SUPPORT THE WORK

# GetWiki

### principle of explosion

ARTICLE SUBJECTS
news  →
unix  →
wiki  →
ARTICLE TYPES
feed  →
help  →
wiki  →
ARTICLE ORIGINS
principle of explosion
[ temporary import ]
- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
1. We know that "All lemons are yellow", as it has been assumed to be true.
2. Therefore, the two-part statement "All lemons are yellow OR unicorns existâ€ must also be true, since the first part is true.
3. However, since we know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true, i.e., unicorns exist.
Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proved, it trivializes the concepts of truth and falsity.WEB
, McKubre-Jordens
, Maarten
, This is not a carrot: Paraconsistent mathematics
, Plus Magazine
, Millennium Mathematics Project
, August 2011
,
,
, January 14, 2017, Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem put much effort into revising set theory to eliminate these contradictions, resulting in the modern Zermeloâ€“Fraenkel set theory.
In a different solution to these problems, a few mathematicians have devised alternate theories of logic called paraconsistent logics, which eliminate the principle of explosion. These allow some contradictory statements to be proved without affecting other proofs.

## Symbolic representation

In symbolic logic, the principle of explosion can be expressed schematically in the following way:
P, lnot P vdash Q
(For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.)

## Proof

Below is a formal proof of the principle using symbolic logic{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"|+ style="background:paleturquoise"! style="width:5%" | Step! style="width:15%" | Proposition! style="width:25%" | Derivation
| Assumption
| Assumption
| Disjunction introduction (1)
| Disjunctive syllogism (2,3)
This is just the symbolic version of the informal argument given in the introduction, with P standing for "all lemons are yellow" and Q standing for "Unicorns exist". We start out by assuming that (1) all lemons are yellow and that (2) not all lemons are yellow. From the proposition that all lemons are yellow, we infer that (3) either all lemons are yellow or unicorns exist. But then from this and the fact that not all lemons are yellow, we infer that (4) unicorns exist by disjunctive syllogism.

### Semantic argument

An alternate argument for the principle stems from model theory. A sentence P is a semantic consequence of a set of sentences Gamma only if every model of Gamma is a model of P. But there is no model of the contradictory set (P wedge lnot P). A fortiori, there is no model of (P wedge lnot P) that is not a model of Q. Thus, vacuously, every model of (P wedge lnot P) is a model of Q. Thus Q is a semantic consequence of (P wedge lnot P).

## Paraconsistent logic

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.

## Use

The metamathematical value of the principle of explosion is that for any logical system where this principle holds, any derived theory which proves âŠ¥ (or an equivalent form, phi land lnot phi) is worthless because all its statements would become theorems, making it impossible to distinguish truth from falsehood. That is to say, the principle of explosion is an argument for the law of non-contradiction in classical logic, because without it all truth statements become meaningless.

## References

{{reflist}}{{Classical logic}}

- content above as imported from Wikipedia
- "principle of explosion" does not exist on GetWiki (yet)
- time: 4:38am EDT - Mon, Apr 22 2019
[ this remote article is provided by Wikipedia ]
LATEST EDITS [ see all ]
GETWIKI 09 MAY 2016
GETWIKI 18 OCT 2015
M.R.M. Parrott
Biographies
GETWIKI 20 AUG 2014
GETWIKI 19 AUG 2014
GETWIKI 18 AUG 2014
Wikinfo
Culture
CONNECT