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<p>Asking for syntactically simple axioms that are equivalent, with plane absolute geometry in the sense of Hilbert as background theory, to Aristotle’s axiom <b>Ar</b> (“The perpendiculars dropped from one side of an angle to the other grow without bound”) and to the <i>Lotschnittaxiom</i> <b>L</b> (“The perpendiculars raised on the sides of a right angle intersect”), we find that: (1) <b>Ar</b> can be expressed as a positive statement in terms of points, betweenness, and equidistance; (2) <b>Ar</b> cannot be expressed positively in terms of points and collinearity; (3) the simplest form of <b>Ar</b>, with respect to quantifier type, in terms of points and collinearity, is <i>AAAAEA</i> (a prenex statement in which four universal quantifiers are followed by an existential quantifier, which in turn is followed by a universal quantifier); (4) <b>L</b> can be expressed as a positive sentence, using only <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\wedge \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>∧</mo> </math></EquationSource> </InlineEquation> as logical connective, in terms of points and collinearity; (5) the simplest forms of <b>L</b> in terms of points and collinearity are prenex statements of quantifier types <i>AAAAAAE</i> and <i>AAAAAEE</i>.</p>

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Simple forms of Aristotle’s axiom and of the Lotschnittaxiom

  • 期刊论文

  • 发表日期：

    2026年7月23日

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  • Sheila K. Miller Edwards,
  • Victor Pambuccian

摘要

Asking for syntactically simple axioms that are equivalent, with plane absolute geometry in the sense of Hilbert as background theory, to Aristotle’s axiom Ar (“The perpendiculars dropped from one side of an angle to the other grow without bound”) and to the Lotschnittaxiom L (“The perpendiculars raised on the sides of a right angle intersect”), we find that: (1) Ar can be expressed as a positive statement in terms of points, betweenness, and equidistance; (2) Ar cannot be expressed positively in terms of points and collinearity; (3) the simplest form of Ar, with respect to quantifier type, in terms of points and collinearity, is AAAAEA (a prenex statement in which four universal quantifiers are followed by an existential quantifier, which in turn is followed by a universal quantifier); (4) L can be expressed as a positive sentence, using only \(\wedge \) ∧ as logical connective, in terms of points and collinearity; (5) the simplest forms of L in terms of points and collinearity are prenex statements of quantifier types AAAAAAE and AAAAAEE.