This custom is also hinted at in the Shabbat zemirot, for example in Ma Y'didut which tells of the need "to set out a number of varieties" of food (La'aroch kama minim).
The source for the MB is the Zohar, evidently the following passage: "And on Shabbat night, a person should eat from all [he has prepared], to show that this canopy of peace comprises all, as long as he does not detract any dish from the day... It is enough if he leaves two dishes."
The way the Zohar envisions tasting a number of dishes is to taste a bit at night of the dishes which are prepared for the day. At the most basic level this is just a practical consideration; if someone has limited resources he may have prepared only one dish for each meal, but even so he can have a variety of dishes by tasting a tiny bit at night of the dish designated for the daytime. At this level, showing that "the canopy of peace encompasses all" just means that the repose and delight of Shabbat encompasses a range of bodily enjoyments, that all worldly delights can be elevated to holiness.
However, the clear intention seems to be to show that specifically the evening meal includes an aspect of the daytime meals. This is after all the specific example given, and besides the "canopy of peace" is a description of Shabbat used specifically at night (in the Sh'ma b'rachot of Shabbat Maariv).
Many sources testify to the fact that each period on Shabbat - evening, morning, and afternoon - represents a different aspect of Shabbat. The passages from the Zohar printed in many siddurim following the Shabbat prayers provide a kabbalistic classification. The wording of the Shabbat prayers in many prayer books provides a simpler but related one: the special benediction for Shabbat in the Amida requests just before the closing that Israel should rest "on it", but at night we say BAH (feminine), in the morning BO (masculine) and at minchah BAM (plural). Night is the feminine aspect, morning the masculine, afternoon the unification.
The Tur, explaining why each Shabbat Amidah prayer has a completely different opening, states that evening refers to the Shabbat of the Creation, morning the Shabbat of the giving of the Torah, and afternoon to the future Shabbat, the aspect of Shabbat which is "like the World to Come" (Tur OC 292).
Basing ourselves on this understanding, the injunction to eat from each meal on Shabbat evening reminds us that the Shabbat of Creation already includes an aspect of the giving of the Torah and the final repose of the World to Come.
The Shabbat of Creation occurred because Hashem rested - He desisted from the work of creation, even though the world was yet incomplete. The Torah states that Hashem blessed and sanctified the seventh day, because on it He rested from all his work "which G^d created to make" (B'reshit 2:3). The Midrash states that "created to make" means that Hashem created the world incomplete, "to make" it complete by human endeavor. Yet this seeming incompleteness really encompasses the world's ultimate perfection. The future giving of the Torah was already "built-in" to the Creation, as Rashi explains only a few verses previously, on B'reshit 1:31.
The "incompleteness" of the original Creation left room for the giving of the Torah, which would enable mankind to perfect the world with our own efforts ultimately leading us to the World to Come. So the Shabbat of the Creation already includes a "taste" of the entire process of the ultimate perfection of the world.
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