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Encyclopedia > Modus ponens

In logic, modus ponens (Latin: mode that affirms; often abbreviated MP) is a valid, simple argument form. It is a very common form of reasoning, and takes the following form: Logic, from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos (the word), is the study of the principles and criteria of valid inference and demonstration. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ... In psychology a conclusion is said to be valid, if and only if, it is based on true premises. ... In logic, the argument form or test form of an argument results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. ...

If P, then Q.
P.
Therefore, Q.

In logical operator notation: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

P → Q
P
⊢ Q

where ⊢ represents the logical assertion ("Therefore Q is true"). The logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. ...

The modus ponens rule may also be written:

$qquadfrac{(P rightarrow Q), P}{Q}$

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 propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

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: In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion. ... Common dictionary definitions of truth mention some form of accord with fact or reality. ... In logic, an argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ...

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. Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ...

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). The metalogic of a system of logic is the formal proof supporting its soundness. ... The Cut-elimination theorem is the central result establishing the significance of the sequent calculus. ... A rule of inference is admissible with respect to a logical system in case: If the rule belongs to the system, every theorem that can be proven making use of the rule can be proven without making use of it; If the rule doesnt belong to the system, then... In proof theory and mathematical logic, the sequent calculus is a widely known deduction system for first-order logic (and propositional logic as a special case of it). ...

For an amusing dialog that problematizes modus ponens, see Lewis Carroll's "What the Tortoise Said to Achilles." Lewis Carroll. ... What the Tortoise Said to Achilles is a brief dialog by Lewis Carroll which playfully problematizes the foundations of logic. ...

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: In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

P → Q
Q → R
P
⊢ R

Results from FactBites:

 Modus ponens - Wikipedia, the free encyclopedia (306 words) 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. A propositional argument using modus ponens is said to be deductive. Modus ponens can also be referred to as affirming the antecedent.
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