Math 2.0

concept

Vacuous truth

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.

Explanations

L1 human

"All unicorns in this room are blue" is true when the room has no unicorns. There is nothing that can break the statement. This convention keeps the rules of logic simple and regular.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 5 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Proposition concept
  2. Implication definition
  3. Predicate definition
  4. Universal quantifier definition
  5. Vacuous truth concept

Sources

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

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