Etymology
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proposition (n.)

mid-14c., proposicioun, "a riddle" (a sense now obsolete); late 14c., in rhetoric, "a setting forth as a topic for discussion or discourse," from Old French proposicion "proposal, submission, (philosophical) proposition" (12c.), from Latin propositionem (nominative propositio) "a setting forth, statement, a presentation, representation; fundamental assumption," noun of action from past-participle stem of proponere "put forth, set forth, lay out, display, expose to view" (see propound). Meaning "action of proposing something to be done, an offered plan of action," is from late 14c. General sense of "matter, problem, undertaking" recorded by 1877. Related: Propositional; propositionally.

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proposition (v.)

"make or present a proposition," 1914, from proposition (n.). The older verb is propose. Specifically of sexual favors by 1936. Related: Propositioned; propositioning.

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theorem (n.)

"demonstrable proposition in science or mathematics," 1550s, from French théorème (16c.) and directly from Late Latin theorema, from Greek theorema "spectacle, sight," in Euclid "proposition to be proved," literally "that which is looked at," from theorein "to look at, behold" (see theory).

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thesis (n.)
late 14c., "unaccented syllable or note," from Latin thesis "unaccented syllable in poetry," later (and more correctly) "stressed part of a metrical foot," from Greek thesis "a proposition," also "downbeat" (in music), originally "a setting down, a placing, an arranging; position, situation," from reduplicated form of PIE root *dhe- "to set, put." Sense in logic of "a formulation in advance of a proposition to be proved" is first recorded 1570s; that of "dissertation presented by a candidate for a university degree" is from 1650s.
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hypothesis (n.)

1590s, "a particular statement;" 1650s, "a proposition, assumed and taken for granted, used as a premise," from French hypothese and directly from Late Latin hypothesis, from Greek hypothesis "base, groundwork, foundation," hence in extended use "basis of an argument, supposition," literally "a placing under," from hypo- "under" (see hypo-) + thesis "a placing, proposition" (from reduplicated form of PIE root *dhe- "to set, put"). A term in logic; narrower scientific sense is from 1640s.

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dilemma (n.)
Origin and meaning of dilemma

1520s in rhetoric (see below), from Late Latin dilemma, from Greek dilemma "double proposition," a technical term in rhetoric, from di- "two" (see di- (1)) + lemma "premise, anything received or taken," from root of lambanein "to take" (see lemma).

A form of argument in which it is shown that whoever maintains a certain proposition must accept one or other of two alternative conclusions, and that each of these involves the denial of the proposition in question. [Century Dictionary]

Loosely, "choice between two undesirable alternatives," from 1580s. It should be used only of situations where someone is forced to choose between two alternatives, both unfavorable to him (the alternatives are called the horns of a dilemma). But even logicians disagree on whether certain situations are dilemmas or mere syllogisms. Related: Dilemmatic.

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rest (n.2)

"remainder, that which is left after a separation," early 15c., from French reste "remnant," from rester "to remain" (see rest (v.2)). Meaning "others, those not included in a proposition" is from 1530s.

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corollary (n.)

late 14c., "a proposition inadvertently proved in proving another," from Late Latin corollarium "a deduction, consequence," from Latin corollarium, originally "money paid for a garland," hence "gift, gratuity, something extra;" and in logic, "a proposition proved from another that has been proved." From corolla "small garland," diminutive of corona "a crown" (see crown (n.)).

Also in Middle English "a follower, a sycophant" (late 14c.). As an adjective, "of the nature of a corollary," mid-15c.

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postulate (n.)

1580s, "a request, demand, petition," from Latin postulātum "demand, request," properly "that which is requested," noun use of neuter past participle of postulare "to ask, demand; claim; require" (see postulate (v.)).

The sense in logic, "proposition proposed for acceptance without proof, something taken for granted," is from 1640s, from a sense in Medieval Latin. The meaning "self-evident practical proposition" is by 1751. The earlier noun in English was postulation "a petition, request" (c. 1400). Middle English also had postulate (adj.) "nominated to a bishopric or archbishopric" (mid-15c.).

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premise (n.)

late 14c., in logic, "a previous proposition from which another follows, a judgment causing another judgment," from Old French premisse (14c.), from Medieval Latin praemissa (propositio or sententia) "(the proposition) set before," noun use of fem. past participle of Latin praemittere "send forward, put before," from prae "before" (see pre-) + mittere "to send" (see mission).

In legal documents it meant "matter previously stated" (early 15c.), which in deeds or wills often was a description of a house or building, hence the extended meaning "house or building, with grounds" (1730).

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