Principle_of_explosion

Principle of explosion

The principle of explosion is the law of classical logic and a few other systems (e.g., intuitionistic logic) according to which "anything follows from a contradiction" - i.e., once you have asserted a contradiction, you can infer any proposition, or its converse. 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.

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)."

The principle of explosion is also known as ex falso quodlibet, ex falso sequitur quodlibet (EFSQ for short), ex contradictione (sequitur) quodlibet (ECQ for short), and ex falso/contradictione (sequitur) (Latin: "from falsehood/contradiction (follows) anything", literally "... what pleases").

Arguments for explosion

There are two basic kinds of argument for the principle of explosion.

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,
  2. :assumption
  3. phi,
  4. :from (1) by conjunction elimination
  5. neg phi,
  6. :from (1) by conjunction elimination
  7. phi vee psi,
  8. :from (2) by disjunction introduction
  9. psi,
  10. :from (3) and (4) by disjunctive syllogism
  11. (phi wedge neg phi) to psi
  12. :from (5) by conditional proof (discharging assumption 1)

Or:

  1. phi wedge neg phi,
  2. :hypothesis
  3. phi,
  4. :from (1) by conjunction elimination
  5. neg phi,
  6. :from (1) by conjunction elimination
  7. neg psi,
  8. :hypothesis
  9. phi,
  10. :reiteration of (2)
  11. neg psi to phi
  12. :from (4) to (5) by deduction theorem
  13. (neg phi to neg neg psi)
  14. :from (6) by contraposition
  15. neg neg psi
  16. :from (3) and (7) by modus ponens
  17. psi,
  18. :from (8) by double negation elimination
  19. (phi wedge neg phi) to psi
  20. :from (1) to (9) by deduction theorem

Or:

  1. phi wedge neg phi,
  2. :assumption
  3. neg psi,
  4. :assumption
  5. phi,
  6. :from (1) by conjunction elimination
  7. neg phi,
  8. :from (1) by conjunction elimination
  9. neg neg psi,
  10. :from (3) and (4) by reductio ad absurdum (discharging assumption 2)
  11. psi,
  12. :from (5) by double negation elimination
  13. (phi wedge neg phi) to psi
  14. :from (6) by conditional proof (discharging assumption 1)

Rejecting the principle

Proponents of paraconsistent logic reject the principle of explosion, and thus must find flaw with both of the arguments above.

As for the semantic argument, 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.

As for the proof-theoretic arguments, they reject some of the assumptions typically including the following: disjunctive syllogism, disjunction introduction, and reductio ad absurdum). See the article on paraconsistent logic.

See also

External links

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