Those claiming to be Christians will make the claim on the basis of what they believe the Scriptures teach. This is so, regardless of how far right or left on the spectrum of thought they find themselves. This is not to say that all interpretations are correct or that as long as they make the claim, they are fine. It's just a starting point.
Today, we might say, "the Scriptures teach" or something similar. This is essentially on par with the idiom, "It is written," recognized as the way Jesus answered His temptations (Matthew 4; Luke 4). The "writings" (i.e., Scriptures) were considered authoritative by Jesus and the Jews of His time. To appeal to what was written was to appeal to authority. Scriptures were considered God's word to man, and "Man shall not live on bread alone, but on every word that proceeds out of the mouth of God" (Matthew 4:4; Deuteronomy 8:3). Though the word of God was more than only what was written down (e.g., Jesus is the Word, the prophets, etc.), what was written down was nevertheless seen as God's word, and if God's word, then it carries the authority of God.
To say, then, that the Scriptures teach something implies that there is something authoritative about them and we should listen. We aren't Christians because we think some self-help book or blog said something important. Christians recognize that there is authority in the Scriptures because of that deeper-held belief that God is behind what is revealed (I Corinthians 2; II Timothy 3:16-17; II Peter 1:20-21). All of this seems simple enough, but clearly, there is more to the issue because we all know it is not good enough just to point to a passage and say, "See, this is what the Bible says." Anyone can do that, but if the passage is being taken out of context or misapplied, then we know there is a problem. Even the devil quoted Scripture to Jesus (Matthew 4:6).
Regardless of who it is interpreting Scripture, there is a basic process, often unspoken, employed by anyone who thinks Scripture is authoritative that cannot be denied without denying fundamental logic or sounding outright silly. This process involves looking directly at what is said, considering examples given, and then inferring from what is said and shown how important and applicable these matters are. Even those who fuss and pejoratively poke at CENI (Command, Example, Necessary Inference) do the same. At the end of the day, if they are going to say, "this is what Christians ought to believe or do," they will only get there through the same means. They will appeal to what Scripture says, to the examples provided, and through a reasoning process (inferring) come to conclusions that they think are important. So it is with everyone. Everyone! People will differ on outcomes, but there is no denying the process of how communication works. It astounds me when some try to deny it.
I've been told that when I speak of all of this in terms of "telling, showing, and implying," that I'm just repackaging the old CENI in new terms. First, I don't deny that this is basically true, though I do deny some of the baggage they attach to it (e.g., that it is a "Church of Christ hermeneutic"). I have long said that "Command, example, necessary inference" is a more formal way of saying that God tells us, shows us, and implies what He expects us to get — something every interpreter will necessarily have to agree with (try denying it). I prefer the latter terminology because I think the former was a little too narrow ("command" is narrower than "tell," and not everything told is a command). I have argued that, perhaps, had these matters been explained more in terms of basic logic and communication, maybe there wouldn't have been such a kickback against it all later. Now we find ourselves having to defend the simple and undeniable. Second, saying it is just repackaging doesn't deal with the issue. If that bothers someone, maybe that person can suggest a better way to communicate than through telling, showing, and implying. Maybe that one thinks that God has communicated in ways other than this. I don't know. When it's all over, though, I guarantee that the person disagreeing with all of this will go to the Bible and point to something that is said, or to some example, or infer something from what is said and shown that he thinks makes his point. Fussing about CENI is a red herring. It isn't the real problem.
Now, none of that is to say that all the particulars are worked out. What I'm defending is a process, not all of the conclusions that have been reached through the process by various interpreters. What people are really fussing about is not the process so much as whether or not some conclusions really are necessary, or whether or not some commands are still binding. If some think they can take the Lord's Supper on a day other than the first day of the week, they will argue that those who teach the latter are binding where God has not and have inferred what is not necessary. But they will still argue their position on what they think is to be inferred from the revealed information because they cannot bypass the process without just making things up out of thin air.
Let's take this issue into the area of grace and law. Some speak against those whom they think put too much stress on God's commands, saying that they don't say enough about grace. They think we put too much on authority and need to allow for more freedom based on grace. But here is the kicker: those who teach their view of grace do so by going to the text and arguing that it's what the authoritative text either states or implies. That is, they go right back to the "Tell-Show-Imply" process to prove their point about grace. To make their case, they must rely every bit as much on the authority of the text as those with whom they disagree. Authority is still at the foundation of any of these discussions. The only other option is to make things up out of the blue.
Further, grace is only possible when one is authoritative enough to grant that grace. Mark 2 shows that Jesus had the authority to forgive sins. Not just anyone can forgive the sins of others. This can only come from God's power. Therefore, to pit grace against authority is fallacious because to talk grace, we must necessarily confess God's authority to grant the grace on His own terms.
We know what we know about grace because of what the authoritative text tells us. We know what we know about God's commands because of what the text tells us. No one knows anything authoritatively apart from what is revealed in the authoritative text. It is certainly possible that we put more stress on one matter over another, and we may indeed fail in our teaching because we ignore what the text teaches. But the standard for judging any of this still needs to be the text and not our personal preferences.
Regardless of which position we take, the authority of the text must be key. Why? Because it is the revelation of God's mind. The only other option is to invent our own doctrines, and then where are we? We may disagree with each other on exactly what the text teaches, but there is absolutely no basis for unity when we give up the authority of the text. "It is written" needs to be the appeal. Without it, we will wallow in self-willed authority with no foundation for anything other than our own desires serving as the standard. I would hope that all of us would emphatically deny that alternative.