axiomn.

Brit.

/ˈaksɪəm/, U.S.

/ˈæksiəm/

Forms: 1500s–1600s axioma, axiome, Middle English– axiom.

Etymology: < French axiome, < Latin axiōma, < Greek ἀξίωμα that which is thought worthy or fit, that which commends itself as self-evident, < ἀξιοῦν to hold worthy, < ἄξιος worthy.

a. A proposition that commends itself to general acceptance; a well-established or universally-conceded principle; a maxim, rule, law.

ΘΚΠ

the mind > mental capacity > knowledge > saying, maxim, adage > self-evident truth, axiom > [noun]

truth1500

maxim?1530

head assertion1531

maximum1563

maxima1564

axiom1578

self-evident1675

truism1714

postulate1751

1578 J. Lyly Euphues f. 35 The Axiomaes of Aristotle.

1604 T. Dekker & T. Middleton Honest Whore iv. iii. 45 Thats an Axiom, a principle.

1651 T. Hobbes Philos. Rudim. i. §2. 3 Which Axiom, though received by most, is yet certainly false.

1757 S. Johnson Rambler No. 175. ⁋1 The axioms of wisdom which recommend the ancient sages to veneration.

1837 J. Harris Great Teacher 389 The axiom known by the name of the golden rule.

1868 W. C. Hazlitt in tr. Paris & Vienne Prol. p. xi An axiom which in Latin expressed: Hoc crede quod tibi verum esse videtur.

1875 H. E. Manning Internal Mission of Holy Ghost ii. 33 It is an axiom of the human reason that God is everywhere.

†b. Specially restricted by Bacon to: An empirical law, a generalization from experience. Obsolete.

ΘΚΠ

the mind > mental capacity > philosophy > empiricism > [noun] > Baconianism > elements of

form1605

organum1620

axiom1626

1626 F. Bacon Sylua Syluarum §2 Led by great Judgement, and some good Light of Axioms.

1627 W. Rawley in F. Bacon Sylua Syluarum To Rdr. sig. A2 True Axiomes must be drawne from plaine Experience, and not from doubtfull.

a1856 W. Hamilton Lect. Metaphysics (1860) IV. xxvi. 47 Empirical rules,—(Bacon would call them axioms).

†2. Logic. A proposition (whether true or false).

ΘΚΠ

the mind > mental capacity > philosophy > logic > logical proposition > [noun]

proposition?a1475

axiom1588

proposite1620

propos1816

1588 A. Fraunce Lawiers Logike ii. i. f. 86^v An axiom or proposition..hath two partes, the bande, and the partes bound.

1656 T. Stanley Hist. Philos. II. viii. 43 Universally negative axioms are those, which consist of an universall negative particle, and a Categorem; as, no man walketh.

1664 H. More Apol. in Modest Enq. Myst. Iniquity 533 Otherwise no man might dispute or pronounce a false Axiome.

1742 in N. Bailey Universal Etymol. Eng. Dict.

3. Logic and Mathematics. ‘A self-evident proposition, requiring no formal demonstration to prove its truth, but received and assented to as soon as mentioned’ (Hutton).

ΘΚΠ

the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > self-evident or accepted without proof

petition1529

request1551

axiom1593

postulate1660

porime1702

postulatum1743

ansatz1936

the mind > mental capacity > philosophy > logic > logical reasoning > [noun] > deductivism or a priori reasoning > a principle or axiom

principlea1387

maximc1450

first principle1525

ground1528

principal1545

principium1550

protasis1572

theorem1588

postulate1590

axiom1593

groundsel1604

postulatuma1620

praecognitum1624

datum1646

self-evident1675

philosopheme1678

dictum of all and none1697

dictum of Aristotle1827

prius1882

ground rule1890

posit1900

1593 R. Hooker Of Lawes Eccl. Politie i. viii. 64 Axiomes or principles more general are such as this, That the greater good is to be chosen before the lesse.

1660 R. Coke Justice Vindicated 16.

1785 T. Reid Ess. Intellect. Powers i. ii Nor are they necessary truths, as mathematical axioms are.

1807 Ld. Byron Thoughts College Exam. in Hours Idleness Happy the youth in Euclid's axioms tried.

1851 H. Spencer Social Statics ii. ix. §6 The axiom that the whole is greater than its part.

Phrases

axiom of reducibility n. Logic (see quot. 1952).

