It’s not so much that Judaism considers dogs unclean, but rather, according to the Oral Law (Talmud), that “evil dogs” should be chained and not kept as pets because their barking may frighten a pregnant woman and cause her to miscarry. There is much debate among Orthodox Jews even today about having a dog as a pet, even the smaller breeds. Cats, on the other hand, are fine. At the same time, Judaism requires that pets be fed before household members because animals do not have the reasoning capacity to delay gratification.
There’s a reason for this. In an Israelite (and later, Jewish) context, dogs were more often encountered in the form of wild scavengers who roamed around cities and the countryside, living off leftovers and worthless food (Exodus 22:31; also cf.
Joseph and Aseneth 10.14, where Aseneth throws her royal dinner “out of the window as food for the dogs”), refuse and dead animals: street dogs for instance ate Jezebel’s body (cf. 2 Kings 9). This is the reason why in the OT, dogs were more often used negatively as examples of unclean habits and a menace (1 Kings 21:19, 23, 24; Psalm 22:17, 21; Isaiah 56:11; Jeremiah 15:3). The book of Tobit is the first example of a biblical literature which shows a dog as a travelling companion - in fact, the fact that the dog which accompanies Tobias gets mentioned twice (6:2; 11:4), all the while doing nothing important to the plot, or for that matter nothing at all, leads some authors to believe that in an earlier version of the story, the dog had a larger role - although Job 30:1 already indicates their presence to guard sheep. This negative image continues to some extent in the NT, where dogs are paired with the definitely-unclean pigs (Matthew 7:6) and heretics ( Philippians 3:2; 2 Peter 2:22; Revelation 22:15). This was of course in contrast to other cultures, where people had a more favorable view of dogs: as deities (cf. the Egyptians and the jackal-headed Anubis), as hunting dogs and bodyguards (cf. Mesopotamian cultures), or even as house pets (cf. Greco-Roman culture). Some attribute the eventual softening of Jewish attitudes towards dogs due to exposure to other peoples.
We shouldn’t deny that this passage is difficult to understand, and that Jesus’ reply to the woman is indeed somewhat harsh. The problematic word here is
kynaria/
kynariois. Many commentators attempt to soften the passage by pointing out how
kynarion is the diminutive form of
kyōn “dog” - thereby producing something like “pups,” “pet dogs,” or “doggies.” However, other commentators are not so certain whether the word should really be understood in the diminutive here - and even if it is, it still doesn’t totally remove the difficulty. Some authors would also say that “dogs” were common terms for non-Jews, based on their negative image in the OT and NT, although this is more likely an oversimplified caricature: while it is indeed a term of disapprobation in Hebrew thought - and even in Greek (cf. the
Cynics, which comes from the adjective meaning “dog-like”) - “dog” in either case is not tied to ethnic identity. Still another proposal interprets it as a proverb, in which case Jesus was giving the woman a cue for further conversation and to test her faith.
IMHO however, the best way to interpret this passage (which acknowledges the harshness of the saying and understands it in the context of Jesus’ view of Hismission) is to look at the text itself, since the Evangelists pretty much already give the answer: Jesus saw Himself as being sent for the Jews - He Himself pretty much says so (Matthew 15:25; cf. 10:5-6): all in all, Jesus’ contacts with non-Jews were few and exceptional. It’s not that gentiles are being excluded or that He did not care about them. Just as the members of the household must be fed before animals (be it the house pet or that random dog on the street), the Jewish people had temporal priority over the gentiles, since it was to them that the Messiah came. In any case, the woman (and Matthew and Mark) apparently do not see Jesus’ words as an insult but as a signal for her to come up with a witty riposte. (As an aside: we don’t know for sure what Jesus meant by “dog” here, but we can assume that the woman assumes it refers to house dogs, as the reference to “under the table” shows!

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I think that the core thing to keep in mind here is that, Jesus Himself oftentimes tended to say something that is somewhat rude to His fellow Jews as well. For example, He tells a would-be follower who begs time to bury his father: “Let the dead bury their dead,” and even orders His disciples when sending them out to “greet no one on the road.” (Which was, while most of us don’t really realize it since most Western cultures don’t really have any custom of greeting total strangers, a very grave
faux pas - apparently the urgency is such that He even denies His disciples from practicing common courtesy!) Both of these sayings have a common line of thinking:
time is of the essence. Jesus’ response to the woman should in my opinion be read in this light.