MODUS PONENS

In logic, 'modus ponendo ponens' (Latin: ''mode that affirms by affirming''; often abbreviated 'MP') is a valid, simple argument form. It is a very common rule of inference, and takes the following form:
:If P, then Q.
:P.
:Therefore, Q.
In logical operator notation:
:
where represents the logical assertion (that Q is true).
The modus ponens rule may also be written:
:
ightarrow Q), P}{Q}.
The argument form has two premises. The first premise is the "if–then" or ''conditional'' claim, namely that P implies Q. The second premise is that P, the ''antecedent'' of the conditional claim, is true. From these two premises it can be logically concluded that Q, the ''consequent'' of the conditional claim, must be true as well. In Artificial Intelligence, modus ponens is often called forward reasoning.
Here is an example of an argument that fits the form ''modus ponens'':
:If today is Tuesday, then I will go to work.
:Today is Tuesday.
:Therefore, I will go to work.
The fact that the argument is valid cannot assure us that any of the statements in the argument are true; the validity of modus ponens tells us that the conclusion must be true if all the premises are true. It is wise to recall that a valid argument within which one or more of the premises are not true is called an ''unsound'' argument, whereas if all the premises are true, then the argument is ''sound''. In most logical systems, modus ponens is considered to be valid. However, the instances of its use may be either sound or unsound:
:If the argument is modus ponens and its premises are true, then it is sound.
:The premises are true.
:Therefore, it is a sound argument.
A propositional argument using modus ponens is said to be deductive.
Modus ponens can also be referred to as 'Affirming the Antecedent' or 'The Law of Detachment'.
In metalogics, modus ponens is the cut rule. The cut-elimination theorem says that the cut is valid (an admissible rule) in some logical calculus (sequent calculus).
For an amusing dialog that problematizes modus ponens, see Lewis Carroll's "What the Tortoise Said to Achilles."
An expanded form of the argument, called 'multiple modus ponens' (often abbreviated 'mmp'), also exists, and has the following form:
:If P, then Q.
:If Q, then R.
:P.
:Therefore, R.
In logical operator notation:
:P → Q
:Q → R
:P
:⊢ R

Contents

See also

See also

★ Hypothetical syllogism

★ Modus tollens

★ Modus tollendo ponens

★ Modus tollendo tollens

★ Affirming the consequent

★ Denying the antecedent

★ Disjunctive syllogism

★ Inference rule