For the sake of clarity, we here follow the careful work of Gerd Theissen and Dagmar Winter, who distinguish between the "criterion of dissimilarity from Judaism” (CDJ) and the “criterion of dissimilarity with Christianity” (CDC). Here we will first look at CDJ, which holds that if an element of the Gospels diverges from Judaism it is more likely to be an authentic saying of Jesus.
The criterion has a long history and is has been linked to the broader issue of anti-Semitism in biblical studies. The logic of the criterion was especially refined in connection with the rise of “New Quest” scholarship, which is largely remembered defining Jesus over and against the Judaism of his day. Of course, this emphasis of New Quest scholarship has been largely rejected by most contemporary scholarship, which is marked with a renewed appreciation for the Jewishness of Jesus’ message.
Moreover, this criterion also has another fatal flaw: identifying elements dissimilar from ancient Judaism requires the possession of a complete knowledge of first-century Judaism, which, surely, we do not have. Morna Hooker articulates the point well:
“Since the method proceeds by eliminating ideas found in Judaism and early Christianity it presupposes a fairly confident knowledge of both areas. To what extent is this justified? Use of this criterion seems to assume that we are dealing with two known factors (Judaism and early Christianity) and one unknown―Jesus; it would perhaps be a fairer statement of the situation to say that we are dealing with three unknowns, and that our knowledge of the other two is quite as tenuous and indirect as our knowledge of Jesus himself.”Essentially, then, the criterion is an argument from silence.
 Gerd Theissen and Dagmar Winter, The Quest for the Plausible Jesus: The Question of Criteria (trans. M. E. Boring; Louisville: Westminster John Knox Press, 2002), 19. See also Stanley E. Porter, The Criteria for Authenticity in Historical-Jesus Research: Previous Discussion and New Proposals (London / New York: T & T Clark, 2000), 114.
 See Norman Perrin, Rediscovering the Teaching of Jesus (London: SCM; New York: Harper & Row, 1967), 15-53. For Perrin’s influence, see W. O. Seal, “Norman Perrin and his ‘School’: Retracing a Pilgrimage,” JSNT 20 (1984): 87-107; W.H. Kelber, “The Work of Norman Perrin: An Intellectuall Pilgrimage,” JR 64 (1984):484-500. However, clearly the roots of this criterion precede Perrin. See Rudolf Bultmann, History of the Synoptic Tradition (trans. B. Marsh; Oxford: Basil Blackwell, 1963), 205: “We can only count on possessing a genuine similitude of Jesus where, on the one hand, expression is given to the contrast between Jewish morality and piety and the distinctive eschatological temper which characterized the preaching of Jesus; and where on the other hand we find no specifically Christian features.”
 Here we cannot delve into this issue. See Theissen and Winter, The Quest for the Plausible Jesus, 1-171.
 This stage of research was sparked by Ernst Käsemann, who was concerned about the growing popularity of Rudolph Bultmann's theology, which he saw as leading to a kind of Docetism. See Ernst Käsemann, “Das Problem des historischen Jesus,” ZTK 51 (1954): 125-53. His challenge to scholars to find the earthly Jesus behind the kerygma was picked up by a number of scholars (Fuchs, Bornkamm, Conzelmann, Fuller, Perrin and Robinson). The term “New Quest” itself was first used by James Robinson, although it seems that it was primarily used at the behest of his editor and not necessarily to his liking. See James Robinson, A New Quest of the Historical Jesus (Studies in Biblical Theology. London: SCM, 1959). Bultmann responded to the project in 1959 [see Rudolf Bultmann, Das Verhältnis der urchristlichen Christusbotschaft zum historischen Jesus (Heidelberg: Winter, 1960)]. This response, in large part, brought an end to the New Quest. For an overview see, e.g., Colin Brown, “Historical Jesus, Quest of,” in Dictionary of Jesus and the Gospels (ed. J. B. Green, et al.; Downers Grove: InterVarsity Press, 1992), 336-37. Of course, the taxonomy of the Jesus research into periods such as “Old Quest”, “New Quest” and “Third Quest” has come under fire by some. While such a schema undoubtedly has weaknesses, it is overall still helpful in identifying specific trends in scholarship. Here I am in virtual total agreement with the analysis offered by Michael F. Bird, “Is There Really a ‘Third Quest’ for the Historical Jesus?” in Scottish Bulletin of Evangelical Theology 24.2 (2006): 195-219.
 For the tendency of New Questers to focus on Jesus’ dissimilarity to the Judaism of his day see Dunn, Jesus Remembered, 87: “In the second quest the principal criterion, the criterion of dissimilarity, tried to make a virtue out of what second questers perceived as a necessity by reconstructing their picture of Jesus out of what distinguished Jesus from his historical context and set him over against his Jewish milieu.” Likewise, see Paul Achtemeier, Joel B. Green, and Marianne Meye Thompson, Introducing the New Testament: Its Literature and Theology (Grand Rapids: Eerdmans, 2001), 60: “… the Second Quest made almost nothing of Jesus the Jew…”
 See Craig A. Evans, “Assessing Progress in the Third Quest of the Historical Jesus,” JSHJ (2006):37-8, who, speaking of this criterion, states: “Such a method… could hardly accommodate a portrait of Jesus that takes into account his Jewish context and the Jewish dimensions of his teaching and activities. Fortunately, this dubious criterion has received the trenchant criticism it deserves. Almost no one today is guided by it.” See also J. T. Sanders, “The Criterion of Coherence and the Randomness of Charisma: Poring through some Aporias in the Jesus Tradition,” NTS 44 (1988): 1-25: “Indeed, if there is one thing on which most scholars who have written on Jesus agree, it is that he was a religious Jew and that he therefore no doubt quoted scripture… This, of course, means that the criterion of dissimilarity is hogwash and everyone who writes on Jesus begins with some a priori assumptions that are never stated and often never examined.”
 The point was made by Morna Hooker, “On Using the Wrong Tool,” Th 75 (1972): 575. “In addition, see idem., “Christology and Methodology,” in New Testament Studies 17 (1970-1): 482. Hooker’s criticism has been picked up by many others. See, for example, Porter, The Criteria for Authenticity in Historical-Jesus Research, 74; Holmén, Tom Holmén, “Authenticity Criteria,” in Encyclopedia of the Historical Jesus (ed. C. A. Evans; New York / London: Routledge, 2008), 53; Theissen and Winter, The Quest for the Plausible Jesus, 21-22; Craig A. Evans, “Authenticity Criteria in Life of Jesus Research,” CSR 19 (1989), 15; John P. Meier, “The Present State of the Third Quest,” for the Historical Jesus: Loss and Gain’, Bib 80 (1999): 476.  See, among others, Hooker, “Using the Wrong Tool,” 575: “As far as Judaism is concerned, the discovery of the Qumran material should be sufficient warning against overconfidence in supposing that we know the whole truth about first-century Judaism. Any comparison between the beliefs of Judaism and the teaching of Jesus which claims to find ideas in the latter unparalleled in the former is inevitably an argument from silence, and should be treated as such.” See also Theissen and Winter, The Quest for the Plausible Jesus, 21-22: “Every claim to dissimilarity is necessarily an argument from silence, for dissimilarity, not to speak of underivability, cannot be verified historically. Such a verification would demand a complete picture of history, but sources always have only a fragmentary character.” Likewise, see Borg, Conflict, Holiness, and Politics in the Teachings of Jesus, 38: “…given our limited knowledge of both ancient Judaism and the ancient church, the claim that a saying is unparalleled (and hence authentic) is based on an argument from silence.”