Highly complicated pattern of use in which the meanings of the forms change according to the person of the subject. 

Shall is used to indicate simple futurity: 

I shall (not will) have to buy another ticket. 

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 …the same sense of futurity is expressed by will: 

The comet will (not shall) return in 87 years.

You will (not shall) probably encounter some heavy seas when you round the point.  

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 …may express:

DeterminationPromiseObligationPermission

…depending on the context.   

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“I will leave tomorrow” indicates that the speaker is determined to leave.   

“You shall leave tomorrow”  or“She shall leave tomorrow”  is likely to be

interpreted as a command.

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“You shall have your money” expresses a promise.

Whereas…

“You will have your money” makes a simple prediction.

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Determination on the part of the speaker, as in issuing a threat: “You shall pay for this!”

Compulsion (obligation), now especially in official documents: “The Tenant shall return the keys to the Landlord.”

Certainty or inevitability “Our day shall come.”

Nonspecific futurity “I don't think I shall ever see her again.” ; “He doubts whether he shall be in tomorrow.”

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The English and some traditionalists about usage are probably the only people who

follow these rules, and then not with perfect consistency.

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 …people who try to adhere to these rules run the risk of sounding pretentious or haughty. (altanero)

Americans normally use will to express most of the senses reserved for shall in English usage.

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…use shall chiefly:

1. in first person invitations 2. questions that request an opinion or

agreement

such as “Shall we go?”  and in certain fixed expressions, such

as “We shall overcome.”Gini Blanco.09/04/23 10

…use shall to express: an explicit obligation

“Applicants shall provide a proof of residence”

 though this sense is also expressed by Must  or should.

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Instead of using “shall” or “will” to make a distinction in meaning, they choose another auxiliary such as must or have to; or use an adverb such as certainly. 

“I will leave tomorrow ” (stressing the auxiliary in BrE; "I intend to leave")

“I must leave tomorrow ”; “I’ll certainly leave tomorrow” (AmE, without stressing the auxiliary)

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…to its sense of obligation, shall also can convey high moral seriousness that derives in part from its extensive use in the King James Bible, as in "Righteousness shall go before him and shall set us in the way of his steps"(Ps 85:13) and "He that shall humble himself shall be exalted" (Mt 23:12). 

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…this may be why Lincoln chose to use shall instead of will in the Gettysburg Address:

“Government of the people, by the people, for the people shall not perish

from the earth." 

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…Shall comes from: Middle English schal. Old English sceal.

The past tense of shall is should.“I told him he should pay for this.”(not advising him, but threatening him.)

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Should has the sense of being more threatening.

If you wanted to make it softer you could

say: “I told him he ought to pay for this" (present tense)

“I told him he ought to have paid for this" (past tense)

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