The founding siman of the laws of doubt: the דבר חשוב (important object — the 7 things, living animals, anything sold by count) never nullifies, even in a thousand (but it is דרבנן, so the doubt is lenient); כל קבוע כמחצה על מחצה (a prohibition fixed in its place = 50/50) versus כל דפריש מרובא פריש (what separates follows the majority — the nine shops); the ספק ספיקא (double doubt) that permits — and its limits (a doubt בגופו or a prohibition בקבוע does not combine) — Shulchan Aruch, Yoreh De'ah 110 — 10 se'ifim
דָּבָר חָשׁוּב אוֹסֵר בְּמִינוֹ בְּכָל שֶׁהוּא, וְהֵם שִׁבְעָה דְּבָרִים, וְאֵלּוּ הֵם: אֱגוֹזֵי פֶּרֶךְ, וְרִמּוֹנֵי בָּדָן, וְחָבִיּוֹת סְתוּמוֹת, וַחֲלָפוֹת תְּרָדִין, וְקִלְחֵי כְרוּב, וּדְלַעַת יְוָנִית, וְכִכָּרוֹת שֶׁל בַּעַל הַבַּיִת. וְכֵן בַּעֲלֵי חַיִּים חֲשׁוּבִים הֵם וְאֵינָם בְּטֵלִים. אֲבָל שְׁאָר דְּבָרִים, אַף עַל פִּי שֶׁדַּרְכָּן לִימָּנוֹת, הֲרֵי אֵלּוּ עוֹלִים כְּשִׁעוּרָן.
An important object forbids within its species in any amount, and these are seven things: Perekh nuts, Badan pomegranates, sealed barrels, beet stalks, cabbage stumps, the Greek gourd and a householder's loaves. Likewise living animals are important and do not nullify. But other things, even though it is customary to count them, nullify in their measure [60].
Shulchan Aruch, Yoreh De'ah 110:1LEVEL 01
Hebrew text of the 10 se'ifim with a fluent English translation. The דבר חשוב that does not nullify, slaughtered animals, the nine shops (קבוע / רוב), the meat of the מקולין and the ספק ספיקא explained with practical cases.
📖 StudyLEVEL 02
In-depth pilpul: the sugya of Chullin, the yesod of דבר שבמנין (Taz / Shach), the source of כל קבוע (Bava Kamma, ספק נפשות), the מחלוקת Rashba / Rosh on פירש ממילא, and the great analysis of ספק ספיקא (two vs three תערובות, the doubt בגופו, the two doubts known together, חזקת איסור).
📖 StudyLEVEL 03
Comparative tables (קבוע / פירש, במתכוין / ממילא, two / three mixtures), golden rules, classic pitfalls (ס״ס בגופו, בקבוע, דשיל״מ) and memorization of the 10 se'ifim.
📖 StudyLEVEL 04
The practical halacha according to the Shach, Taz, Pri Megadim and Pitchei Teshuva, then the Sephardic poskim (Yabia Omer, Yalkut Yosef, Or LeTzion) and Ashkenazi poskim. Note: the Shulchan Aruch HaRav does not deal with this siman — this is a level of psak, not "Daat HaRav".
📖 StudyAccording to the Shulchan Aruch (YD 110:1), certain objects are so important that they never nullify, even in a thousand: these are seven things (Perekh nuts, Badan pomegranates, sealed barrels, beet stalks, cabbage stumps, the Greek gourd and a householder's loaves) as well as living animals (בעלי חיים). The Rama adds that, by custom, anything sold by count (דבר שבמנין) does not nullify. But this importance is only דרבנן, so in doubt one is lenient (אזלינן בספיקו לקולא), and it depends on what is important in that place and time. For practical halacha, consult your Rav.
This is the heart of se'if 3. If nine shops sell kosher slaughtered meat and one sells נבילה, and one took from one of them not knowing which, it is forbidden: as long as the prohibition is fixed in its place, we say כל קבוע כמחצה על מחצה דמי (the fixed is treated as half-and-half, we do not follow the majority). But meat found in the market, which has already separated from its place, is permitted by כל דפריש מרובא פריש (what separates follows the majority). The Rama specifies: only if it did not separate before us (לפנינו); separated before our eyes, it is still קבוע. For practical halacha, consult your Rav.
A ספק ספיקא — two independent doubts known together — can permit, even a prohibition מן התורה (se'ifim 4, 8). But it has limits: it does not combine when the first doubt is בגופו (in the very body of the object, like a ספק טריפה in se'if 9) nor when the prohibition was בקבוע (in its place); a mere doubt of mixture added to an already certain prohibition מן התורה is not called ספק ספיקא (Rama, se'ifim 4 and 9). For practical halacha, consult your Rav.