Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese

definition - Modus_ponens

definition of Wikipedia

   Advertizing ▼


Modus ponens


In propositional logic, modus ponendo ponens (Latin for "the way that affirms by affirming"; often abbreviated to MP or modus ponens[1][2][3][4]) or implication elimination is a valid, simple argument form and rule of inference.[5] It can be summarized as "P implies Q; P is asserted to be true, so therefore Q must be true."

While it 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"[6] 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 ever-lengthening string of symbols; for this reason modus ponens is sometimes called the rule of detachment.[7] Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",[8] 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 premiss [sic]; it is the dissolution of an implication".[9]

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".[10] 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.[11] An example is:

If it's raining, I'll meet you at the movie theater.
It's raining.
Therefore, I'll meet you at the movie theater.

Modus ponens can be stated formally as:

\frac{P \to Q, P}{\therefore Q}

where the rule is that whenever an instance of "P \to Q" and "P" appear by themselves on lines of a logical proof, "Q" can validly be placed on a subsequent line.

It is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms such as affirming the consequent, denying the antecedent, and evidence of absence. Constructive dilemma is the disjunctive version of modus ponens. Hypothetical syllogism is closely related to modus ponens and sometimes thought of as "double modus ponens."


  Formal notation

The modus ponens rule may be written in sequent notation:

P \to Q, P \vdash Q

where \vdash is a metalogical symbol meaning that Q is a syntactic consequence of P \to Q and P in some logical system;

or as the statement of a truth-functional tautology or theorem of propositional logic:

((P \to Q) \land P) \to Q

where P, and Q are propositions expressed in some logical system.


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 chaining.

An example of an argument that fits the form modus ponens:

If today is Tuesday, then John will go to work.
Today is Tuesday.
Therefore, John will go to work.

This argument is valid, but this has no bearing on whether any of the statements in the argument are true; for modus ponens to be a sound argument, the premises must be true for any true instances of the conclusion. An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is not only sound on Tuesdays (when John goes to work), but valid on every day of the week. A propositional argument using modus ponens is said to be deductive.

In single-conclusion sequent calculi, modus ponens is the Cut rule. The cut-elimination theorem for a calculus says that every proof involving Cut can be transformed (generally, by a constructive method) into a proof without Cut, and hence that Cut is admissible.

The Curry-Howard correspondence between proofs and programs relates modus ponens to function application: if f is a function of type P → Q and x is of type P, then f x is of type Q.

  Justification via truth table

The validity of modus ponens in classical two-valued logic can be clearly demonstrated by use of a truth table.

p q p → q

In instances of modus ponens we assume as premises that p → q is true and p is true. Only one line of the truth table—the first—satisfies these two conditions (p and p → q). On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true.

  Formal proof

Proposition Derivation
P\rightarrow Q Given
P\,\! Given
\neg P\or Q Material implication
\neg\neg P Double Negation
Q\,\! Disjunctive syllogism

  See also


  1. ^ Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.. 
  2. ^ Copi and Cohen
  3. ^ Hurley
  4. ^ Moore and Parker
  5. ^ Enderton 2001:110
  6. ^ Alfred Tarski 1946:47. Also Enderton 2001:110ff.
  7. ^ Tarski 1946:47
  8. ^ Enderton 2001:111
  9. ^ Whitehead and Russell 1927:9
  10. ^ Russell 1927:9
  11. ^ Jago, Mark (2007). Formal Logic. Humanities-Ebooks LLP. ISBN 978-1-84760-041-7. 


  • Alfred Tarski 1946 Introduction to Logic and to the Methodology of the Deductive Sciences 2nd Edition, reprinted by Dover Publications, Mineola NY. ISBN 0-486-28462-X (pbk).
  • Alfred North Whitehead and Bertrand Russell 1927 Principia Mathematica to *56 (Second Edition) paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.
  • HerbertB. Enderton, 2001, A Mathematical Introduction to Logic Second Edition, Harcourt Academic Press, Burlington MA, ISBN-13: 978-0-12-238452-3.

  External links



All translations of Modus_ponens

sensagent's content

  • definitions
  • synonyms
  • antonyms
  • encyclopedia

Dictionary and translator for handheld

⇨ New : sensagent is now available on your handheld

   Advertising ▼

sensagent's office

Shortkey or widget. Free.

Windows Shortkey: sensagent. Free.

Vista Widget : sensagent. Free.

Webmaster Solution


A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !

Try here  or   get the code


With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.

Business solution

Improve your site content

Add new content to your site from Sensagent by XML.

Crawl products or adds

Get XML access to reach the best products.

Index images and define metadata

Get XML access to fix the meaning of your metadata.

Please, email us to describe your idea.


The English word games are:
○   Anagrams
○   Wildcard, crossword
○   Lettris
○   Boggle.


Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.


Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !

English dictionary
Main references

Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).


The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.


Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.

last searches on the dictionary :

4360 online visitors

computed in 0.062s

I would like to report:
section :
a spelling or a grammatical mistake
an offensive content(racist, pornographic, injurious, etc.)
a copyright violation
an error
a missing statement
please precise:



Company informations

My account



   Advertising ▼