I want to criticize a version of the ontological argument which proposes atheism and does a reductio ad absurdum. I have also recently heard of "Gödel's ontological argument", but I'm not really into logic and I haven't found it with words so I can't criticize it yet. I would also like to know if the criticism I will do right now can work for Gödel's argument, and why / why not.
This is the version of the ontological argument I propose:
Axioms:
1. God is the most perfect (greatest) possible being.
2. A being which doesn't exist (possible) is less perfect than a being which exists (real).
Premises:
1. God doesn't exist in reality.
2. Therefore, there could be something greater than God: God existing in reality.
3. Therefore, there is something greater than the greatest possible being (contradiction).
Conclusion: God must as well exist in reality. (In more logical terms: If God is possible, he's necessary.)
If this version has something deeply different from Anselm's version then tell me so I can correct it.
Criticism:
It follows from axiom 2 that God being possible is less perfect than God existing in reality. Or, in another case, that they are both equally perfect possibilities.
Then, premise 2 is a non sequitur from premise 1. There could be something greater than the possibility of God, namely God existing in reality. This is non-contradictory, and it follows naturally from axiom 2.
3. Would read: "There is something greater than the possibility of the greatest possible being (The greatest being itself)."
There could be God existing in reality, but there could be not as well (just possible), that's why God is more perfect than his possibility. If God doesn't exist in reality, then there could be something more perfect than the possibility of God, not greater than God himself. There is no contradiction.
Aside, when I define God as the "greatest possible being" or "most perfect possible being", how perfect/great is that being? If I define my unicorn as "the equine with the longest horn", but it's impossible for equines to have horns, then the word "unicorn" would become a name for a horse whose horn measured 0 cm - the longest possible horn length in equines.
Questions:
1. Is my criticism right, if the argument is in fact a valid ontological argument version? (I hope)
2. Can this criticism work for Gödel's argument? Why / Why not? Can someone explain the differences in Gödel's argument without modal logic to me, in words?
3. Is it possible for a being like God to exist? What do we mean with this?
Definition of God: conscious entity that created the universe.
Ontological argument criticism
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Post #2
There are a number of these types of proofs of God. From where I sit they all share the same failing as the Berry paradox:
Consider the expression:
But the above expression is only ten words long, so this integer is defined by an expression that is under eleven words long; it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under eleven words), there cannot be any integer defined by it.
Substitute greatest or most perfect for defined by an expression that is under eleven words long in the Berry paradox and you have a close approximation of an ontological argument.
[Joke]
Consider the expression:
- The smallest positive integer not definable in under eleven words.
But the above expression is only ten words long, so this integer is defined by an expression that is under eleven words long; it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under eleven words), there cannot be any integer defined by it.
Substitute greatest or most perfect for defined by an expression that is under eleven words long in the Berry paradox and you have a close approximation of an ontological argument.
[Joke]
- Nothing is better than God.
- A sandwich is better than nothing.
- Therefore, a sandwich is better than God.
Examine everything carefully; hold fast to that which is good.
First Epistle to the Church of the Thessalonians
The truth will make you free.
Gospel of John
First Epistle to the Church of the Thessalonians
The truth will make you free.
Gospel of John
Post #3
McCulloch wrote:Substitute greatest or most perfect for defined by an expression that is under eleven words long in the Berry paradox and you have a close approximation of an ontological argument.
Thank you, really interesting, didn't think about it! It seems to me that each person can find a different flaw in ontological arguments, there are just so many!... Even David Hume's statement that a being cannot be shown to exist a priori should suffice.
McCulloch wrote:[Joke][/Joke]
- Nothing is better than God.
- A sandwich is better than nothing.
- Therefore, a sandwich is better than God.

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Post #4
This is not correct. Why should the mere infinity of integers require that any of them need eleven or more words to describe? The error in this seems to be the pairing off of each phrase with an integer but a phrase can describe more than one integer.McCulloch wrote:Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words.
Post #5
A single phrase can only define one integer. "Two plus two" for "four", for example. Otherwise, it would be ambiguous and therefore not defining any integer at all.mgb wrote:This is not correct. Why should the mere infinity of integers require that any of them need eleven or more words to describe? The error in this seems to be the pairing off of each phrase with an integer but a phrase can describe more than one integer.
Show us a phrase that defines two integers simultaneously.
In fact, the reverse of what you said is true. An integer can be defined by more than one out of the finitely many phrases.
"two plus two", "two times two", "square root of four", "five minus three"...
There aren't infinitely many numbers definable under 11 words. There is a large, finite number.
Last edited by Ragna on Tue Mar 08, 2011 2:47 pm, edited 3 times in total.
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Post #6
A phrase which describes more than one integer does not define any integer.
Examine everything carefully; hold fast to that which is good.
First Epistle to the Church of the Thessalonians
The truth will make you free.
