Editing Disjunctive syllogism

{{Short description|Logical rule of inference}}
{{Infobox mathematical statement
| name = Disjunctive syllogism
| type = [[Rule of inference]]
| field = [[Propositional calculus]]
| statement = If <math>P</math> is true or <math>Q</math> is true and <math>P</math> is false, then <math>Q</math> is true.
| symbolic statement = <math>\frac{P \lor Q, \neg P}{\therefore Q}</math>
}}
{{Transformation rules}}

In [[classical logic]], '''disjunctive syllogism'''<ref>{{cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |title=Introduction to Logic |publisher=Prentice Hall |year=2005 |page=362 }}</ref><ref>{{cite book |title=A Concise Introduction to Logic 4th edition |last=Hurley |first=Patrick |year=1991 |publisher=Wadsworth Publishing |pages=320–1 |isbn=9780534145156 |url=https://archive.org/details/studyguidetoacco00burc|url-access=registration }}</ref> (historically known as '''''modus tollendo ponens''''' ('''MTP'''),<ref>[[E. J. Lemmon|Lemmon, Edward John]]. 2001. ''Beginning Logic''. [[Taylor and Francis]]/CRC Press, p. 61.</ref> [[Latin]] for "mode that affirms by denying")<ref>{{cite book |last=Stone |first=Jon R. |year=1996 |title=Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language |location=London |publisher=Routledge |isbn=0-415-91775-1 |page=[https://archive.org/details/latinforillitera0000ston/page/60 60] |url=https://archive.org/details/latinforillitera0000ston |url-access=registration }}</ref> is a [[Validity (logic)|valid]] [[logical form|argument form]] which is a [[syllogism]] having a [[Logical disjunction|disjunctive statement]] for one of its [[premise]]s.<ref>Hurley</ref><ref>Copi and Cohen</ref>

An example in [[English language|English]]:
# I will choose soup or I will choose salad.
# I will not choose soup.
# Therefore, I will choose salad.

==Propositional logic==
In [[propositional calculus|propositional logic]], '''disjunctive syllogism''' (also known as '''disjunction elimination''' and '''or elimination''', or abbreviated '''∨E'''),<ref>Sanford, David Hawley. 2003. ''If P, Then Q: Conditionals and the Foundations of Reasoning''. London, UK: Routledge: 39</ref><ref>Hurley</ref><ref>Copi and Cohen</ref><ref>Moore and Parker</ref> is a valid [[rule of inference]]. If it is known that at least one of two statements is true, and that it is not the former that is true; we can [[inference|infer]] that it has to be the latter that is true. Equivalently, if ''P'' is true or ''Q'' is true and ''P'' is false, then ''Q'' is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step [[argument]], and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's ''disjuncts''.  The rule makes it possible to eliminate a [[logical disjunction|disjunction]] from a [[formal proof|logical proof]]. It is the rule that

:<math>\frac{P \lor Q, \neg P}{\therefore Q}</math>

where the rule is that whenever instances of "<math>P \lor Q</math>", and "<math>\neg P</math>" appear on lines of a proof, "<math>Q</math>" can be placed on a subsequent line.

Disjunctive syllogism is closely related and similar to [[hypothetical syllogism]], which is another rule of inference involving a syllogism. It is also related to the [[law of noncontradiction]], one of the [[Law of thought#Three traditional laws: identity, non-contradiction, excluded middle|three traditional laws of thought]].

== Formal notation ==
For a [[formal system|logical system]] that validates it, the ''disjunctive syllogism'' may be written in [[sequent]] notation as

: <math> P \lor Q, \lnot P \vdash Q </math>

where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>Q</math> is a [[logical consequence|syntactic consequence]] of <math>P \lor Q</math>, and <math>\lnot P</math>.

It may be expressed as a truth-functional [[tautology (logic)|tautology]] or [[theorem]] in the object language of propositional logic as

:<math> ((P \lor Q) \land \neg P) \to Q</math>

where <math>P</math>, and <math>Q</math> are propositions expressed in some [[formal system]].

== Natural language examples ==
Here is an example:
# It is red or it is blue.
# It is not blue.
# Therefore, it is red.

Here is another example:
# The breach is a safety violation, or it is not subject to fines.
# The breach is not a safety violation.
# Therefore, it is not subject to fines.

==Strong form==
''Modus tollendo ponens'' can be made stronger by using [[exclusive disjunction]] instead of inclusive disjunction as a premise:

:<math>\frac{P \underline\lor Q, \neg P}{\therefore Q}</math>

==Related argument forms==
Unlike ''[[modus ponens]]'' and ''[[modus ponendo tollens]]'', with which it should not be confused, disjunctive syllogism is often not made an explicit rule or axiom of [[logical system]]s, as the above arguments can be proven with a combination of [[reductio ad absurdum]] and [[disjunction elimination]].

Other forms of syllogism include:
*[[hypothetical syllogism]]
*[[categorical syllogism]]

Disjunctive syllogism holds in classical propositional logic and [[intuitionistic logic]], but not in some [[paraconsistent logic]]s.<ref>Chris Mortensen, [http://plato.stanford.edu/entries/mathematics-inconsistent/ Inconsistent Mathematics], ''Stanford encyclopedia of philosophy'', First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008</ref>

== See also ==
* [[Stoic logic]]
*[[Syllogism#Other types|Type of syllogism (disjunctive, hypothetical, legal, poly-, prosleptic, quasi-, statistical)]]

==References==
{{reflist}}

[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]
[[Category:Classical logic]]
[[Category:Paraconsistent logic]]