False Statement implies Every Statement
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Theorem
If something is false, then it implies anything.
Formulation 1
\(\ds \neg p\) | \(\) | \(\ds \) | ||||||||||||
\(\ds \vdash \ \ \) | \(\ds p \implies q\) | \(\) | \(\ds \) |
Formulation 2
- $\vdash \neg p \implies \paren {p \implies q}$
This apparent paradox can be 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