According to Euclid's "first common notion", "things which are equal to the same thing are equal to each other." This post examines two very different cases in which the same (fictional) person is referred to as both a "Witch" and as a "wise woman".
The self-evident conclusion that we can draw from these examples is that if the same person is being referred to as both a "Witch" and a "wise woman", then these two terms are synonymous, or at least they overlap significantly in their meanings.
An important consideration when applying the transitive property of equality in the realm of semantics in this way, is that one must be certain that the two labels are being applied not only to the same person, but to that person acting in the same capacity. Just because one person can be described as both a truck driver and an expert marksman does not mean that "truck driver" is synonymous with "expert marksman". However, it is obvious that "truck driver" and "lorry operator" are synonymous, because they do in fact refer to the same person in the same capacity, and the same is true of the phrases "expert marksman" and "very good shot". It will be shown in both of the examples below that the same person acting in the same capacity is referred to as both a Witch and as a wise woman (and in the second example, that of the legend of the Lambton Worm, she is also referred to as a "white Witch" and as a "sibyl").
The Witch of Brentford in the Merry Wives of Windsor
Fast forward to Act III, Scene III, in which Falstaff believes he is having a secret rendezvous with Mrs. Ford. Suddenly, Mrs. Page "arrives", although in fact she has been there all along, and announces that Mr. Ford is on the way, and in a fury at the news of his wife's dalliance, which he has discovered. The merry wives convince Falstaff to hide in a laundry basket before he is discovered, and then they proceed to cover him in filthy laundry, and then they have the servants dump the contents of the basket into a nearby muddy creek.
The following day, Falstaff has once more been lured to yet another unsecret assignation, which is interrupted by the announcement that master Ford is once again on his way. Falstaff refuses to get into the laundry basket again, and so it is suggested that he disguise himself as the maid's Aunt, visiting while on errands, and in that way to make his escape right under Mr. Ford'd nose. What Falstaff doesn't know is that master Ford despises this particular woman, whom he is convinced is a Witch, and has sworn to beat her if he ever sees her again.
At this point it is important to focus in on the repeated use of the word "witch" to refer to the maid's Aunt, who is also known as "the old woman of Brentford". First, Mrs. Ford informs Mrs. Page, behind Falstaff's back, that her husband "cannot abide the old woman of Brentford; he swears she's a witch." A few lines later, Mrs. Page says, "let's go dress him [Falstaff] as the witch of Brentford." Then when Mr. Ford arrives and is told that the maid's Aunt is in the house he flies into a rage and declaims:
A witch, a quean, an old cozening quean! Have I not
forbid her my house? She comes of errands, does
she? We are simple men; we do not know what's
brought to pass under the profession of
fortune-telling. She works by charms, by spells,
by the figure, and such daubery as this is, beyond
our element we know nothing. Come down, you witch,
you hag, you; come down, I say!
At which point he begins to thrash Falstaff, believing him to be "the witch of Brentford."
Falstaff eventually manages to escape and retreats to his room at the Garter Inn, still wearing his "disguise". When he later exits his room, in his normal attire, he is questioned about his lady friend, who was seen entering Falstaff's room just a little while ago. Sir John is asked by Simple (a servant) "was't not the wise woman of Brentford?" To which Falstaff responds, "what would you of her?" And then Simple explains that his master has pending business with the wise woman, concerning the identification of a suspected thief.
Not only does all of this very clearly attest to the equivalence of "witch" and "wise woman" in late 16th century English (The Merry Wives was probably written around 1598), but in addition, the Bard was kind enough to provide us with very telling details about the nature of the magical services that the Witch/wise-woman in question provided to her clients. Mr. Ford says that the "witch of Brentford" was involved in "fortune-telling", and that "she works by charms, by spells, by figure ...". Simple, on the other hand, states that the "wise woman of Brentford" has been contracted by his master for her ability to magically identify thieves. Very significantly, neither when she is referred to as a "witch" nor when she is referred to as a "wise woman" is there any mention of her using her reputed magical abilities to cause harm to anyone.
The Lambton Worm
The earliest telling of the legend of the Lambton Worm on record appears to be the 1820 prose version attributed to Robert Surtees (http://www.washingtonlass.com/LambtonWormSurtees). According to this version, when John Lambton had already failed several times to defeat the Dragon, he decided to "add policy to courage", and, to accomplish this he "went to consult a witch or wise woman" and thanks to "her judicious advice" the young Lord Lambton was able to finally slay the Dragon. Surtees' version of the story was first published in the 1820 volume of The History and Antiquities of the County Palatine of Durham (see previous link). It also appeared later in An historical, topographical, and descriptive view of the county Palatine of Durham in 1834 (link), edited by Eneas McKenzie and Marvin Ross, and also in Surtees' Memoir published in 1852 (link). A very interesting but brief background piece on Surtees and the legend was published by The Northern Echo on march 24, 2011: The Historian and the Lambton Worm. According to the story, Surtees first heard the tale from a woman with the rather suggestive name of Sybil Elizabeth Cockburn. At the Herrington-Heritage.Org.Uk website there is even more background provided on Surtees in an article titled The Surtees Connection.
