Principle of explosion

The principle of explosion, (Latin: 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 and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction.[1] 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 "\vdash" symbolizes the relation of logical consequence):

\{ \phi , \lnot \phi \} \vdash \psi
or
\bot \to P.

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

Contents

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 in nature. A sentence \psi is a semantic consequence of a set of sentences \Gamma only if every model of \Gamma is a model of \psi. But there is no model of the contradictory set \{\phi , \lnot \phi \}. A fortiori, there is no model of \{\phi , \lnot \phi \} that is not a model of \psi. Thus, vacuously, every model of \{\phi , \lnot \phi \} is a model of \psi. Thus \psi is a semantic consequence of \{\phi , \lnot \phi \}.

The proof-theoretic argument

The second type of argument is proof-theoretic in nature. Consider the following derivations:

  1. \phi \wedge \neg \phi\,
    assumption
  2. \phi\,
    from (1) by conjunction elimination
  3. \neg \phi\,
    from (1) by conjunction elimination
  4. \phi \vee \psi\,
    from (2) by disjunction introduction
  5. \psi\,
    from (3) and (4) by disjunctive syllogism
  6. (\phi \wedge \neg \phi) \to \psi
    from (5) by conditional proof (discharging assumption 1)

This is just the symbolic version of the informal argument given above, with \phi standing for "all lemons are yellow" and \psi 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. \phi \wedge \neg \phi\,
    hypothesis
  2. \phi\,
    from (1) by conjunction elimination
  3. \neg \phi\,
    from (1) by conjunction elimination
  4. \neg \psi\,
    hypothesis
  5. \phi\,
    reiteration of (2)
  6. \neg \psi \to \phi
    from (4) to (5) by deduction theorem
  7. ( \neg \phi \to \neg \neg \psi)
    from (6) by contraposition
  8. \neg \neg \psi
    from (3) and (7) by modus ponens
  9. \psi\,
    from (8) by double negation elimination
  10. (\phi \wedge \neg \phi) \to \psi
    from (1) to (9) by deduction theorem

Or:

  1. \phi \wedge \neg \phi\,
    assumption
  2. \neg \psi\,
    assumption
  3. \phi\,
    from (1) by conjunction elimination
  4. \neg \phi\,
    from (1) by conjunction elimination
  5. \neg \neg \psi\,
    from (3) and (4) by reductio ad absurdum (discharging assumption 2)
  6. \psi\,
    from (5) by double negation elimination
  7. (\phi \wedge \neg \phi) \to \psi
    from (6) by conditional proof (discharging assumption 1)

Addressing the principle

Paraconsistent logics 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 \{\phi , \lnot \phi \} 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 paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including disjunctive syllogism, disjunction introduction, and reductio ad absurdum.

See also

References

  1. ^ Carnielli, W. and Marcos, J. (2001) "Ex contradictione non sequitur quodlibet" Proc. 2nd Conf. on Reasoning and Logic (Bucharest, July 2000)