Seems like sound reasoning to me.
Of the leaders of all major religions, Jesus was the only one to claim divinity. Either he was mad, a liar, or he was who He claimed to be. Since no other actions indicate madness or bad character, we can assume the latter with moral certainty.
Lewis could have strengthened his argument with the following line of reasoning. First, without faith, one can accept the Bible as the best attested-to historical document of its era, with the existence of more than 500 copies dating from before the year 300 A.D. Within these recorded writings, Jesus made verifiable predictions, most notably, that he would establish a church against which the gates of hell would not prevail.
Looking at history, we can see that the Catholic Church has existed since Pentecost. This Church possesses an unbroken, non-contradictory body of dogmatic teaching, and an unbroken line of successors to Peter. There exists no comparable institution in world history, thus supporting Jesus' prophecy. We can then conclude, without resorting to fideism, that Jesus was who he claimed to be, and that the Cathoilc Church is the church that Christ founded. This Church affirms the inspiration of Scripture, and thus the validity of all of Jesus' statements. We reach these conclusions without resorting to circular reasoning. This proof is based on the simple premise that the Bible is an historical document.
As additional evidence of the nature of the Catholic Church, there exist today ongoing miracles, like the eucharistic miracle of Lanciano, that are associated with Christ's Church.
“Seems like sound reasoning to me.”
The reasoning ignores two utterly simple points that liberals would quickly make. First is the possibility that the historical Jesus didn’t say all the things that were attributed to him. Second is that brilliant insights are made by deeply flawed, self-important men all the time.
I of course believe Jesus is the Son of God. What I object to is the laughable reasoning. If you want to make an evidentiary argument based on historical analysis and so on, that’s one thing. But you could drive a truck through the holes in Lewis’s logic.