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| {{Transformation rules}}
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| In [[propositional calculus|propositional logic]], '''''modus ponendo ponens''''' ([[Latin]] for "the way that affirms by affirming"; often abbreviated to '''MP''' or '''''modus ponens'''''<ref>{{cite book | last=Stone | first=Jon R. | year=1996 | title=Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language | publisher=London, UK: Routledge: 60.}}</ref><ref>Copi and Cohen</ref><ref>Hurley</ref><ref>Moore and Parker</ref>) or '''implication elimination''' is a [[validity|valid]], simple [[argument form]] and [[rule of inference]].<ref>Enderton 2001:110</ref> 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.<ref>[[Susanne Bobzien]] (2002). The Development of Modus Ponens in Antiquity, ''Phronesis'' 47.</ref>
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| 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".<ref>Alfred Tarski 1946:47. Also Enderton 2001:110ff.</ref> ''Modus ponens'' allows one to eliminate a [[material conditional|conditional statement]] from a [[formal proof|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'''.<ref>Tarski 1946:47</ref> Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",<ref>Enderton 2001:111</ref> 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".<ref>Whitehead and Russell 1927:9</ref> | |
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| 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".<ref>Whitehead and Russell 1927:9</ref> In other words: if one [[statement (logic)|statement]] or [[proposition]] [[material conditional|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.<ref>{{cite book | last=Jago | first=Mark | title=Formal Logic | publisher=[http://www.humanities-ebooks.co.uk/ Humanities-Ebooks LLP] |year= 2007 |isbn=978-1-84760-041-7 }}</ref> An example is:
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| :If it's raining, I'll meet you at the movie theater.
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| :It's raining.
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| :Therefore, I'll meet you at the movie theater.
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| ''Modus ponens'' can be stated formally as:
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| :<math>\frac{P \to Q,\; P}{\therefore Q}</math>
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| where the rule is that whenever an instance of "''P'' → ''Q''" and "''P''" appear by themselves on lines of a logical proof, ''Q'' can validly be placed on a subsequent line; furthermore, the premise ''P'' and the implication "dissolves", their only trace being the symbol ''Q'' that is retained for use later e.g. in a more complex deduction.
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| 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 [[Logical disjunction|disjunctive]] version of modus ponens. [[Hypothetical syllogism]] is closely related to modus ponens and sometimes thought of as "double modus ponens."
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| == Formal notation ==
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| The ''modus ponens'' rule may be written in [[sequent]] notation:
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| :<math>P \to Q,\; P\;\; \vdash\;\; Q</math>
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| where ⊦ is a [[metalogic]]al symbol meaning that ''Q'' is a [[syntactic consequence]] of ''P'' → ''Q'' and ''P'' in some [[formal system|logical system]];
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| or as the statement of a truth-functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic:
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| :<math>((P \to Q) \land P) \to Q</math>
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| where ''P'', and ''Q'' are propositions expressed in some logical system.
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| == Explanation ==
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| The argument form has two premises (hypothesis). The first premise is the "if–then" or [[Logical conditional|conditional]] claim, namely that ''P'' implies ''Q''. The second premise is that ''P'', the [[Antecedent (logic)|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]].
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| An example of an argument that fits the form ''modus ponens'':
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| :If today is Tuesday, then John will go to work.
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| :Today is Tuesday.
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| :Therefore, John will go to work.
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| This argument is valid, but this has no bearing on whether any of the statements in the argument are [[Truth|true]]; for ''modus ponens'' to be a sound argument, the premises must be true for any true instances of the conclusion. An [[Logical argument|argument]] can be valid but nonetheless [[Soundness|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 logic|propositional]] argument using ''modus ponens'' is said to be [[deductive logic|deductive]].
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| In single-conclusion [[sequent calculus|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 rule|admissible]].
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| 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''.
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| == Relation to Modus Tollens ==
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| Any Modus Ponens rule can be proved using a Modus Tollens rule and transposition.
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| :The proof is as follows.
