>\n
This set is inconsistent (5\u22276\u2192\u00ac7)1<\/sup><\/a>. But it seems clear to me that we ought to give up 7.<\/p>\nNotes<\/h3>\n