Hi Daniel!

I'm a lawyer, not a logician, but I understood what you meant; hence I wrote that you were misrepresentingmy argument. The trickery of your "refutation" is pretty evident: you cannot assume that " A is of the formnecessarily B", since by " possible" I mean "non-contradictory and not necessary ", as I already clarified.

Definition 1 says: x is possible iff x does not entail a contradiction. There is nothing in definition 1 thatprevents a possible thing from also being necessary. Allowing -- for the sake of argument -- that we cananalyse possibility in terms of the absence of contradiction, Definition 1 is what we should go for.

Definition 2 says: x is impossible iff x entails a contradiction. (This definition is redundant; it followsdirectly from Definition 1.)

Definition 3 says: x is necessary iff ~x entails a contradiction. (Again, this definition is redundant: it followsdirectly from Definition 1, together with the standard equivalence between "necessary" and "not possibly

not".)

If you want to replace Definition 1 with

Definition 1*: x is possible iff (1) x does not entail a contradiction; and (2) x is not necessary

then you will get a consequent revision of Definition 2:

Definition 2*: x is impossible iff either (1) x entails a contradiction or (2) x is necessary.

But Definition 2* is absurd: nothing that is necessary is impossible.

So it is not true that you have clarified how it can be that "possible" means "non-contradiction and notnecessary".

Of course, there is another term to be defined:

Definition 8: x is contingent iff it is possible that x and it is possible that ~x

Given Definition 8 and Definition 1, we can infer:

Theorem 1: x is contingent iff neither x nor ~x entails a contradiction.

Definition 7 says: B is an opposite of A iff it is not possible that A and B. (It follows from this thatthe negation of A is an opposite of A. If you want to say that ~A is the opposite of A, then we don't need thedefinition: we are already making use of the notion of negation.)

It follows straightforwardly from Theorem 1 that A is contingent iff ~A is contingent. However, it does notfollow from Theorem 1 and Definition 7 that, if A is contingent, and B is an opposite of A, then B iscontingent. In particular, if B is impossible, then, while A&B is impossible, B is not contingent.

So, if your Axiom 1 is reinterpreted in terms of contingency, we have to reject it: it is not necessarily truethat an opposite of something that is contingent is contingent.

And, as I have already shown -- and as you seem to agree -- if Axiom 1 is interpreted in terms of possibility(as ordinarily understood), then we also have to reject it, because it is subject to counterexamples of thekind that I gave previously.

In sum: even if we accept the claim -- rejected pretty much universally by contemporary professionalphilosophers -- that x is possible just in case x does not entail a contradiction -- we should still reject your Axiom 1.

Proposition 1 says: Any universe is the opposite of nothingness. This requires some interpretation.Suppose we accept that

Definition 5*: x contains a universe iff x contains a maximal fusion of mass-energy.

Definition 4*: x is empty iff x contains no mass-energy

It follows from these definitions that it is not possible both that x contains a universe and that x is empty.(I'll now take this -- it is not possible both that x contains a universe and that x is empty -- as an acceptablestatement of Proposition 1.)

Proposition 2 says: it is possible that something is empty.

You try to prove this as follows:

Any universe is the opposite of nothingness (by Proposition 1). The opposite of what is possible is also possible (by Axiom 1). Any non-contradictory universe is possible (by Definition 1). Therefore, nothingness is possible.

Proposition 1 gives us that it is not possible both that @ contains a universe and that @ is empty. Sincethere is a universe, we know that it is possible that @ contains a universe. (Whatever is actual is possible.)But, from these two claims -- that it is not possible both that @ contains a universe and that @ is empty,and that it is possible that @ contains a universe -- it simply does not follow that it is possible that @ isempty. (Moreover, as we have already seen we cannot appeal to Axiom 1 to bridge the gap, because Axiom1 is false.)

Cheers,

Graham

Professor Graham OppySchool of Philosophical, Historical and International StudiesMenzies Building20 Chancellor's WalkMonash University VIC 3800