Is there an argument that God, as a self-proving entity, is necessary for logic? I recall hearing an argument for God (or, really, that being a non-Christian was equivalent to being a nihilist) that in order for logic to be valid, one could not have circular arguments, but since you can't prove your premises, all arguments had to be unfounded or circular unless you introduced God as some sort of self-proving entity (both premise and proof thereof).

Is anyone familiar with this argument?  I have forgotten the details (e.g. why it is okay for God to effectively be a circular argument but not for anything else to be) and have not yet been able to find it anywhere.  Does this argument have a name, and is it (or at least a summary) written somewhere?
I think you're referring to The Transcendental Argument for the Existence of God (TAG). It was first formulated by Immanuel Kant, but has been refined, disputed and defended over the years.

The argument

One formulation of the argument is,


  Prove A: The Christian God exists.
  Step 1 ~A: (Assume the opposite of what we are trying to prove): The Christian God does not exist.
  Step 2 (~A--> B): If God does not exist, then there is no intelligible experience since God is the precondition of intelligibility
  Step 3 (~B): There is intelligible experience (Contradiction!)
  Step 4 (~ ~A): It is not the case that God does not exist (Modus Tollens on 2 and 3)
  Step 5 (A): --> God does exist (Law of negation.)
  
  Source: Reformed Apologist (blog)  


The finer details of whether this argument actually is valid, of course, need a more thorough explanation. The argument isn't fundamentally based on inductive nor deductive, but  transcendental reasoning. Kant describes transcendental as follows:


  I entitle transcendental all knowledge which is occupied not so much with objects as with the mode of our knowledge of objects insofar as this mode of knowledge is to be possible a priori.


Thus to properly examine and understand the argument, I suspect a more in-depth studying is required. I'm not a philosopher and don't claim to understand TAG, so I'll just point you elsewhere:

Further reading

Butler, Michael R. The Transcendental Argument for God's Existence.