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| :1. P → Q
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| :2. P /∴ Q
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| :3.~Q → ~P 1 Transposition
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| :4.~~P 2 Double Negation
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| :5.~~Q 3,4 Modus Tollens
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| :6. 5 Double Negation
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| ==Justification via truth table==
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| The validity of ''modus ponens'' in classical two-valued logic can be clearly demonstrated by use of a [[truth table]].
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| {| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:45%"
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| |+ ''' '''
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| |-
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| ! style="width:15%" | ''p''
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| ! style="width:15%" | ''q''
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| ! style="width:15%" | ''p'' → ''q''
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| |-
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| | bgcolor=#FFFF00 | T
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| | bgcolor=#00FF00 | T
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| | bgcolor=#00FFFF | T
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| |- style="color:gray"
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| | bgcolor=#FFFF00 | T || F || F
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| |- style="color:gray"
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| | F || T
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| | bgcolor=#00FFFF | T
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| |- style="color:gray"
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| | F || F
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| | bgcolor=#00FFFF | T
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| |}
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| <br> | |
| 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. | |
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| ==See also==
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| *[[What the Tortoise Said to Achilles]]
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| *[[Condensed detachment]]
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| == References ==
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| {{Reflist}}
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| == Sources ==
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| *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).
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| *[[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.
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| *Herbert B. Enderton, 2001, ''A Mathematical Introduction to Logic Second Edition'', Harcourt Academic Press, Burlington MA, ISBN 978-0-12-238452-3.
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| == External links ==
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| *{{springer|title=Modus ponens|id=p/m064570}}
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| *{{PhilPapers|search|modus_ponens}}
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| * ''[http://mathworld.wolfram.com/ModusPonens.html Modus ponens]'' at Wolfram MathWorld
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| {{DEFAULTSORT:Modus Ponens}}
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| [[Category:Rules of inference]]
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| [[Category:Latin logical phrases]]
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| [[Category:Theorems in propositional logic]]
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| [[Category:Classical logic]]
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Getting started online through having an Internet industry is not difficult, but you'll need some direction and training if you wish to be positive. It took me about per annum to make any money when I first came online in 2005, and which was because I have been trying to achieve it myself. While i began working with a mentor, put some software in place, and began serving one main niche, that is when I started making money and feeling like a honest entrepreneur.
Match level. You will likely record 10-15 songs for an album, with each will have its own level of quality. By matching levels, that doesn't signify each song has to sound the same, but it surely does imply you are looking for the heart of each track promote it be different minus the imperfections that will seep towards a taped performance. Again, you don't want 10-15 identical songs. But you do want songs that each bear a conventional of quality and professionalism and reliability.
Now, let's take a closer look at what I had suggested earlier on. That is developing your personal personal product. Performing this so, lowering the end up having no competition. In addition, anyone were to set up an associate at work program, find out then supply the affiliate marketers doing the promotion for you. This way, you would end up making sales not since a outcomes of your own efforts additionally as a consequence of the promotion how the affiliate marketers would be doing regular on your behalf. This way, I had managed to increase my personal income by four times as a result of my first product start off.
It may take a few days to move on. it may even take each and every year or two to turn a significant profit. but as long as you remain focused and consistently grow, you'll causes it to be.
If you compare a pair of paragraphs and think about third world countries, dazzling the opportunities there to order second career - the stuff we keep on the subject of. They have low to no technology right now, consider where they are in a couple years. hmmm can you hear opportunity knocking using your door right now?
In my personal experience, in case you are trying produce a profitable business with zero start off money and zero experience, you're going to have to operate very hard to get it going.
The moon also rules the 5th house in the Indian horocope. Love and Romance will be on the growth in 2011. Indians who are single discover their real love. Those that married can have more romantic movie.
In picture printers consumption use any paper size; you should use the perfect specified one, which isn't case with inkjet. Even if you want to print an incredibly small photo, a picture printer still uses total film. If dust gets into it chances are they'll show vertical scratches in regards to the pictures. A major difference is the price. Photographs printer costs $$ in order to inkjet inkjet printer. A difference which most buyers receive!
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