Editing Principle of explosion (section)

== Proof ==

Below is the '''Lewis argument''',<ref name="MacFarlane2021"/> a formal proof of the principle of explosion using [[symbolic logic]].
	
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none; width: 45%"
|-
! style="width:5%" | Step
! style="width:15%" | Proposition
! style="width:25%" | Derivation
|- 
| 1 || <math>P\land\neg P</math> || Premise{{efn|Burgess2005 uses 2 and 3 as premises instead of this one}}
|-
| 2 || <math>P</math>        || [[Conjunction elimination]] (1)
|-
| 3 || <math>\neg P</math>   || [[Conjunction elimination]] (1)
|-
| 4 || <math>P \lor Q</math> || [[Disjunction introduction]] (2)
|-
| 5 || <math>Q</math>        || [[Disjunctive syllogism]] (4,3)
|-  
|}

This proof was published by [[C. I. Lewis]] and is named after him, though versions of it were known to medieval logicians.<ref>{{cite book 
| last1=Lewis 
| first1=C I 
| last2=Langford 
| first2=C H 
| title=Symbolic Logic 
| edition=2nd 
| date=1959 
| publisher=Dover 
| pages=250
| ISBN=9780486601700}}
</ref><ref name="Burgess2005">
{{cite book 
| last1=Burgess 
| first1=John P 
| title=The Oxford Handbook of Philosophy of Mathematics and Logic (ed Stewart Shapiro) 
| date=2005 
| publisher=Oxford University Press |page=732| ISBN=9780195325928}}
</ref><ref name="MacFarlane2021">{{cite book 
| last1=MacFarlane 
| first1=John 
| title=Philosophical Logic: A Contemporary Introduction 
| date=2021 
| publisher=Routledge 
| page=171 
| ISBN=978-1-315-18524-8}}</ref>

This is just the symbolic version of the informal argument given in the introduction, with <math>P</math> standing for "all lemons are yellow" and <math>Q</math> 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 <math>P</math> is a ''[[semantic consequence]]'' of a set of sentences <math>\Gamma</math> only if every model of <math>\Gamma</math> is a model of <math>P</math>. However, there is no model of the contradictory set <math>(P \wedge \lnot P)</math>. [[A fortiori]], there is no model of <math>(P \wedge \lnot P)</math> that is not a model of <math>Q</math>. Thus, vacuously, every model of <math>(P \wedge \lnot P)</math> is a model of <math>Q</math>. Thus <math>Q</math> is a semantic consequence of <math>(P \wedge \lnot P)</math>.