Since we're talking about truth, let me be honest at the outset: I will be unable to prove ultimate truth satisfactorily under the rules of formal logic.
In the end, this assertion is rooted in faith, which is another word for trust.
But I think it's a logical faith. In Romans 12:2, for example, Paul talks about worshiping God with our whole lives, something I talk about here. One of our English translation, rightly, says that such self-surrender is our "spiritual worship." Literally, the original Greek says, logike latreia, meaning logical service. There is a sense in which, after a consideration of the facts, our experiences, and the well-documented human penchant for worshiping something, that it makes sense that we yield to the greater truth of God.
Let me give you what I consider to be two logical indicators for the existence of ultimate truth.
First: There is the amazing consensus that exists throughout history across varying cultures and belief systems about what is right and wrong. C.S. Lewis writes this:
If anyone will take the trouble to compare the moral teachings of, say, the ancient Egyptians, Babylonians, Hindus, Chinese, Greeks, and Romans, what will really strike him [sic] will be how very like they are to each other and to our own...I need only ask the reader to think what a totally different morality would mean. Think of a country where people were admired for running away in battle, or where a man felt proud of double-crossing all the people who had been kindest to him. You might just as well try to imagine a country where two and two made five. Men [sic] have differed as regards what people you ought to be unselfish to--whether it was only your own family, or your fellow countrymen, or everyone. But they have always agreed that you ought not to put yourself first. Selfishness has never been admired...It seems that all human beings sense the existence of an ultimate truth toward which it is best to conform one's life, although we all fail.
The Bible says that God's truth is written into our hearts and we sense its call on our lives, although we have the freedom to rebel and with our psyches clouded by the distortion of self that we call sin--a condition before it results in actions, we try to bury it.
Paul, in the New Testament book of Romans, speaks of "the wrath of God," which is not an act of punishment by God, but the consequence of routinely ignoring the ultimate truth we observe in daily life. (Wrath then is, as I tell my Catechism students, like what happens if we stick our fingers in an electrical socket. The socket has nothing against us. But the electrical system is set up so that if we "cross the line" of God's laws for good living, there are consequences. They may not be immediately evident. But there are consequences. More on that presently.)
For the wrath of God is revealed from heaven against all ungodliness and wickedness of those who by their wickedness suppress the truth. For what can be known about God [the Author of truth] is plain to them, because God has shown it to them. Ever since the creation of the world His eternal power and divine nature, invisible though they are, have been understood and seen through the things He has made...(Romans 1:18-20)The moral system, founded on twin pillars, which appears inbuilt in the human psyche is one which seems to pervade all human attempts to live good lives or establish virtue. Those twin pillars, summarized not only in the Great Commandment articulated by Jesus, but also in the two tables of the Ten Commandments from the Old Testament and in the two sacraments instituted by Jesus, are these: love God and love neighbor.
Even moral systems not founded on monotheism acknowledge the need to bow to Someone or something greater than self.
Even moral systems that differ on the implications of neighbor love agree that it is an ultimate value, the violation of which is a great wrong.
This leads to the second logical indicator for the existence of ultimate truth. It's quite simple: God has revealed it and has done so ultimately, in the Person of Jesus Christ.
Human beings have a desire to bury the truth that there is ultimate right and wrong and that it has an Author worthy of our surrender. After all, if we can do that, then we have license to be our own gods, accountable only to ourselves.
What child hasn't fantasized about or actually tried, gorging on all the cookies in the jar when Dad or Mom wasn't looking? (After all, they reason, I want those cookies!) What married man or woman hasn't fantasized or actually tried, taking sexual intimacies from outside their marital relationship? (After all, they reason, I want those cookies!) This inclination is really a desire to bury God and, in the words of Genesis, "be like God."
If one is successful in burying God and the ultimate truth around which He says life is good, we erase wrath (the consequences of acting like gods to ourselves), or so we think. (It undoubtedly is why the world killed and buried Jesus. Fortunately for us, Jesus wouldn't stay dead. But I'm jumping ahead of myself.)
God, being gracious and as the Bible puts it, "slow to anger and abounding in steadfast love," is unwilling to see us victimized by our worship of self. God refuses to give us up to wrath without fighting for us. That's why Jesus, the ultimate revelation of God, came into the world. The wages of sin may be death, as Romans 6:23 says, "but the free gift of God is eternal life in Christ Jesus our Lord."
Jesus reveals ultimate truth. He shows us what ultimate truth is by the life that He lived while walking this earth. It's a life of absolute love for God and neighbor. In this life, Jesus accomplished something which no human being, in spite of the sense of right and wrong that courses through the veins of all of us, had previously been able to do and something that nobody since has pulled off either. His is a life so perfect that He was able to act as the sacrifice, the payment, for our sins and so destroy the power of wrath over the lives of those who turn from sin and entrust their existences--past, present, and future--to Him.
"I am the way, the truth, and the life," Jesus tells us. (John 14:6)
Although I believe that these two things--the inbuilt sense of right and wrong and the Person of Jesus--are logical indicators of the existence of ultimate truth, I realize that they don't prove my assertion.
But this is what I know: If you will allow yourself to surrender to Jesus Christ, you will also believe in ultimate truth. I dare you to try that.
It's what I did back in my atheist days. I found my mind, heart, and will stirred by something as I spent time with these Christians, everyday people who never claimed to be better than others, but who found hope, strength, and encouragement in Christ. I was further stirred as I started to read the Bible in a translation that was accessible to me.
Against my will, against my previous derision of faith, of Christians, and of the weakness of those who surrender, I found myself loving Jesus Christ.
I finally said something like, "God, I don't know if You're there and I don't know exactly what's happening to me. But if Jesus is Who You are, I want You in my life. I don't want to the be the ultimate authority of my life because I know that I'll only screw up."
I haven't always obeyed God's ultimate truth. I do things that deserve wrath. (In fact, one of my frequent daily prayers is, "Thank You, God, for not killing me as I deserve.") But I have found God to be just like He reveals Himself to be through Jesus: a Truth of rightness, love, charity, and forgiveness on Whom I can build my life!
I hope that this helps.