If we want to know what the Bible says about an issue, what methods can help us to find all the relevant scriptures? Particularly if someone is looking for a "clear, unambiguous, direct statement in Scripture" (source). A sola scriptura perspective is sought.
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The following tools are good for determining what the bible says about a matter. Take, for example, this question about pets going to heaven. If you wanted to know the answer, you should consult:
1. A Concordance
The first tool that just about any pastor is going to use to answer a question such as this is a concordance. A concordance is nothing more than an index of words in Scripture, arranged in dictionary form. The best concordance, of course, is the one in our heads. The next best one is one in a book. The classic work here is Strong's Concordance, which originally indexed the KJV bible. Other editions look at the Greek and Hebrew words underneath, or use other tricks to better index the words. Using google, by the way, is another easy way to find verses you sort of know, but don't know where they are. Searching for 'pets' you will find nothing. Searching for 'dogs,' you will find that it is usually an insult to a person, not anything about the animal. (Clearly, the writers misguided cat-o-philes, I suppose. ) Regardless, the point is that if you look up any of the words you might think are useful, you just won't find anything.
One word of warning: It is tempting to think a concordance can answer everything. It can't. And, references always need to be read in context. But, it is a good place to start.
2. A Topical Dictionary of the Bible
Sometimes, a word might not appear in Scripture, but an idea does. The word 'Trinity' for example, appears nowhere in Scripture - BUT the idea certainly permeates the whole. For concepts, a topical dictionary like Nave's or one of these are useful.
In these dictionaries, key words will be explored, and centuries of Christian knowledge will be summarized, as a useful starting point for where you are heading. Again, you need to read what they write in context, but it will show you where to focus your study.
Nave's lists Pestilence (hey some people keep spiders as pets!), Peter, and Pethiah - so, yeah, maybe not.
3. A Systematic Theology
Finally, for larger topics, one should consult a systematic theology. Several (if not most!) theologians will write books that seek to put a propositional order to everything in Scripture. Typically, this means they will start with questions like "The Nature of God," "The Nature of Man," "The Nature of the Bible," etc... If you know what transubstation or predestination ante praevisa merita is, you jump straight there and learn a lot - but more likely, you need to be exposed to the terms and directed as to why these things are important. That's what a systematic theology would do for you.
In this case, looking at the nature of the soul, the curse of the law, and the need for redemption only when there is transgression, one could begin to formulate questions around whether or not animals have souls, whether or not the "image of God" is a necessity for heaven, and the like. But that takes work!
And, that ultimately, is where all of the tools need to be placed in perspective. Each of these tools is a map to a microscope. If you don't have an overall picture, these tools won't provide a lot of light. But, with regular reading, the overall map takes shape and you will be able to find these answers.
There are no shortcuts to the process you appear to be asking about. I say that through personal experience, and from many Bible enthusiasts I have known and collaborated with over the years.
There are some excellent aids to enhance your Bible study free on the internet, some are online Bible study, and for the novice there are some excellent interactive sites available. A simple Google search will give you thousands of sites for each of these, but I suggest that you add any qualifiers you may think appropriate, otherwise the return list can be quite extensive (as in the millions).
For someone who is fairly familiar with Scriptures, as I feel I was, after having read the Bible from Genesis through Revelation approximately ten times maybe more. I have downloaded two excellent free Bible Study Programs to my computer and use both extensively in my study and for research for this site. Even though they are free I have found them so valuable that I have donated to keep them free.
Along with the study programs there are many Bible translations, commentaries, and books that can be downloaded for the programs.
There are also some useful aids that can be downloaded to your computer to enhance your study, one that I have found very helpful is 'What the Bible says. It is free to try, and very inexpensive to buy. This program has a list of topics on the left and on the Right it lists all of the Scriptures which address that topic.
Absolute certainty probably isn't attainable (whether absolute certainty is or should be considered attainable by humans at all might be a worthwhile question), but as brought up in Affable Geek's answer to the inciting question, you can get pretty close through use of a good concordance and topical Bible dictionary.
Concordances are lists of words used. You can think of them as a giant index at the back of a book, or the Bible reorganized so that each word is alphabetically ordered. Often, a concordance will distinguish between the words in their original language and assign a number to the original term. Here's an example of a concordance entry for the Hebrew word that's often translated as "stranger" or "foreigner." Keep in mind that when you're looking based on English words, there is a disconnect between the English translation and the original language, and sometimes the same work in (for example) Greek can be translated different ways, even in the same Bible translation. A concordance in English is going to be based on a particular translation/version of the Bible, too, so make sure you pay attention. A New International Version concordance is not going to coincide as well with a King James Version of the Bible! (Side note: A lot of the work of a concordance is crunching through the data. There's been some really good work on getting computers to help out with that.)
A topical dictionary, on the other hand, is an attempt by its creators to lump together passages by topic. There's more interpretation done by the creators of the dictionary here. I've seen a topic like "beauty" that only included references to physical human beauty, for example.
I'll also throw in that a major reason for academics to publish and circulate papers comes from an older tradition, seen in seminaries as well as classical Greek dialogues, of viewing any particular argument as part of a larger discussion, in which we engage with others, trusting that "Iron sharpens iron, and one man sharpens another." (Proverbs 27:17).
A program which you might try is http://bibleresources.org/bible-answers/ it has research in a variety of Bible including Hebrew and Greek plus several other languages