Math 2.0

Mathematical logic and foundations Concept

Vacuous truth

Builds on ImplicationUniversal quantifier

Summary

A vacuous truth is an implication or a universal statement that is true because its hypothesis is never true. An implication p→qp \to q with a false pp is true, and a statement ∀x∈X, P(x)\forall x \in X, \, P(x) with an empty XX is true.

Remarks

Example: "each element of the empty set is an even number" is vacuously true. Vacuous truth explains why the empty set is a subset of every set.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionVacuous truth, concept, no evidence levelVacuous truthconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Implication definition
  3. Predicate definition
  4. Universal quantifier definition
  5. Vacuous truth concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Vacuous_truth, lead section
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