Old English sceal, Northumbrian scule "I owe/he owes, will have to, ought to, must" (infinitive sculan, past tense sceolde), from *skulanan, a common Germanic preterite-present verb (along with can, may, will), from Proto-Germanic *skul- (source also of Old Saxon sculan, Old Frisian skil, Old Norse and Swedish skola, Middle Dutch sullen, Old High German solan, German sollen, Gothic skulan "to owe, be under obligation"). This is said to be related, via a past tense form, to Old English scyld "guilt," German Schuld "guilt, debt;" also Old Norse Skuld, name of one of the Norns.
These Germanic words are reconstructed (Watkins, Pokorny) to be from a PIE root *skel-(2) "to be under an obligation." The basic sense of the Germanic word probably was "I owe," hence "I ought." Cognates outside Germanic include Lithuanian skelėti "to be guilty," skilti "to get into debt;" Old Prussian skallisnan "duty," skellants "guilty." But Boutkan gives the group no PIE etymology and writes that the alleged root, limited as it is to Germanic and Balto-Slavic, "is likely to represent an innovation on the basis of North European substrate material."
Shall survives as an auxiliary. The original senses are obsolete; the meaning shifted in Middle English from obligation to include futurity. It has no participles, no imperative, and no infinitive. Its past-tense form has become should (q.v.) and has acquired special senses of its own.
The Germanic words are from PIE root *wel- (2) "to wish, will" (source also of Sanskrit vrnoti "chooses, prefers," varyah "to be chosen, eligible, excellent," varanam "choosing;" Avestan verenav- "to wish, will, choose;" Greek elpis "hope;" Latin volo, velle "to wish, will, desire;" Old Church Slavonic voljo, voliti "to will," veljo, veleti "to command;" Lithuanian velyti "to wish, favor," pa-velmi "I will," viliuos "I hope;" Welsh gwell "better").
Compare also Old English wel "well," literally "according to one's wish;" wela "well-being, riches." The use as a future auxiliary was already developing in Old English. The implication of intention or volition distinguishes it from shall, which expresses or implies obligation or necessity. Contracted forms, especially after pronouns, began to appear 16c., as in sheele for "she will." In early use often -ile to preserve pronunciation. The form with an apostrophe ('ll) is from 17c.
also sha'n't, by 1660s, representing a colloquial contraction of shall not.
Scottish form of northern English moun "must," from Old Norse man, first- and third-person singular of munu "shall, will." Related: Maunna "must not."
an auxiliary verb in future indicative, now archaic or dialectal, "shall, will," late 12c., from Old Norse monu, a future tense auxiliary verb ultimately meaning "to intend," ultimately from the PIE root *men- (1) "to think."
