x is contingent if POS(!x)
x is necessary if !POS(!x)
The car might have been blue, but is in fact red. (Contingent)
If some proposition implies another proposition and the first proposition is true, the second proposition is true. (Necessary).
Every set of contingent truths is also contingent:
Proof by contrary supposition:
Suppose there exists a SET of contingent truths which is necessary. Every proposition in a conjunction is true if the conjunction is true. (def of "conjunction).
Let C be the conjunction and P be any proposition in the conjunction C. Suppose C is necessary. C implies P (since if !P -> !C). Therefore C -> P. Supposing C is necessary, then P is necessary too.
Principle of Sufficient Reason -> Every contingent proposition has a sufficient explanation for it's truth.
A(P), if (P) -> (E(G) G->P and G).
In the minimal case where P is necessary, P = G, and therefore P -> P. To say that P is not necessary (viz contingent) is to say that the contrary, that P !=G.
"The Dog is on the ground" is implied by "The Dog is on the ground", however, There exist other "reasons" (gravity, the existence of dogs, the earth, etc.) such that the conjunction of those reasons implies that the dog is on the ground. Since the conjunction of those reasons is not necessary (there need not be any earth), the dog being on the ground is not necessary either.
The set W of all contingent propositions exists by definition.
The set W is contingent (by 1)
There exists a truth G such that G implies W (by 2)
G is not contingent (by def of contingent)
Thus there exists some necessary fact such that it implies the existence of the set of contingent truths.
Attribution of CHOICE to G.
A fact P is chosen if Pos( ! P) & E(N) N->P but not necessary (P).
If contingent P -> Pos(!P) (definition)
The set W of all contingent facts is itself contingent (see above).
Thus contingent (W) -> Pos(!W)
If possible !W and W then the sufficient reason for W (from 2 above) effective decides W as opposed to !W and is therefore a choice by Definition.
Attribution of Goodness to G:
This one is tougher, there are two main issues:
A) we don't have a good definition of good, but let's try these things:
(a) a thing is good IFF it is well intended
(b) what it does in fact is -on the whole- beneficial for others
and PERFECT if
(c) There is no other thing that it could have done that would have been more beneficial to others.
Let's assume (a) for the moment (since it would be fruitless to try to guess in this case).
(b) is often denied, however, it's clear that -many good things- have happened in the course of history - from pleasant lunches to overthrowing of tyrants.
Thus it's clear that G is Good.
However, the further question, is G PERFECT implies that there is no other possible course of action for G that G might have taken that would have, on the whole, been better for everyone.
That's a tall order, I suggest only the following considerations:
a) People want to CHOOSE what they do, implying that sometimes they will choose bad things, causing suffering.
b) A world in which people weren't free or didn't exist to enjoy it would not be Good.
c) therefore it's reasonable to expect some BAD things in the world.
d) The AMOUNT of bad things in the world from AIDS epidemics and cancer to earthquakes and rapists is hard to understand nevertheless, but there are further considerations here too:
The production of, in particular, Patience, Kindness, Honor, Love, Justice and, in the end, other Good Beings therefore, would be the most beneficial possible thing for G to do, and the world we see exhibits exactly those characteristics - a world in which Goodness in Others is encouraged for its own sake.
And while it's possible to say "God could have done X to make this world better" without a proof of that proposition, the speculation is irrelevant.
Thus, G is both Free and Good.