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Modus ponens
In propositional logic, modus ponendo ponens
(Latin for "the way that affirms by affirming";
often abbreviated to MP or modus ponens) or
implication elimination is a valid, simple
argument form and rule of inference. It can be
summarized as "P implies Q; P is asserted to be
true, so therefore Q must be true."
The history of modus ponens goes back to
antiquity.
While modus ponens is one of the most commonly used
concepts in logic it must not be mistaken for a logical law;
rather, it is one of the accepted mechanisms for the
construction of deductive proofs that includes the "rule
of definition" and the "rule of substitution". Modus
ponens allows one to eliminate a conditional statement
from a logical proof or argument (the antecedents) and
thereby not carry these antecedents forward in an everlengthening string of symbols; for this reason modus
ponens is sometimes called the rule of detachment.
Enderton, for example, observes that "modus ponens can
produce shorter formulas from longer ones", and Russell
observes that "the process of the inference cannot be
reduced to symbols. Its sole record is the occurrence of
⊦q [the consequent] . . . an inference is the dropping of a
true premise; it is the dissolution of an implication".
A justification for the "trust in inference is the
belief that if the two former assertions [the
antecedents] are not in error, the final assertion
[the consequent] is not in error". In other words:
if one statement or proposition implies a second
one, and the first statement or proposition is
true, then the second one is also true. If P
implies Q and P is true, then Q is true. An
example is:
If it is raining, I will meet you at the theater. It is
raining. Therefore, I will meet you at the theater.