see... a VALID argument is an argument where it is IMPOSSIBLE for all the premises to be true and the conclusion false at the same time. That is to say... IF the argument is VALID and the premises are all true then the conclusion MUST BE TRUE. Alternatively, on all possible worlds where the premises are true the conclusion must be true. If an argument is valid and you want to deny the conclusion then you must show one of the premises to be false.

It doesn't matter how many OTHER premises you add. Hence, whether an argument is valid or not isn't dependent on our finite cognitive capacity.

E.g.,

P1) It is raining and it is wet
________________________________
C) It is raining

The proof is like this:

P . Q
therefore P

Now we have four possible worlds (possible assignments of truth values):

P = T, Q = T
P = T, Q = F
P = F, Q = T
P = F, Q = F

In order for the first premiss to be true we are considering a world where P and Q are both T. That means line one is relevant (and the other lines are not). You will notice that in that world (that line of the truth table) P is T. That means that in any possible world where the premiss is true the conclusion is true as well and thus the argument is valid.

Lets add something to the argument

P1) It is raining and it is wet
P2) Frogs are purple
________________________________
C) It is raining

Now we are going to need more lines to symbolise this:
P . Q
R
___________
P

P = T; Q = T; R = T
P = T; Q = T; R = F
P = T; Q = F; R = T
P = T; Q = F; R = F
P = F; Q = T; R = T
P = F; Q = T; R = F
P = F; Q = F; R = T
P = F; Q = F; R = F

Are all the possible worlds (all the possible combinations of truth values for three variables P, Q, and R). The relevant worlds where the premises are true are the worlds where P, Q, and R are true. That is the first line of the truth table again. Note that on that world the conclusion is true again. Hence the argument is still valid. But we know that R is false. That means that the second line of the truth table is relevant. Notice that adding a falsehood to the premises of a valid argument does not affect its validity. The conclusion is true on that world too. Hence it doesn't matter if our cognitive capacity means that there are factors that we haven't considered. If we have deductive proof (proof by contradiction, for example) then extra information is irrelevant. That is an interesting feature of deductive reasoning (it doesn't apply to inductive reasoning, however).

How does proof by contradiction work? Try to make the argument invalid. The only way you can do this is to assign contradictory truth values to the variables. For example... 'Sure god can make a rock so big he can't move it, though of course he can move it because he can do anything, though of course he can't move it because he can do anything etc etc).

Why are people so determined that God can do contradictory stuff? Because the bible says God can do anything? The bible says bats are birds (they aren't they are mammals)... The bible says insects have four legs (they have 6)... I don't see what is wrong with the 'greatest possible being' conception. I thought people would have seen this as an interesting fact about Gods nature.

I don't even really see what is to argue about...

People will be saying 2+2=5 in a minute... Or that it is all a matter of opinion or personal preference or something...

Do people really think their GOd wants them to abandon reason?