Editing Disjunctive syllogism (section)

==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]].