How can a Christian make a case for the inerrancy of Scripture? I'd like to sketch some apologetic strategies.
1. Before doing that, permit me to draw a some preliminary distinctions:
i) What makes something true may be different from how we prove it true. Suppose I see a sign that a pet owner posted about their missing dog. It has a description. Let's say it's a border collie. Now, it can't be a collie unless it's a dog, and it can't be a dog unless it's a mammal. Suppose I saw missing the dog, and I notify the owner. However, I don't need to first prove that the animal is a mammal, then prove that the mammal is a dog, to be justified in believing that I saw a border collie fitting that description. So the logical or ontological order in which something must be the case needn't mirror the order in which we prove that it's true. A multi-staged argument may be artificial in that regard.
ii) By the same token, we can believe something for different reasons than the reasons we give to justify our belief. Or there may be some overlap.
For instance, I may believe that a certain high school once existence, even though the location is now a vacant lot, because I attended that school when I was a teenager. That may be all I know, but that's that's all I need to know. And it would be reasonable for a second party to accept my testimony.
On the other hand, I could prove the existence of the high school by producing photographs or public records. I don't need that kind of evidence for me to know or be justified in my belief. But were I proving it to someone else, who didn't have my firsthand knowledge, I might resort to corroborative evidence for his sake.
iii) By the same token, a person might believe in Christianity for very personal reasons. Maybe he has firsthand experience of a Christian miracle or answered prayer to Jesus. Or a trusted friend or family member relayed to him an experience in kind.
When, however, we make a case for Christianity, we generally confine ourselves to publicly available evidence, since an outsider isn't privy to our personal experience. Hence, I may have greater warrant for my faith than the evidence I adduce in mounting an argument for Christianity, because I'm confining myself to kinds of evidence accessible to outsiders. Even though I'm using probabilistic arguments, that doesn't necessarily mean the basis for my own belief is reducible to the evidence I present to persuade others.
2. The inerrancy of Scripture is logically grounded in the inspiration of Scripture.
3. Apropos (2), the argument from prophecy is an argument for divine inspiration. And that's a paradigm of Biblical inspiration. I've discussed how we can justifiably extrapolate from the prophetic paradigm to the inspiration of Scripture in general:
And Michael Kruger comes at the same issue from a different angle:
4. Likewise, we can argue for and from the the general historical reliability of the Gospels. That, in turn, ratifies the inspiration of the OT via the testimony of Jesus. And the OT is a paradigm of inspiration that applies perforce to the NT (see above).
At this stage of the argument, an apologist is not assuming the inspiration or infallibility of the Gospels. Rather, that will be the conclusion of his argument.
Keep in mind that this is a logical strategy. A stepwise argument. It doesn't mean the apologist is in a state of suspense regarding the outcome of his argument. He isn't waiting to find out what the answer will be. He believes in the inspiration and inerrancy of Scripture going in. So this is just a way of formulating the argument incrementally. We leave the inspiration and inerrancy of Scripture in abeyance until we reach the conclusion for the sake of argument.
5. Apropos (4), the Gospels give redundant testimony regarding the person and work of Christ. So the conclusion doesn't turn on any particular verse. No single passage is crucial to the conclusion. The Gospels could be fallible, but even if (ex hypothesi) they contain mistakes, there's a wide margin for error giving redundancy and multiple attestation.
6. Apropos (5), if the Gospels bear witness to Jesus as God Incarnate, and if Jesus bears witness to the historicity and inspiration of the OT, then we can derive the inerrancy of Scripture through a chain of inference: general reliability of the Gospels>identity of Jesus>dominical attestation.
Notice that I'm not inferring inerrancy directly from general historical reliability. Rather, the argument is indirect. If it's demonstrable that the Gospels are generally historically reliable, then that affords a reliable account of who Jesus is and what he thought of the OT. That in turn validates the OT. And that in turn validates the NT, inasmuch as the NT is a continuation and completion of the OT paradigm.
7. On a related note is the argument from miracles. If the NT is a trustworthy account regarding the miracles of Jesus and the apostles, then that's authenticates the divine mission of the messenger.
8. There are multiple lines of evidence for the historical reliability of the NT:
i) Undesigned coincidences
Suppose a biographer interviews friends and relatives of the person whose life he's narrating. What one informant says will often overlap with what another informant says. Not only is that multiple attestation, but in some cases what one informant says will fill in gaps left by what another informant says. So you have independent, interlocking lines of evidence. And that phenomenon has been documented in the Gospels and the record of Paul:
Lydia McGrew, Hidden in Plain View: Undesigned Coincidences in the Gospels and Acts (DeWard, 2017)
ii) Archeological corroboration.
The Gospels often correspond with extrabiblical evidence for what was going on at that time and place. Some critics attempt to dismiss this by saying that's consistent with historical fiction, where a narrator will sprinkle his account with enough realistic details to give it an air of verisimilitude.
However, that objection is at cross-purposes with another objection critics raise: namely, their claim that Gospels were written by narrators far removed from the time and place of the events they purport to record. But the critic can't have it both ways. Either the narrators were out of touch with time and place of Jesus, in which case they'd be too ignorant to write good historical fiction–or else they were conversant with the facts on the ground, in which case you can't chalk up the corroborative evidence to the artifice of historical fiction.
On a related note is the historicity of Acts. Because the historical purview of Acts is more cosmopolitan than the provincial focus of the Gospels, it intersects with more 1C history. As a result, the Book of Acts enjoys greater historical corroboration. Yet the author of Acts was the same person as the author of Luke's Gospel. That goes to show that Luke is a conscientious historian with many informants.
Monographs that collate this kind of evidence include:
Paul Barnett, Finding the Historical Jesus (Eerdmans, 2009)
Richard Bauckham, Jesus and the Eyewitnesses (Eerdmans, 2nd ed., 2017)
Craig Blomberg, The Historical Reliability of the New Testament (B&H Academic, 2016)
Craig A. Evans, Jesus and His World: The Archeological Evidence (WJK, 2012)
Colin Hemer, The Book of Acts in the Setting of Hellenistic History (Eisenbrauns, 1990)
Craig Keener, Acts: A Exegetical Commentary, 4 vols. (Baker Academic, 2012-2015)
Stanley Porter, John, His Gospel, and Jesus (Eerdmans, 2016)
Bruce. W. Winter, The Book of Acts in Its First Century Setting, vols. 1-4 (Eerdmans)
9. Another line of evidence is the traditional authorship of the Gospels. That includes both internal and external lines of evidence;
i) Patristic testimony.
ii) Textual evidence for the originality or extreme antiquity of the titles (e.g. Martin Hengel, Richard Bauckham).
iii) By process of elimination, the narrator of John's Gospel is an eyewitness disciple–in all probability the Apostle John.
iv) If Mark was a younger contemporary of Jesus, as well as a native of Jerusalem (Acts 12:12), then it's likely that his account is based in part on firsthand observation.
10. Assuming that Matthew and Luke make use of Mark, that gives us an opportunity to check on how they handle source material. By comparing Matthew and Luke with Mark, we can see that Matthew and Luke are quite conservative in their use of Mark. Very faithful to their source. So that's a good reason to think they are trustworthy when they supplement Mark with independent sources of information.