Logic Example: The Rock Paradox

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:

5. GodIsOmnipotent

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:

9. GodIsOmnipotent
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.

Comments

  1. hey what's up?<br><br>now about the ROCK paradox! ...

    by ANONYMOUS on THURSDAY, OCTOBER 30, 2008 at 11:20AM
    hey what's up?

    now about the ROCK paradox! its just goes against logic! so why call it logic? I have studied logic, and one way of sloving logical problems is to find an equivalent mathematical form for it, here we go;

    1- God can do anything (create anything) is equivalent to (lets say) x = 5.

    2- Can God create a rock that he can not carry??! ( prove that x + 2 = 10)
    can u? knowing that x = 5???!

    did u got it?

    I will tell u something pal! Forget about religion for a while, and about logic.

    Do ur logic accept the theory of relativity?? that time and distance are not absolute and that speed of light is the absolute?? or can ur logic accept the theory of quantum mechanics?? if scientists believed in logic only they would have rejected both theories.

    lsn logic is a powerful tool, but it has its limits.

    U can not learn about God just using this tool, nor u can deny him using it!

    See, now u just do not start with a religion! Try to communicate with this God! He can not be predicted nor seen, but he is the creator of this world.

    Now if u saw something strange, but u knows it was created by someone, the first thing u will ask is, what is its purpose?

    Why God did create us? There should be a reason, u should look for it. See I believe God will help u if u asked him to. Just be honest, do not think about Jesus nor Christianity, Just GOD!

    So why are u strict to Christianity, read about different religions that believe in Monotheism. Jews believe that only people whos mother is jew can be Jew. See here:

    http://www.jewfaq.org/whoisjew.htm

    and see the part about who is Jew!

    Muslims although they are stereo-typed for being terrorist are more opened. I mean after all 1.5 billion are muslim. if they were terorists, then one of every 4 persons is bad guy (assuming that there rest 4.5 billion are all good) but I was really surprised that Islamic countries have the least crimes rates world wide.

    So now I am doing readings about it! do not read commercial books nor the internet. Use books that were published by professors. They are objective!

    read about Judaism too! and tell me what u think!

    read about Judaism too! and tell me what u think! they are the only two religions I would trust after Christianity!
  2. thanks stu!<br><br>i lived for two years with a gu...

    by GARETH on TUESDAY, AUGUST 29, 2006 at 7:32PM
    thanks stu!

    i lived for two years with a guy, a christian, who woudl answer that one with the nonsensical "yes, and he could lift it", which betrays the problem with much of thsi country's experience of christianity: that it will not stand up to proper scrutiny, and that its own adherents are the proof of that in their dogmatic and unquestioning plod towards teh end of the world.

    i am glad that you confirm the answer i've always given, "don't talk such nonsense!" as being the correct one to the paradox you quote!!! and i am also glad that your blog may be helping us believers engage our brains a bit more, in our own belief and also in our dealing with those who have no faith. it's very important to me that we as christians don't treat those who aren't as idiots!!!
  3. Stu,<br><br>Your post says that the resolutions ar...

    by ANONYMOUS on MONDAY, FEBRUARY 13, 2006 at 7:58PM
    Stu,

    Your post says that the resolutions are too simplistic (which isn't a test for veracity).

    So, you're happy enough to say that there is no problem for a logician to talk of God's omipotence?

    Tom
  4. I know it's resolvable. That's what my post says....

    by STU SHERWIN on MONDAY, FEBRUARY 13, 2006 at 2:47PM
    I know it's resolvable. That's what my post says.
  5. Hi Stu,<br><br>Can you just run me off a one line ...

    by ANONYMOUS on SUNDAY, FEBRUARY 12, 2006 at 11:09PM
    Hi Stu,

    Can you just run me off a one line definition of omnipotence please?

    Have you read Alvin Plantinga on this?

    This is a good one because. It's a completely resolvable problem.

    Tom
  6. Hi Barry, thanks for posting.<br><br>Firstly, I'd ...

    by STU SHERWIN on SUNDAY, FEBRUARY 12, 2006 at 10:00PM
    Hi Barry, thanks for posting.

