Disjunctive syllogism
inference rule in logics : with "A or B" and "not A" deduce "B"
In classical logic, disjunctive syllogism (historically known as modus tollendo ponens (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.
An example in English:
I will choose soup or I will choose salad.
I will not choose soup.
Therefore, I will choose salad.
Propositional logic
In propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), 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 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 disjunction from a logical proof. It is the rule that
P
∨
Q
,
¬
P
∴
Q
{\displaystyle {\frac {P\lor Q,\neg P}{\therefore Q}}}
where the rule is that whenever instances of "
P
∨
Q
{\displaystyle P\lor Q}
", and "
¬
P
{\displaystyle \neg P}
" appear on lines of a proof, "
Q
{\displaystyle Q}
" can be placed on a subsequent line.
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