This was originally part of the previous post, but I thought it was a bit too long, so I split it into two.
We can demonstrate the power of logic by resolving a so-called problem of God's omnipotence. Most people have encountered the question 'Can God create a rock he cannot lift?' The paradox is obvious: if God is omnipotent, he can create anything. So then God can create a rock he can't lift. But then he can't lift it, so there's something he can't do, so he's not omnipotent after all. But then if God can't create an object he cannot lift, then that's something God can't do, so he's not omnipotent. Conclusion: God is not omnipotent; but if God was not omnipotent, he couldn't be God; so God as we define him doesn't exist.
Let's see if we can help God get out of this mess, using logic. First, we can define omnipotence in terms of lifting and creating things. If God is omnipotent, he can create anything. We can formulate this as follows:
1. GodIsOmnipotent ⇒ ∀x : GodCanCreate(x)
This simply means: The statement 'God is omnipotent' implies that for all (∀ means 'for all') objects x, God can create x.
Next: if God is omnipotent, he can lift anything.
2. GodIsOmnipotent ⇒ ∀x : GodCanLift(x)
One of the rules of logic is this: (A ⇒ B ∧ A ⇒ C) ⇒ (A ⇒ B ∧ C). This means, if A implies B, and A implies C, then A implies both B and C. We can use this rule to combine the two statements above:
3. GodIsOmnipotent ⇒ ∀x : GodCanCreate(x) ∧ ∀x : GodCanLift(x)
We can use another rule of logic to combine two 'for all' statements: (∀x : P(x) ∧ ∀x : Q(x)) ⇒ (∀x : P(x) ∧ Q(x)), giving us:
4. GodIsOmnipotent ⇒ ∀x : GodCanCreate(x) ∧ GodCanLift(x)
Now we can make our first statement:
We can use our rules of logic on this (if A is true, and A implies B, then B is true) to get:
6. ∀x : GodCanCreate(x) ∧ GodCanLift(x)
Now we can make our next statement: God can create a rock that he cannot lift. We can express it in this way:
7. ∃c : GodCanCreate(c) ∧ ¬GodCanLift(c)
This means: There exists (∃ means 'there exists') a particular object c (our rock), such that it is the case that God can create c, but it is not the case that God can lift c. This is our contentious statement. Using another rule of logic, we can move from the general to the particular (∀x : P(x) ⇒ ∃c : P(c)). Using this on statement 6, we get:
8. ∃c : GodCanCreate(c) ∧ GodCanLift(c)
But now we have a contradiction! Statement 7 contradicts statement 8. But we know that if we come to a contradiction, we have made a meaningless set of statements. So the statement 'God is omnipotent and can create a rock that he cannot lift' is, in fact, gibberish.
How about the statement 'God is omnipotent and cannot create a rock that he cannot lift'? This can be expressed as:
10. ∃c : ¬GodCanCreate(c) ∧ ¬GodCanLift(c)
Which obviously leads to another contradiction. So this statement is meaningless too. The correct conclusion is not that God is not omnipotent, but that it does not make sense to talk of a rock that God can create but cannot lift. There is no such rock. This is not saying that God cannot create it, but saying that such a rock cannot exist, as much as a 'square circle' cannot exist.