Math 2.0

definition

Implication

Definition

Let pp and qq be propositions. The implication from pp to qq is the proposition p→qp \to q, read "if pp, then qq". The proposition pp is the hypothesis, and the proposition qq is the conclusion. The proposition p→qp \to q is false when pp is true and qq is false. In all other cases, p→qp \to q is true.

pp qq p→qp \to q
T T T
T F F
F T T
F F T

Remarks

Another notation for p→qp \to q is p⇒qp \Rightarrow q. The proposition p→qp \to q has the same truth value as ¬p∨q\neg p \lor q. When pp is false, p→qp \to q is true. This is vacuous truth. The converse of p→qp \to q is q→pq \to p. It can have a different truth value.

Explanations

L1 human

Think of p→qp \to q as a promise: "if it rains, I take an umbrella". You break the promise only when it rains and you take no umbrella. When it does not rain, you keep the promise, whatever you do.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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Proposition, concept, no evidence levelPropositionconceptImplication, definition, no evidence levelImplicationdefinition
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  1. Proposition concept
  2. Implication definition

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Sources

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

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