    Firstly, I'd like to say that I am aware that many 'resolutions' have been proposed to the many dilemmas and contradictions that intelligent thinkers have found within Christianity. I have looked into many of these and found a lot of them too simplistic. Please be aware that I have been thinking about these questions for a while, while I still regarded myself as a Christian, and it was only after finding many of these so-called resolutions to the issues I was facing unsatisfactory that I realised that stepping out of the 'box' of Christianity was for me the only way to view these problems objectively.

    As to the debate on Hell, of course I am aware that there are many different views on the subject, having looked into many of them myself. My post was geared towards a particular view on Hell which has been the orthodox view for hundreds of years, and which many (if not most) Christians still believe today. My aim was to highlight some of the problems of believing this view, and to show how I find it to be unsatisfactory.

    In every doctrinal area of Christianity there have been a plethora of conflicting views and opinions, many of these have been regarded as acceptable for a Christian to hold, and many have been condemned as heresy. The doctrine of Hell is no different. The reason there is, as you say, 'a very real debate' about Hell is that many Christians, very possibly using similar arguments to that which I used in my post, have come to the conclusion that the 'orthodox' doctrine of Hell as a place of permanent, eternal suffering, is in need of serious revision. In this I would say I have to agree, and if you're one of them, then we're both arguing to the same end, but from different sides of the border. My point is that if all the problems and contradictions in Christianity were ironed out in this way, then we would have something quite unlike the Christianity of the past 2000 years, and quite possibly one which I would have no problem in believing.
  7. Stu,<br><br>The greatest mistake you made was in s...

    by BLACKNAD on MONDAY, DECEMBER 19, 2005 at 12:44AM
    Stu,

    The greatest mistake you made was in suppressing your doubts. You were gifted with a brain and then thought that as a Christian you shouldn't use it. How long can someone exist in such a state of intellectual limbo. Many people do and by the time the tension causes a snap, great damage has been done, because whereas it could have been a journey of resolving one issue at a time, you are left with a hundred different doubts that create a pervasive feeling that Christianity is a nonsense.

    There are reasonable answers to your doubts - and in some cases there are reasonable answers that show why a particular issue is not subject to reason - even St Paul accepted that we see through a glass darkly. We cannot reference all possible variables so sometimes the answer will elude us. But I bet it is far less than you imagine.

    You also need to accept that many of the things you held to be dear about Christianity were in fact misconceptions and misrepresentations, also accepted blindly by people in your church because they too were scared of doubt and afraid of 'going there' because it might rock their faith.

    I can tell you that asking the difficult and awkward questions is liberating. I am glad to se you are now doing so - but I hope it is not too late for you.

    An example is your understanding of Hell. It is one of the least well understood aspects of Christianity and you probably take a view, and do not realise that there is a real debate about it's permanence.

    Regards,

    Blacknad (Barry Stephenson).
  8. Jon is correct.<br><br>Logic is 'the relationship ...

    by BLACKNAD on SUNDAY, DECEMBER 18, 2005 at 11:53PM
    Jon is correct.

    Logic is 'the relationship between elements and between an element and the whole in a set of objects, individuals, principles, or events'

    Logic is simply the reasoning method that allows us to understand (in this case) what is a reasonable proposition about God and what is nonsense. The question 'can he make a rock that is so heavy he can't lift it' is sheer nonsense. The two ideas are irreconcilable.

    Omnipotence is surely understood to mean all-powerful within the realm of what is ultimately possible.

    I do not actually believe that God can create sentient beings with free will that do not have the ability to do things contrary to his will - the two things are logically incompatible. But I still don't have a problem with him being omnipotent and having the ability to impose his will upon those beings at any particular moment - it would simply mean that in that instance he is over-riding their free will.

    We have to be wary about what form of scrutiny we subject God to. Trying to Logic Chop to come to an understanding of Him is interesting but ultimately limiting. We may not like it, but it is evidently so.
    Also humanity is fallen and therefore corrupted - this extends to our reasoning. This is why God has to reveal himself - because our reasoning does not lead us to him.

