1.12.3 Disequality

Finally, let us also say something about disequality, which is negationMathworldPlanetmath of equality:11We use “inequality” to refer to < and . Also, note that this is negation of the propositional identity type. Of course, it makes no sense to negate judgmental equality , because judgments are not subject to logical operationsMathworldPlanetmath.


If xy, we say that x and y are unequal or not equal. Just like negation, disequality plays a less important role here than it does in classical mathematics. For example, we cannot prove that two things are equal by proving that they are not unequal: that would be an application of the classical law of double negation, see §3.4 (http://planetmath.org/34classicalvsintuitionisticlogic).

Sometimes it is useful to phrase disequality in a positive way. For example, in Theorem 11.2.4 (http://planetmath.org/1122dedekindrealsarecauchycomplete#Thmprethm1) we shall prove that a real number x has an inversePlanetmathPlanetmathPlanetmathPlanetmath if, and only if, its distance from 0 is positive, which is a stronger requirement than x0.

Title 1.12.3 Disequality