Following on from yesterday’s post, here is the answer from Roy Sorenson’s Cabinet of Philosophical Curiosities:
“The elderly scientist is certainly correct. The reason is that any assertion of an impossibility is equivalent to a statement of possibility. ‘It is impossible that p’ is equivalent to ‘It is possible that it is impossible that p’: ~ p ↔ ~ p. So Clarke would have to assign a low probability to the impossibility statement and a high probability to the possibility statement. It would be impossible for Clarke’s two probability assignments to be both correct.
Proof of the biconditional: ~ p ↔ ~ p. The left-to-right direction, ~ p → ~ p, follows from the principle that whatever is actual is possible.
The right-to-left side, ~ p → ~ p, follows from the principle that whatever is possible is necessarily possible: p → p. (This is the characteristic formula of the popular modal system S5.) The contrapositive of this formula is ~ p → ~ p To say something is not necessary, ~ , is equivalent to saying it is possibly not the case, ~. So the contrapositive can be rewritten as ~ p → ~ p.
Conjoining the two conditionals establishes the equivalence ~ p ↔ ~ p.
Séamus Sweeney 