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 False Statement implies Every Statement.

Dilbert

The apparent paradox False Statement implies Every Statement 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

• Paradoxes of Material Implication, in which category this result is grouped

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)