This blog post is going to be about a very particular style of argument that I have heard from the mouth of a Buddhist and I have read in C.S.Lewis's "Mere Christianity". I find the arguments interesting because they are essentially mathematical in nature. This makes them amenable to the rigorous treatment of mathematical logic.
Firstly a rough template of the arguments:
* A thing may be x like.
* Some things are more x like than other things.
* For this to be the case there must be a perfectly x like thing (call it X) which the other things are judged against.
* If X is perfectly x like then for any Y we must have that Y is less x like than X or that Y is in fact X.
This argument has been used with morality. Then the things in discussion are moral actions and they can be good or bad to differing extents. This argument can be used with minds which can be more or less knowledgeable than each other (or more or less enlightened than each other). The argument can be used with entities which can be more or less powerful than each other.
In each of these cases the arguments are meant to conclude respectively that there is a universal morality, that there is a state of universal enlightenment and that there is a unique all powerful entity (which I guess you hope is also good).
There are many problems with these arguments including:
A) The arguments need to show that something exists (or could exist in the case of enlightenment) in the world NOT as a mathematical abstraction (with the exception of the case of the universal morality). The argument above is an argument for mathematical existence not physical existence.
B) It simply is not the case that whenever you have many different options some of which can be ranked higher than each other that you must have a unique best option.
Here is a simple game called "My daddy's richer than your daddy!". Two players write down a number privately on a piece of paper. They show the numbers to each other. Whoever has the highest number wins. It is always better to write a higher number down but there is no highest number that can be written down. A religious person might object by inventing quantities greater than normal numbers. A mathematician cannot complain because that is essentially what mathematicians call ordinals. However, then we get the Burali-Forti paradox that there is no highest ordinal (I'm simplifying here for clarity) and we are left with a cast iron instance of this argument style failing. Notice that what we really need rather than a top number is a concept of quantity. The concept of quantity is not itself a number. In the same way to judge morality we need an ethical theory not a perfectly moral entity.
There are further problems including the problem that one might not be able to compare which of two things is more x like.
I think, however that what is really going on here is that people are leaving out assumptions that would make the arguments stand up properly. I shall give the simplest assumptions I can think of that would suffice to make the above argument work and then we can look at how appropriate they are in the above cases.
1) If we have a set of things that are x like there is something that is at least as x like as all of them.
2) There is a sensible way of ordering things as to how x like they are in the first place.
3) The quantity of objects we are considering is bounded (but possibly infinite).
The problem for this style of argument is that (1) and (2) are generally just as contentious as the results the arguments seek to prove. (3) is never really going to be true when the argument seeks to prove something interesting. In all the cases above (1) and (2) are contentious and (3) is false.
It is my opinion that there are other theological arguments that we would do well to examine from the perspective of mathematical logic.