In 1835, the story of the Lambton Worm was included in William Andres Chatto's Rambles in Northumberland and on the Scottish Border (http://books.google.com/books?id=jBcHAAAAQAAJ). Chatto's version also makes reference to "a witch, or a wise woman".
Next comes a versified version of the story that appeared in Tait's Edinburgh Magazine in 1840 (http://books.google.com/books?id=tS0tAAAAYAAJ). In that version, titled "The Legend of the Lambton Worm", the woman who helps John Lambton to defeat the Dragon is identified only as "a witch". This version also appears in Joseph Watson's 1877 pamphlet, The Wonderful Tradition of the Lambton Worm.
Then in 1866 we find a version referring to a "sybil or wise woman". This one is found in William Henderson's Notes on the Folklore of the Northern Counties of England and the Borders (http://books.google.com/books?id=o2cAAAAAMAAJ).
In 1867, C. M. Leumane came out with a popular song called "The Lambton Worm" and there is no mention of the Witch (or wise woman, or sybil) in the song's lyrics. (http://www.washingtonlass.com/LambtonWormSong)
Yet another variation occurs in 1868 in the March 14 issue of the illustrated literary weekly "Once a Week", which was edited by Eneas Sweetland Dallas at the time (http://books.google.com/books?id=0RbnAAAAMAAJ). This version makes no mention of a "Witch" or of a "wise woman", but rather refers to a "neighboring sibyl". Interestingly, just two months previously "Once a Week" featured a piece on "Witchcraft in Devon", in which the phrase "white witch" appears numerous times.
The same 1877 pamphlet by Joseph Watson mentioned above (because it contains the 1840 poetic version of the tale), also contains a prose description of the legend in which it is stated that Lord Lambton "consulted a Sibyl on the best means to be pursued to slay the monster." (http://books.google.com/books?id=gKoWAAAAQAAJ)
In 1888 another version of the story appeared in The Monthly Chronicle of North-Country Lore and Legend (http://www.washingtonlass.com/LambtonWormLegend1888.html). In this version there is a whole subsection entitled simply "The Witch", describing how John Lambton sought out and received guidance from said Witch.
A still later version of the story appeared in Baily's Magazine of Sports and Pastimes in 1895. In that 1895 version, John Lambton is assisted by a "wise woman", and there is no mention of a "Witch". (http://books.google.com/books?id=y5cbAQAAIAAJ)
Finally, yet another retelling of the story appeared in the Gentleman's Magazine in 1904 as part of an article by Barbara Clay Finch on "Reptile Lore." (http://books.google.com/books?id=vOIIAAAAIAAJ) In that version, the young Lord Lambton is told how to kill the Dragon by "a notable white witch".
In 1584 Reginald Scot wrote: "At this day it is indifferent to say in the English tongue, 'she is a witch,' or 'she is a wise woman.'" About a century later Henry More wrote in a letter to his friend Joseph Glanvill:
"As for the words Witch and Wizzard, from the Notation of them, they signifie no more than a wise Man, or a wise Woman. In the word Wizzard, it is plain at the very first sight. And I think the most plain and least operose deduction of the name Witch, is from Wit, whose derived Adjective might be Wittigh or Wittich, and by contraction afterwards, Witch; as the Noun wit is from the Verb to weet, which is, to know. So that a Witch, thus far, is no more than a Knowing woman; which answers exactly to the Latine word Saga ...."We need not concern ourselves over the validity (or operosity) of More's etymological analysis, but only with his perception that "Witch" and "wise woman" are essentially synonymous.
The equivalence stated explicitly by Scot and More is demonstrated in practice by no less an authority on the English language than William Shakespeare, in his Merry Wives of Windsor. If these three data points were all we had, there would still be a strong case for the claim that the word "Witch" in the 16th and 17th centuries was not, as some have claimed, an unambiguous designation for malevolent workers of malefic magic who were hated by their neighbors. Rather, these three sources, and a great many more, all attest to the use of the word "Witch" to refer to workers of beneficial magic who were valued and sought out by others for their magical services.
Moving to the case of the Lambton Worm, we find that as the same story is told and retold over the course of 84 years, that the same character is alternatively referred to as a "Witch", a "wise woman", a "sybil" and even a "white witch". In many of the redactions the story teller goes out of his or her way to use both "witch" and either "wise woman" or "sibyl". Not only does this Witch play a very positive role in the story, she provides the knight/hero, who already possesses strength and courage, with the key thing that he is missing, without which it is not possible to defeat the "Worm". This missing ingredient is precisely that which Henry More cited as the defining feature of the "Witch": knowledge. It only remains to add that according to sources from Jacob Grimm's Teutonic Mythology (link) and Joseph Bosworth's Anglo-Saxon Dictionary (link), both published in 1835, up to the Dictionary of English Folklore by Jacqueline Simpson and Steve Roud, published in 2000: "The Old English word ‘witch’ meant ‘one who casts a spell’. Intrinsically neutral, it could be applied to those using magic helpfully." (link)
In conclusion there is no room for reasonable doubt concerning the meaning of the word "Witch" from pre-Conquest times to the 21st century. This word has from the beginning been used to refer to "those using magic helpfully", and there has never been a period of time over the last 1000 years when this was not the case. Those who claim otherwise are either ignorant of the most basic relevant facts, or they are engaged in a systematic attempt to misrepresent those facts.