What is the difference between the two?
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Common Philosophical Beginnings
A full answer will require us to take a historical detour to 15th century Europe. Until that time, hermeneutics followed fairly straightforwardly the traditions of the church handed down from one generation to the next. The guiding principle was apostolic succession so the first step to discovering what a passage in the Bible means would be a search of early church fathers. Since the allegorical approach of the Alexandrian school gained dominance in the European church, christological interpretations dominated Western Christianity's understanding of the Bible.
Further development of the allegoical technique produced the Fourfold hermeneutic:
Of the four, the final three depended heavily on traditional interpretations. Meanwhile, Jewish hermeneutics exhibited a similar fourfold scheme around that time and, of course, also put great weight on precedent. This particular combination of tradition handed from one generation to the next and the metaphysical view that everything meant more than what is easily observed were the core of medieval philosophy.
Exactly how philosophy moved away from tradition and allegory is more of a historical question than a hermeneutical one, but a good case can be made that the first breech came when the original works of Aristotle were rediscovered. Up to that point, most scholars had access to ancient Greek philosophy as distilled through Boëthius. When it was discovered that the ancients had something to offer independent of what was passed down via tradition, the path was laid for Biblical hermeneutics based solely on sensus historicus.
Divergence of Basic Assumptions
Once you begin questioning the veracity of tradition, the first issue must be how far back do you go to get to the truth. The dividing line between "historical-grammatical hermeneutics" and "historical criticism" lies along the location where we can be certain truth may be found. Two schools emerged:
From these labels we can glean that the Historical-Grammatical method holds sway among Protestants and the Historical-Critical method is most often seen among secular scholars. Skeptics included Lorenzo Valla, René Descartes, Desiderius Erasmus , and Baruch Spinoza, while the reformers were men like Martin Luther, John Calvin, John Knox, and Huldrych Zwingli. All were opposed to the traditional hermeneutical approach, but for divergent philosophical motivations. Generally speaking the skeptics were committed to a rationalistic program and the reformers blended rationalism and church traditions.
Historic Criticism's Assumption
The assumption of the historical critic is: "Reason and reason alone is the source of all knowledge." Therefore, anything that cannot be directly accessed by pure reason is up for examination. As a practical consequence, even the texts themselves are open to suspicion. For this reason, critics have innovated new subdisciplines such as form criticism, source criticism, redaction criticism, and textual criticism. In theory, the Biblical texts ought to be interpreted using the same tools used for any other text—sacred or secular.
The Historical-Grammatical method employs a wider range of assumptions that are summarized by Raymond F Surburg in "The Presuppositions of the Historical-Grammatical Method as Employed by Historic Lutheranism":
(I leave off the last 8 presuppositions that seem to be specific to a certain branch of the method: namely Lutheranism. That said, many of the remaining assumptions are ubiquitous within the Protestant movement.)
It's no longer accurate to identify Historic Criticism with philosophical skeptics or the Historical-Grammatical method with Protestants. Each has achieved some level of independence as a general approach to interpreting the Bible. For instance, with the modification of point 1 above to:
I've been able to successfully use the inductive variation of the method with non-believers. Similarly, many believers (including Desiderius Erasmus, Lorenzo Valla and René Descartes listed above) have used the critical method without harming their own faith.
In the comments to this answer, Bruce Alderman notes:
I will add N. T. Wright, who uses all available critical tools in his works such as The Resurrection of the Son of God and Judas and the Gospel of Jesus, is a former Bishop of the Church of England.
It is entirely possible to use the different approaches without buying into the modern versions of the philosophies each was born from. Philosophy and Biblical Hermeneutics have drifted into different spheres over time.
While the two methods share a common origin as a reaction to what Luther called the "wax nose" of allegorical intrepretations, they differ on their commitment to the text as Scripture: Critics tend to take a wait-and-see attitude while reformers are philosophically committed to truth of the Bible.
Pre-Enlightenment: The questions of how one should interpret Scripture, and how focused the interpreter should be on the historical details of Scripture have been asked for thousands of years. None of these discussions are new to the modern era.
Enter "Historical-Critical": The Historical-Critical method was one of the results of the Enlightenment and its push toward carefully examining the evidence in order to determine truth -- and against blindly accepting dogmatic claims. Essentially these scholars would want to encourage a careful examination of historical evidence.
What "Historical-Critical" came to be associated with: The Historical-Critical method could be (and was) employed by both Christians and non-Christians. For many non-Christians, the result was often the discrediting of the accuracy of Scripture. For some Christians, the result was, at times, an overemphasis of historical details and an underemphasis on the message of Scripture. There do seem to have been some Christians who learned responsibly and found value in Historical Criticism, but in general, "Historical Criticism" came to be associated (especially amongst Christians) with a rejection of the authority of Scripture.
Enter "Historical-Grammatical" The Historical-Grammatical method seems to have been a reaction to what "Historical Criticism" was becoming. (In other words, the "Historical-Critical" label seemed to have been hijacked by the skeptics, so conservative Christian interpreters needed a new label to call their own.) This method was originally an attempt to focus on the meaning of the text (hence, "grammatical"), with an acknowledgment of the historicity of the text (hence, "historical").
What "Historical-Grammatical" has become: Modern conservative Christian interpreters will almost always identify themselves as "Historical-Grammatical" interpreters -- despite the wide variety of modern Christian views on hermeneutics. Many conservative Christian interpreters are now finding value in exploring the extra-Biblical history surrounding the events of Scripture. For example, this might help the interpreter understand the context of a story, or literary references, or genre, etc. In some cases conservative Christians are even reassessing their interpretations based on external evidence. As a result, the word is now used in a slightly different way than it was originally.
There are Christians today who will encourage "Historical Criticism", and there are "Historical-Grammatical" interpreters today who will question the reliability of Scripture. It just depends on who you are talking to and what they mean by their usage of the words. But, in general: