The principle of explosion, (Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

: ex falso quodlibet or ex contradictione sequitur quodlibet, "from a contradiction, anything follows") or the principle of Pseudo-Scotus, is the law of classical logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

 and intuitionistic

Intuitionistic logic

Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...

 and similar systems of logic, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (or its negation) can be inferred from it. In symbolic terms, the principle of explosion can be expressed in the following way (where "" symbolizes the relation of logical consequence):

or .

This can be read as, "If one claims something is both true () and not true (), one can logically derive any conclusion ()."

Arguments for explosion

An informal statement of the argument for explosion is this: Consider two inconsistent statements, “All lemons are yellow” and "All lemons are not yellow", and suppose for the sake of argument that both are true. We can then prove anything, for instance that Santa Claus exists: Since the statement that "All lemons are yellow and all lemons are not yellow" is true, we can infer that all lemons are yellow. And from this we can infer that the statement “Either all lemons are yellow or Santa Claus exists” is true (one or the other has to be true for this statement to be true, and we just showed that it is true that all lemons are yellow, so this expanded statement is true). And since either all lemons are yellow or Santa Claus exists, and since all lemons are not yellow, (this was our first premise), it must be true that Santa Claus exists.

In more formal terms, there are two basic kinds of argument for the principle of explosion, semantic and proof-theoretic.

The semantic argument

The first argument is semantic or model-theoretic

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

in nature. A sentence is a semantic consequence of a set of sentences only if every model of is a model of . But there is no model of the contradictory set . A fortiori, there is no model of that is not a model of . Thus, vacuously, every model of is a model of . Thus is a semantic consequence of .

The proof-theoretic argument

The second type of argument is proof-theoretic

Proof theory

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...

in nature. Consider the following derivations:

1. 

   assumption

2. 

   from (1) by conjunction elimination

3. 

   from (1) by conjunction elimination

4. 

   from (2) by disjunction introduction

   Disjunction introduction

   Disjunction introduction or Addition is a valid, simple argument form in logic:or in logical operator notation: A \vdash A \or B The argument form has one premise, A, and an unrelated proposition, B...

   
   

5. 

   from (3) and (4) by disjunctive syllogism

   Disjunctive syllogism

   A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion....

   
   

6. 

   from (5) by conditional proof

   Conditional proof

   A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent....

   
    (discharging assumption 1)

This is just the symbolic version of the informal argument given above, with standing for "all lemons are yellow" and standing for "Santa Claus exists". From "all lemons are yellow and all lemons are not yellow" (1), we infer "all lemons are yellow" (2) and "all lemons are not yellow" (3); from "all lemons are yellow" (2), we infer "all lemons are yellow or Santa Claus exists" (4); and from "all lemons are not yellow" (3) and "all lemons are yellow or Santa Claus exists" (4), we infer "Santa Claus exists" (5). Hence, if all lemons are yellow and all lemons are not yellow, then Santa Claus exists.

Or:

1. 

   hypothesis

2. 

   from (1) by conjunction elimination

3. 

   from (1) by conjunction elimination

4. 

   hypothesis

5. 

   reiteration of (2)

6. 

   from (4) to (5) by deduction theorem

   Deduction theorem

   In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...

   
   

7. 

   from (6) by contraposition

   Contraposition

   In traditional logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality . For its symbolic expression in modern logic see the rule...

   
   

8. 

   from (3) and (7) by modus ponens

   Modus ponens

   In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...

   
   

9. 

   from (8) by double negation elimination

10. 

    from (1) to (9) by deduction theorem

    Deduction theorem

    In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...

    
    

Or:

1. 

   assumption

2. 

   assumption

3. 

   from (1) by conjunction elimination

4. 

   from (1) by conjunction elimination

5. 

   from (3) and (4) by reductio ad absurdum

   Reductio ad absurdum

   In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

   
    (discharging assumption 2)

6. 

   from (5) by double negation elimination

7. 

   from (6) by conditional proof

   Conditional proof

   A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent....

   
    (discharging assumption 1)

Addressing the principle

Paraconsistent logic

Paraconsistent logic

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

s have been developed that allow for sub-contrary forming operators. Model-theoretic paraconsistent logicians often deny the assumption that there can be no model of and devise semantical systems in which there are such models. Alternatively, they reject the idea that propositions can be classified as true or false. Proof-theoretic

Proof-theoretic semantics

Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system...

 paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including disjunctive syllogism

Disjunctive syllogism

A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion....

, disjunction introduction

Disjunction introduction

Disjunction introduction or Addition is a valid, simple argument form in logic:or in logical operator notation: A \vdash A \or B The argument form has one premise, A, and an unrelated proposition, B...

, and reductio ad absurdum

Reductio ad absurdum

In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

.

See also

• Dialetheism
  Dialetheism
  Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
  
   – belief in the existence of true contradictions
• Law of excluded middle
  Law of excluded middle
  In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
  
   – every proposition is either true or not true
• Law of noncontradiction
  Law of noncontradiction
  In classical logic, the law of non-contradiction is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g...
  
   – no proposition can be both true and not true
• Paraconsistent logic
  Paraconsistent logic
  A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
  
   – a modal logic
  Modal logic
  Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
  
   used to address contradictions
• Paradox of entailment – a seeming paradox derived from the principle of explosion
• Reductio ad absurdum
  Reductio ad absurdum
  In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
  
   – concluding that a proposition is false because it produces a contradiction
• Trivialism
  Trivialism
  Trivialism is the theory that every proposition is true. A consequence of trivialism is that all statements, including all contradictions of the form "p and not p" , are true.- Further reading :***...
  
  – the belief that all statements of the form "P and not-P" are true