ΘΚΠ

the mind > mental capacity > philosophy > logic > predicate or propositional logic > [noun] > set theory or extension > other elements of

isomorphism1892

axiom of reducibility1908

non-emptiness1936

1908 B. Russell in Amer. Jrnl. Math. 30 241 (heading) The axiom of reducibility.

1910 A. N. Whitehead & B. Russell Principia Mathematica I. 168 (heading) The hierarchy of types and the axiom of reducibility.

1910 A. N. Whitehead & B. Russell Principia Mathematica I. 174 We assume, then, that every function of one variable is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes; at any rate, it retains as much of classes as we have any use for... We will call this assumption the axiom of classes, or the axiom of reducibility.

1930 L. S. Stebbing Mod. Introd. Logic xxiii. 463 Ramsey has suggested a reconstruction of the system of Principia Mathematica in which the axiom of reducibility is no longer needed.

1942 D. D. Runes Dict. Philos. 266/2 As an indication or rough description of the axiom of reducibility, it may be said that it cancels a large part of the restrictive consequences of the prohibition against impredicative definition..and, in approximate effect, reduces the ramified theory of types to the simple theory of types.

1952 S. C. Kleene Introd. Metamath. iii. 44 To escape this outcome, Russell postulated his axiom of reducibility, which asserts that to any property belonging to an order above the lowest, there is a coextensive property (i.e. one possessed by exactly the same objects) of order 0.

1963 W. V. Quine Set Theory xi. 251 The axiom of reducibility regales us after all with attributes unspecifiable except by quantifying over attributes whose order is as high as their own.

1982 W. S. Hatcher Logical Found. Math. iv. 127 The axiom of reducibility lets us reduce the order of any term to the lowest possible order consistent with well-formedness (which is thus predicative by definition).

2006 Bull. Symbolic Logic 12 604 No matter what interpretation of the axiom of reducibility is adopted on a philosophical level, the mathematical consequences are the same.

This entry has not yet been fully updated (first published 1885; most recently modified version published online December 2021).

