Principle of explosion
From a contradiction, any statement can be proven.
Wikipedia / Wikimedia Commons
The principle of explosion is a theorem in classical logic, intuitionistic logic, and similar logical systems, stating that any statement can be proven from a contradiction. Its proof was first given by the 12th-century French philosopher William of Soissons. The principle demonstrates that from a contradiction, any proposition—including its negation—can be inferred, a phenomenon known as deductive explosion.
- field
- Logic
- known_for
- Principle of explosion (first proof)
- first_proved_by
- William of Soissons
- century
- 12th century
- nationality
- French
Lore & Background
The principle of explosion was first proved by the 12th-century French philosopher William of Soissons. In classical logic, intuitionistic logic, and similar systems, it is the theorem that from a contradiction, any proposition (including its negation) can be inferred. This is known as deductive explosion.
Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics threatened the entire structure of mathematics. Mathematicians including Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem revised set theory to eliminate these contradictions, resulting in modern Zermelo–Fraenkel set theory.
As a demonstration, consider the contradictory statements 'All lemons are yellow' and 'Not all lemons are yellow.' If both are true, anything can be proven—for example, that unicorns exist—using disjunctive syllogism. The procedure can be repeated to prove that unicorns do not exist, causing an explosion of provable statements.
Reader's Guide
The principle of explosion is significant because it shows that any contradiction in a formal axiomatic system is disastrous: since any statement—true or not—can be proven, it trivializes the concepts of truth and falsity. This principle motivated the development of modern set theory to eliminate contradictions like Russell's paradox. In response, some mathematicians have devised paraconsistent logics, which allow some contradictory statements to be proven without affecting the truth value of all other statements. The metamathematical value of the principle is that for any logical system where it holds, any derived theory that proves a contradiction is worthless, as all statements become theorems. This serves as an argument for the law of non-contradiction in classical logic. The reduction in proof strength of logics without the principle is discussed in minimal logic.
Did You Know?
- The proof of the principle of explosion was first given by 12th-century French philosopher William of Soissons.
- The principle states that from a contradiction, any proposition (including its negation) can be inferred.
- Around the turn of the 20th century, contradictions such as Russell's paradox threatened the foundations of mathematics.
- Some mathematicians have devised paraconsistent logics to allow some contradictory statements without affecting the truth value of all other statements.
Frequently Asked Questions
Who is Principle of explosion?
The Principle of Explosion is a foundational theorem in classical and intuitionistic logic, first formally proved by the 12th-century French philosopher William of Soissons. It establishes that once a contradiction exists within a system, every possible statement becomes derivable from it.
What are Principle of explosion's powers/role?
Its 'power' is that it lets you derive any arbitrary proposition—even its own negation—from a single contradictory premise. In practice, this means a system containing a contradiction becomes useless for distinguishing truth from falsehood.
When did Principle of explosion first appear?
The first known proof was provided by William of Soissons, a French thinker working in the 12th century. This makes the principle one of the oldest formally established results in the history of formal logic.
Why is Principle of explosion important?
It serves as a critical warning that consistency is non-negotiable for any logical system to be meaningful. Without it, a single error or contradiction would let you 'prove' anything, collapsing the entire structure of valid inference.
How does Principle of explosion's story end?
Rather than ending, it persists as a standing constraint in virtually every standard logical framework, including both classical and intuitionistic systems. Its lasting role is to remind logicians that avoiding contradiction is the price of admission for meaningful reasoning.
More in Set Theory & Logic 1-19
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record