Well not entirely, it's also about evidence and rhetoric. But logic is the hinge around which the chapter swings.
Jesus begins the chapter by comparing himself to a good shepherd, who "lays his life down" for the sheep, but also by comparing himself negatively to the hirelings, who run away when the wolves and bandits come, and positively to the gate by which the sheep come and go. "My sheep recognize my voice."
This metaphor is a logical argument as well as an observation. I often notice that not just a voice pattern, but a pattern of thought and expression, can be recognized from different writers. (Just the other day, reading an article at National Review without noting the byline, I said to myself, "This is from Jay N --", both because of its logical weaknesses, and because of its tone. And it was.)
The difficulty here is that Jesus often sounds rather different in John as opposed to the Synoptics. Yet there are underlying commonalities . . . People making up words for Jesus, as in the Gnostics, never approach the sheer genius, still less the personality, of the real Jesus, for example.
Then a debate breaks out among onlookers. "He's a madman. He has a devil! Why listen to him!"
Others reply along two lines of evidence which Pascal and other modern apologists also follow:
"These are not the words of a madman." No, Jesus' words are at a bare minimum those of a great genius, and not an evil genius, either -- of a transcendent sage. Evidence One.
"Can a devil open the eyes of the blind?" A rhetorical question, perhaps suggesting, "And even if he could, would he?" Jesus' actions show he has power. And furthermore (this goes much deeper, see my Jesus and the Religions of Man), the nature of the supernatural works recorded in the gospels is not that of magicians or even the sort of wonders which modern skeptical philosophers like Stephen Law often compare them to. It is not just that Jesus has extraordinary power, it is that in his miracles the divine nature of that power is revealed. Shree Baba vomiting out stone phallic objects may be the work of a devil or a con man. But there is a purity in Jesus' miracles which set them apart from such lame analogies. Evidence Two (or Two and Three).
Now Jesus is in the temple, and his critics surround him and demand, "Stop playing around! Tell us if you're really the Messiah."
And Jesus replies, "I told you! And if I hadn't, these works I am doing should give you a clue! But you're not among my sheep. And BTW, I and the Father are one."
The critics pick up rocks to stone Jesus, not because of his good works (which they do not deny), but because of this alleged blasphemy. (Here they are, at least, making a logical distinction.) Jesus responds with a bold and complex bit of logic, appealing to the authority of the Bible: Psalm 82 says, in an interesting rhetorical twist, 'you are gods.' Why, then, deny the title "Son of God" to one who does works such as those you see me do?
Liar, Lunatic, or Lord. That's the problem for Jesus' listeners. But the first two options are rendered problematic both by Jesus' miracles, and by his transcendent teachings. (Could this be an Undesigned Coincidence? Jesus' teachings in the Synoptics are often even more luminescent. Sermon on the Mount would make this problem especially piquant.)
After Jesus escapes a stoning, John points out that "many" followed him, because while the Baptist "did no miracle, everything he said of this man is true." So there's a third evidential thread, also noted by Pascal: predictive prophesy. This is one of the keynotes of the gospels: Jesus fulfills the promises not only of John the Baptist, but of the prophets and ancient Hebrew writings in general. (I expand that to Chinese writings, etc, in True Son of Heaven, and to Indian, Greek, and Norse writings, in How Jesus Passes the Outsider Test.)
John 10, in short, is a remarkable unity of Aristotle's elements of rhetoric: ethos, logos, and pathos. But especially the first two.