单词	axiom
释义	axiomn. Brit. /ˈaksɪəm/, U.S. /ˈæksiəm/ Forms: 1500s–1600s axioma, axiome, Middle English– axiom. Etymology: < French axiome, < Latin axiōma, < Greek ἀξίωμα that which is thought worthy or fit, that which commends itself as self-evident, < ἀξιοῦν to hold worthy, < ἄξιος worthy. 1. a. A proposition that commends itself to general acceptance; a well-established or universally-conceded principle; a maxim, rule, law. ΘΚΠ the mind > mental capacity > knowledge > saying, maxim, adage > self-evident truth, axiom > [noun] truth1500 maxim?1530 head assertion1531 maximum1563 maxima1564 axiom1578 self-evident1675 truism1714 postulate1751 1578 J. Lyly Euphues f. 35 The Axiomaes of Aristotle. 1604 T. Dekker & T. Middleton Honest Whore iv. iii. 45 Thats an Axiom, a principle. 1651 T. Hobbes Philos. Rudim. i. §2. 3 Which Axiom, though received by most, is yet certainly false. 1757 S. Johnson Rambler No. 175. ⁋1 The axioms of wisdom which recommend the ancient sages to veneration. 1837 J. Harris Great Teacher 389 The axiom known by the name of the golden rule. 1868 W. C. Hazlitt in tr. Paris & Vienne Prol. p. xi An axiom which in Latin expressed: Hoc crede quod tibi verum esse videtur. 1875 H. E. Manning Internal Mission of Holy Ghost ii. 33 It is an axiom of the human reason that God is everywhere. †b. Specially restricted by Bacon to: An empirical law, a generalization from experience. Obsolete. ΘΚΠ the mind > mental capacity > philosophy > empiricism > [noun] > Baconianism > elements of form1605 organum1620 axiom1626 1626 F. Bacon Sylua Syluarum §2 Led by great Judgement, and some good Light of Axioms. 1627 W. Rawley in F. Bacon Sylua Syluarum To Rdr. sig. A2 True Axiomes must be drawne from plaine Experience, and not from doubtfull. a1856 W. Hamilton Lect. Metaphysics (1860) IV. xxvi. 47 Empirical rules,—(Bacon would call them axioms). †2. Logic. A proposition (whether true or false). ΘΚΠ the mind > mental capacity > philosophy > logic > logical proposition > [noun] proposition?a1475 axiom1588 proposite1620 propos1816 1588 A. Fraunce Lawiers Logike ii. i. f. 86^v An axiom or proposition..hath two partes, the bande, and the partes bound. 1656 T. Stanley Hist. Philos. II. viii. 43 Universally negative axioms are those, which consist of an universall negative particle, and a Categorem; as, no man walketh. 1664 H. More Apol. in Modest Enq. Myst. Iniquity 533 Otherwise no man might dispute or pronounce a false Axiome. 1742 in N. Bailey Universal Etymol. Eng. Dict. 3. Logic and Mathematics. ‘A self-evident proposition, requiring no formal demonstration to prove its truth, but received and assented to as soon as mentioned’ (Hutton). ΘΚΠ the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > self-evident or accepted without proof petition1529 request1551 axiom1593 postulate1660 porime1702 postulatum1743 ansatz1936 the mind > mental capacity > philosophy > logic > logical reasoning > [noun] > deductivism or a priori reasoning > a principle or axiom principlea1387 maximc1450 first principle1525 ground1528 principal1545 principium1550 protasis1572 theorem1588 postulate1590 axiom1593 groundsel1604 postulatuma1620 praecognitum1624 datum1646 self-evident1675 philosopheme1678 dictum of all and none1697 dictum of Aristotle1827 prius1882 ground rule1890 posit1900 1593 R. Hooker Of Lawes Eccl. Politie i. viii. 64 Axiomes or principles more general are such as this, That the greater good is to be chosen before the lesse. 1660 R. Coke Justice Vindicated 16. 1785 T. Reid Ess. Intellect. Powers i. ii Nor are they necessary truths, as mathematical axioms are. 1807 Ld. Byron Thoughts College Exam. in Hours Idleness Happy the youth in Euclid's axioms tried. 1851 H. Spencer Social Statics ii. ix. §6 The axiom that the whole is greater than its part. Phrases axiom of reducibility n. Logic (see quot. 1952). ΘΚΠ the mind > mental capacity > philosophy > logic > predicate or propositional logic > [noun] > set theory or extension > other elements of isomorphism1892 axiom of reducibility1908 non-emptiness1936 1908 B. Russell in Amer. Jrnl. Math. 30 241 (heading) The axiom of reducibility. 1910 A. N. Whitehead & B. Russell Principia Mathematica I. 168 (heading) The hierarchy of types and the axiom of reducibility. 1910 A. N. Whitehead & B. Russell Principia Mathematica I. 174 We assume, then, that every function of one variable is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes; at any rate, it retains as much of classes as we have any use for... We will call this assumption the axiom of classes, or the axiom of reducibility. 1930 L. S. Stebbing Mod. Introd. Logic xxiii. 463 Ramsey has suggested a reconstruction of the system of Principia Mathematica in which the axiom of reducibility is no longer needed. 1942 D. D. Runes Dict. Philos. 266/2 As an indication or rough description of the axiom of reducibility, it may be said that it cancels a large part of the restrictive consequences of the prohibition against impredicative definition..and, in approximate effect, reduces the ramified theory of types to the simple theory of types. 1952 S. C. Kleene Introd. Metamath. iii. 44 To escape this outcome, Russell postulated his axiom of reducibility, which asserts that to any property belonging to an order above the lowest, there is a coextensive property (i.e. one possessed by exactly the same objects) of order 0. 1963 W. V. Quine Set Theory xi. 251 The axiom of reducibility regales us after all with attributes unspecifiable except by quantifying over attributes whose order is as high as their own. 1982 W. S. Hatcher Logical Found. Math. iv. 127 The axiom of reducibility lets us reduce the order of any term to the lowest possible order consistent with well-formedness (which is thus predicative by definition). 2006 Bull. Symbolic Logic 12 604 No matter what interpretation of the axiom of reducibility is adopted on a philosophical level, the mathematical consequences are the same. This entry has not yet been fully updated (first published 1885; most recently modified version published online December 2021). < n.1578
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