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Information about 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:



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

See also

Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
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Latin}}} 
Official status
Official language of: Vatican City
Used for official purposes, but not spoken in everyday speech
Regulated by: Opus Fundatum Latinitas
Roman Catholic Church
Language codes
ISO 639-1: la
ISO 639-2: lat
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validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false).
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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.
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In logic, a rule of inference is a function from sets of formulae to formulae. The argument is called the premise set (or simply premises) and the value the conclusion.
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In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction.

Explanation

A sequent has the form


where both Γ and Σ are sequences of logical formulae (i.e.
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The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function ⊃ from truth-values to truth-values.
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artificial intelligence (or AI) is "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions which maximizes its chances of success.
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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.

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validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false).
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truth extends from honesty, good faith, and sincerity in general, to agreement with fact or reality in particular.[1] The term has no single definition about which the majority of professional philosophers and scholars agree.
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Only a valid argument with true premises must have a true conclusion.
The validity of an argument depends on its form, not on the truth or falsity of its premises and conclusions. Logic seeks to discover the forms of valid arguments.
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This article is about the soundness notion of informal logic. For soundness in mathematical logic, see soundness theorem.


A logical argument is sound if and only if
  1. the argument is valid

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This article is about the soundness notion of informal logic. For soundness in mathematical logic, see soundness theorem.


A logical argument is sound if and only if
  1. the argument is valid

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In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules
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The metalogic of a system of logic is the formal theory of the formal logic. Results in metalogic will consist of such things as formal proofs demonstrating the soundness of the logic.
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The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen 1934 in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical
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admissible with respect to a logical system in case:
  1. 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;

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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). The system is also known under the name LK
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Charles Lutwidge Dodgson (IPA: /ˈdɒdsən/) (January 27 1832 – January 14 1898), better known by the pen name Lewis Carroll (/ˈkærəl/
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"What the Tortoise Said to Achilles" is a brief dialogue by Lewis Carroll which playfully problematises the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race.
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In logic, a hypothetical syllogism has two uses. In propositional logic it expresses a rule of inference, while in the history of logic, it is a short-hand for the theory of consequence.
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In logic, Modus ponendo tollens (Latin for "mode that affirms by denying") is the formal name for indirect proof or proof by contraposition (contrapositive inference), often abbreviated to MT.
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Modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form:

P or Q
Not P
Therefore, Q


An English language example:


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Modus tollendo tollens (Latin: the way that denies by denying) is a valid rule of inference. It is closely related to Modus ponens and modus tollens. It takes the form:

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Affirming the consequent is a formal fallacy, committed by reasoning in the form:

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

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Denying the antecedent is a logical fallacy, committed by reasoning in the form:

If P, then Q.
Not P.
Therefore, not Q.

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A disjunctive syllogism, is a classically valid, simple argument form:

P or Q
Not P
Therefore, Q


In logical operator notation:
¬

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