[shal; unstressed shuhl]
auxiliary verb, present singular 1st person shall, 2nd shall or (Archaic) shalt, 3rd shall, present plural shall; past singular 1st person should, 2nd should or (Archaic) shouldst or shouldest, 3rd should, past plural should; imperative, infinitive, and participles lacking.
plan to, intend to, or expect to: I shall go later.
will have to, is determined to, or definitely will: You shall do it. He shall do it.
(in laws, directives, etc.) must; is or are obliged to: The meetings of the council shall be public.
(used interrogatively in questions, often in invitations): Shall we go?

before 900; Middle English shal, Old English sceal; cognate with Old Saxon skal, Old High German scal, Old Norse skal; compare German soll, Dutch zal

can, may, shall, will (see usage note at can)(see usage note at the current entry)(see synonym study at will).

The traditional rule of usage guides dates from the 17th century and says that to denote future time shall is used in the first person (I shall leave. We shall go) and will in all other persons (You will be there, won't you? He will drive us to the airport. They will not be at the meeting). The rule continues that to express determination, will is used in the first person (We will win the battle) and shall in the other two persons (You shall not bully us. They shall not pass). Whether this rule was ever widely observed is doubtful. Today, will is used overwhelmingly in all three persons and in all types of speech and writing both for the simple future and to express determination. Shall has some use in all persons, chiefly in formal writing or speaking, to express determination: I shall return. We shall overcome. Shall also occurs in the language of laws and directives: All visitors shall observe posted regulations. Most educated native users of American English do not follow the textbook rule in making a choice between shall and will. See also should. Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
shall (ʃæl, (unstressed) ʃəl)
vb (takes an infinitive without to or an implied infinitive) (esp with I or we as subject) (with you, he, she, it, they, or a noun as subject) , past should
1.  Compare will used as an auxiliary to make the future tense: we shall see you tomorrow
2.  a.  used as an auxiliary to indicate determination on the part of the speaker, as in issuing a threat: you shall pay for this!
 b.  used as an auxiliary to indicate compulsion, now esp in official documents: the Tenant shall return the keys to the Landlord
 c.  used as an auxiliary to indicate certainty or inevitability: our day shall come
3.  (with any noun or pronoun as subject, esp in conditional clauses or clauses expressing doubt) used as an auxiliary to indicate nonspecific futurity: I don't think I shall ever see her again; he doubts whether he shall be in tomorrow
usage  The usual rule given for the use of shall and will is that where the meaning is one of simple futurity, shall is used for the first person of the verb and will for the second and third: I shall go tomorrow; they will be there now. Where the meaning involves command, obligation, or determination, the positions are reversed: it shall be done; I will definitely go. However, shall has come to be largely neglected in favour of will, which has become the commonest form of the future in all three persons

shouldst or shouldest (ʃʊdst, ˈʃʊdɪst)
vb (used with the pronoun thou or its relative equivalent)
archaic, dialect or a form of the past tense of shall
shouldest or shouldest

Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009
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Word Origin & History

O.E. sceal "I owe/he owes, will have to, ought to, must" (infinitive sculan, pt. sceolde), a common Gmc. preterite-present verb, from P.Gmc. *skal-, *skul- (cf. O.S. sculan, O.N., Swed. skola, M.Du. sullen, O.H.G. solan, Ger. sollen, Goth. skulan "to owe, be under obligation;" related via past tense
form to O.E. scyld "guilt," Ger. Schuld "guilt, debt;" also O.N. Skuld, name of one of the Norns). Ground sense probably is "I owe," hence "I ought." The sense shifted in M.E. from a notion of "obligation" to include "futurity." Its past tense form has become should (q.v.). Cognates outside Gmc. are Lith. skeleti "to be guilty," skilti "to get into debt;" O.Prus. skallisnan "duty," skellants "guilty."
Online Etymology Dictionary, © 2010 Douglas Harper
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