For the moment, let me leave out the fact that his denial of the existence of God is, in itself, a "personal idea" that is "ungrounded". I tried to make the point for him that his thoughts were never solely "rational" (relying only on technical logical thought) since he had his own presuppositions (whether he knew it or not). It was this point that he could not get. His obsession with the idea that "rational thought" was always perfectly neutral had blinded him to the limitations of "rational thought". Idols tend to do things like that.
It is true that we are supposed to use our rational thought processes when we think about things. The Church has never denied this. Yet, it is also true that our minds are not perfectly logical machines (sin effects every part of our being). We can (and often do) make mistakes. As I have said before, science (supposedly a "perfect" exercise in rationality) changes so often that it should be evident that it cannot be relied on to help us with anything more than a "good guess" (and those are often errant!).
In one sense it would be completely accurate to say that I am not exactly "rational" because I do not use just my rational thought to think, but also my faith, which is not necessarily "rational". When we think of what "rational" means, it is evident why atheists, pagans, and heathens would want to imagine that "rationality" has such super powers. Yet, for those of us who know that there is more to the world than what we can see with our eyes, we realize that rationality is severely limited. We realize that faith must come into play.
So my "irrationality" is something that I will wear proudly. I refuse to limit myself to a fallen mental process that is insufficient to discover all that this created universe has to show us. Instead I choose, willingly and gladly, to submit my "rationality" to what faith can discover. As St. Paul said "We are fools for Christ’s sake" (1 Cor 4:10). I will happily be considered a "fool" if it means that I have refused the wisdom of this world for the sake of the perfect wisdom and knowledge of Christ my Lord. Let us be thankful that the world considers us to be "fools".
When I came to you, brethren, I did not come proclaiming to you the testimony of God in lofty words or wisdom. For I decided to know nothing among you except Jesus Christ and him crucified. And I was with you in weakness and in much fear and trembling; and my speech and my message were not in plausible words of wisdom, but in demonstration of the Spirit and of power, that your faith might not rest in the wisdom of men but in the power of God. Yet among the mature we do impart wisdom, although it is not a wisdom of this age or of the rulers of this age, who are doomed to pass away. But we impart a secret and hidden wisdom of God, which God decreed before the ages for our glorification. None of the rulers of this age understood this; for if they had, they would not have crucified the Lord of glory (1 Corinthians 2:1-8).