Gospel of John
First Epistle to the Church of the Thessalonians
The truth will make you free.
Gospel of John
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Post #7
There is a problem with the way this 'paradox' is stated.
There is confusion between a definition and an
existence argument. Proving that the said smallest
integer exists and defining it are not the same thing.
There is also a problem with the word 'define'. What does
it mean in this context? If the integer is to be completely
defined we should know the digits of that integer. If it
is loosely defined or merely shown to exist, then the whole
argument is suspect and open to endless debate concerning
linguistics and the meaning of words.
The 'definition' that is given is "not definable in under
eleven words". But this is not a complete definition. It
only expresses a property of the number. If we are allowed
such vagueness we can 'define' all integers easily:
Prime numbers in under eleven words is defined as-
"Zero, one or next prime."
All other numbers as-
"Product of subset of defined primes."
So, the argument is not logically airtight and
therefore cannot be seen as a true paradox.
Afterthought
Let's define Set E as "Integers definable in less than eleven words"
We need to apply the same degree of rigour to all integers discussed.
Otherwise we are just fooling around with the meaning of 'define'.
"not definable in under eleven words" is the maximal rigour required.
This translates to-
"Not in set E". That is sufficient as a definition.
We now define all those integers - one by one - that are not in Set E as
"Next smallest integer not in Set E"
This definition is less than eleven words so there are no integers
outside Set E. All are 'defined' by the required degree of rigour;
"Smallest integer not in Set E".
Since there are none all are in Set E and there is no paradox.
There is confusion between a definition and an
existence argument. Proving that the said smallest
integer exists and defining it are not the same thing.
There is also a problem with the word 'define'. What does
it mean in this context? If the integer is to be completely
defined we should know the digits of that integer. If it
is loosely defined or merely shown to exist, then the whole
argument is suspect and open to endless debate concerning
linguistics and the meaning of words.
The 'definition' that is given is "not definable in under
eleven words". But this is not a complete definition. It
only expresses a property of the number. If we are allowed
such vagueness we can 'define' all integers easily:
Prime numbers in under eleven words is defined as-
"Zero, one or next prime."
All other numbers as-
"Product of subset of defined primes."
So, the argument is not logically airtight and
therefore cannot be seen as a true paradox.
Afterthought
Let's define Set E as "Integers definable in less than eleven words"
We need to apply the same degree of rigour to all integers discussed.
Otherwise we are just fooling around with the meaning of 'define'.
"not definable in under eleven words" is the maximal rigour required.
This translates to-
"Not in set E". That is sufficient as a definition.
We now define all those integers - one by one - that are not in Set E as
"Next smallest integer not in Set E"
This definition is less than eleven words so there are no integers
outside Set E. All are 'defined' by the required degree of rigour;
"Smallest integer not in Set E".
Since there are none all are in Set E and there is no paradox.
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Post #9
mgb, I think you are missing the point of the paradox. I raised this paradox as an analogy to the ontological argument, not as a discussion of the paradox itself. It has been well discussed by philosophers of mathematics and its resolution is well understood.
I think that the word definition in the paradox should have its normal meaning. That is, to uniquely identify a specific item from a set of items.
The paradox is that if the smallest integer that cannot be defined in less than eleven words does exist, it can be defined in less than eleven words, those being "The1 smallest2 positive3 integer4 not5 definable6 in7 under8 eleven9 words10 . "
The parallel with the ontological argument has to do with defining God as the most perfect possible being. As you correctly point out, the difficulty with both arguments, is in ambiguity with regard to the word defining .
Anyway, is this argument somewhat moot? Is there anyone who seriously supports any flavor of the ontological argument for the existence of God?
I think that the word definition in the paradox should have its normal meaning. That is, to uniquely identify a specific item from a set of items.
The paradox is that if the smallest integer that cannot be defined in less than eleven words does exist, it can be defined in less than eleven words, those being "The1 smallest2 positive3 integer4 not5 definable6 in7 under8 eleven9 words10 . "
The parallel with the ontological argument has to do with defining God as the most perfect possible being. As you correctly point out, the difficulty with both arguments, is in ambiguity with regard to the word defining .
Anyway, is this argument somewhat moot? Is there anyone who seriously supports any flavor of the ontological argument for the existence of God?
Examine everything carefully; hold fast to that which is good.
First Epistle to the Church of the Thessalonians
The truth will make you free.
Gospel of John
First Epistle to the Church of the Thessalonians
The truth will make you free.
Gospel of John
Post #10
I think you're on the right track here. Even though I'm not a fan of Anselm's formulation, many (most? all?) of its critiques are hardly "airtight."mgb wrote:...The 'definition' that is given...is not a complete definition. It only expresses a property of the number...So, the argument is not logically airtight and therefore cannot be seen as a true paradox...