Communication is one of many things that sets us apart from the animals. Whereas they can barely communicate beyond basic "words" or actions, we take it to a whole new level, being able to write to each other.
Obviously, this does have its limits. It is possible to misinterpret even the clearest of texts, and that problem increases when the one writing it is a bad communicator. Let's look at an example phrase: "I didn't say we should kill him". Every word in this phrase has the potential to change the interpretation, depending on which word is emphasised. Observe:
I: Someone else may have said we should kill him, but it wasn't me.
Didn't: I'm denying I ever said "we should kill him".
Say: I may have implied it, but I didn't say it.
We: I could have said someone else should, but not us.
Should: I didn't specify it's something we should do, but maybe we could, or want to?
Kill: I could have said we should hit him, or bury him, or something like that.
Him: Wrong victim!
You see how that one little sentence is loaded with possible interpretations. 7 simple words, 7 majorly different applications. We need more context clues to correctly interpret it.
When we're talking about Scripture, we have a lot of context clues. The 66 books of the Bible contain more than 31,000 verses, many of them longer than our example sentence. It is very hard to interpret it wrong, but some might argue it is also hard to interpret it right. In fact, a great many Churches argue that without them, you cannot interpret it right.
The error in these claims is self evident. It is self evident from the fact the very purpose of communication is that it be interpreted correctly. It is self evident from the fact even the Apostles submitted to the test of Scripture. But most importantly, it is self evident from the fact that even those who claim it 1. don't believe it, and 2. don't act like they believe it until their interpretation is obviously wrong.
As a broken clock is right twice a day, so also are heretical denominations often right about theology. It takes no effort to convince a Roman Catholic, for example, that Mary was a virgin when she birthed the Lord, that Christ was crucified and raised bodily, or that God is triune. They affirm these things wholeheartedly, and can defend them from Scripture. Why? Because Scripture does teach these things.
What it doesn't teach, however, is the Perpetual virginity of Mary. Nor does it teach Purgatory. It is very explicitly against adding works to salvation. You will search in vain for references to a Pope. The list goes on, but the reliance on Scripture, sadly, does not. Where Scripture ceases to agree with the Catholic Church, Catholics cease to treat Scripture as the written work it is. Instead, they use a number of strange excuses (such as "we compiled the Bible") to try to claim they, and they alone, have the authority to interpret Scripture.
Now, I'll be the first to admit that some Scriptures are difficult to interpret. They may have more than one possible, and even more than one probable interpretation. However, language, by its very nature, is possible to interpret without "authority" to do so. Thus, we can hold Catholic feet to the fire here simply by refusing to grant, and yet also refusing to ignore, the Catholic claim to authority.
See, if the Catholic Church really does have the authority it claims to wield, it should be able to consistently and reliably prove the validity of their interpretations. Personally, I find this very easy, even without claiming there even is an authoritative body with the right to interpret Scripture for everyone else. I believe I am as qualified as any literate man to interpret Scripture, and yet when I believe someone is in error, I am able to show them why I believe they are in error without once claiming to be more than their equal.
The example I like to give is Isaiah 43:10: "Ye are my witnesses, saith the Lord, and my servant whom I have chosen: that ye may know and believe me, and understand that I am he: before me there was no God formed, neither shall there be after me." Three simple questions: Based on this verse, how many Gods were there, are there, and will there be? 1, 1, and 1. You can't have more than one God, the God, existing at any one time.
"Not so", says the Mormon Church. To them, there is an infinite regression of gods before God, and an infinite progression of gods, including those who are good Mormons today, into the future. How do they get around this verse? The only come back I've ever heard is when the Mormon missionaries I spoke with told me that He's only talking about gods we need to bother with.
Now, read Isaiah 43:10 again. You can also read it in other versions on Bible Gateway, completely free. Is there anything in there that fits the Mormon interpretation? Obvious answer: No. If God only meant "you never need to worry about other gods", He would have said "you never need to worry about other gods". Instead, He says "before me, there was no God formed". Other gods don't exist. Indeed, this is why the Trinity is so important. If there is only one God, yet the Father, Son, and Holy Spirit, are all identified as one God, therefore God must be triune in nature.
I believe I can read. I also believe a Mormon can read. I don't claim any authority over Scripture that a Mormon does not also have. Why, then, am I, as a Mormon's equal, able to say that a Mormon is wrong? Because the Bible is the authority. The Bible says what I say it says, not because I am the one who says it, but because I say it based on what it says. A Mormon, by contrast, is bound by the authority of his Church. They tell him there were gods formed before God, and that if he obeys their teachings, he will also be formed into a god, and so when the Bible says "before me there was no God formed, neither shall there be after me", the Mormon has no option but to interpret Scripture contrary to its obvious sense.
As the Mormon interpretation of Isaiah 43:10 is invalid, so also are the unique interpretations of every "authoritative" body usually invalid. The whole reason they rely on their claims to authority is precisely because they know that. All things being equal, their interpretations fall flat. This, in itself, proves their claims to authority false. Let's present a simple syllogism:
P1: If you have authority to interpret Scripture, your interpretations must be valid.
P2: Your interpretations are not valid.
C. You do not have authority to interpret Scripture.