False Statement implies Every Statement

Theorem

If something is false, then it implies anything.

Formulation 1

\(\ds \neg p\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds p \implies q\) \(\) \(\ds \)

Formulation 2

$\vdash \neg p \implies \paren {p \implies q}$


Examples

Two-Headed Elephant

If elephants have two heads, then cats can walk on water

is an example of .


Dilbert

The apparent paradox can perhaps be intellectually reconciled by considering the figure of speech in natural language:

If Dilbert passes his Practical Management exam I'll eat my hat.

That is, if statement $p$ is so absurdly improbable as to be a falsehood for all practical purposes, then it can imply an even more absurdly improbable conclusion $q$.


Also see


Sources

  • 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.8$: Implication or Conditional Sentence
  • 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.3$: Basic Truth-Tables of the Propositional Calculus
  • 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.2$: Conditional Statements
  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): implication: 1. (material implication)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): implication: 1. (material implication)
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