    I suppose we could play along with the nonsense and say - 'yes - of course God can create a rock he can't lift... and he can also lift it at the same time'. - and it is just our small feeble minds that can't comprehend how that can be so.

    Regards,

    Blacknad
  9. OK, here's the revised proof, starting from point ...

    by STU SHERWIN on TUESDAY, OCTOBER 25, 2005 at 6:42PM
    OK, here's the revised proof, starting from point 6

    6. ∀x : GodCanCreate(x) ∧ GodCanLift(x)

    7. ⇒ ∀x : GodCanLift(x)

    8. ⇒ ¬∃c : ¬GodCanLift(c)

    Now we can make our next statement: God can create a rock that he cannot lift. We can express it in this way:

    9. ∃c : GodCanCreate(c) ∧ ¬GodCanLift(c)

    10. ⇒ ∃c : ¬GodCanLift(c)

    Now we have a contradiction between point 8 and point 10. QED
  10. Hi Thorwald. What you've said is quite right, the ...

    by STU SHERWIN on SUNDAY, OCTOBER 23, 2005 at 7:17PM
    Hi Thorwald. What you've said is quite right, the contradiction is really between points 6 and 7. Point 8 is actually unnecessary. What I was doing in point 8 was trying to apply the general rule in point 6 to the specific instance c in point 7, so we have two contradictory statements about that specific instance c, one that God can lift it, and one that God cannot lift it. So I was trying to get the statements to apply to the same object.

    The point I'm making is that if you have a 'forall' statement about all objects x, and have a statement there is a specific instance object c for which the 'forall' statement is not true, then obviously you have a contradiction. My formal logic may be a little rusty in places, but the general argument holds.
  11. Sorry Stu, but I can't agree with your logic. The ...

    by THORWALD on FRIDAY, OCTOBER 14, 2005 at 1:46AM
    Sorry Stu, but I can't agree with your logic. The contradiction you mention between point 7 and 8 isn't there. Basically, 7 says there exists something that God can create, and (but?) that he cannot lift. 8 states that there exists something that God can create and that he can lift. Those two things don't have to be the same objects, and can easily coexist for both statements to be true. I think the contradiction is between 6 and 7, though I'm not sure if the rules of logic are satisfied well enough to end the proof (by contradiction) there. Despite the logic, I do agree with the outcome!
  12. yeah, the paradox. It <i>seems</i> to make a point...

    by JON on TUESDAY, SEPTEMBER 06, 2005 at 8:15PM
    yeah, the paradox. It seems to make a point but does not.
  13. &gt;&gt;So this proves nothing, in the end.<br><br...

    by STU SHERWIN on TUESDAY, SEPTEMBER 06, 2005 at 3:41PM
    >>So this proves nothing, in the end.

    What proves nothing, the paradox or my post? ;-)
  14. I thought for a minute that wo people were faster ...

    by JON on MONDAY, SEPTEMBER 05, 2005 at 6:26PM
    I thought for a minute that wo people were faster than me on the old RSS feeds grabbing your latest posts, but turns out it's spammers. Oh well, kill them, Stu.

    I'm going to reply to this one rather than the first post, both are well written again, and raise some interesting ideas. I totally agree with your points on logic: logic is a tool that can be used to prove the consistency of an argument, its validity, if you will; not its truth content, its veracity - a logical argument is literally just letters and symbols.

    Now to the rocks... What your argument shows is that the rock itself is meaningless. It seems to me that the problem (from your first post) is in the definitions - that it is logically incosistant to pit one of God's faculties against another. Can we ask "Is God stronger than God?" - no, because God is one, and as Jesus said, a house divided against itself cannot stand. God cannot compete with himself, that is a contradiction of His nature as One. Therefore it is nonsensical to pit one ability of God against another, just as it is nonsensical to ask if God can make a square-circle.

    God can make anything that can logically exist, and you showed that a rock that God (as OneWhoCanLiftAnything) cannot lift cannot exist, so since God follows logic, He cannot create it. That is not a limit on omnipotence, but a fact of existance.

    So this proves nothing, in the end. Logic is a useful, relevant tool in trying to understand God. Its weakness is that we forget too esily that an argument is only as strong as its premises.

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