Mathematical induction - Revision history

en>Jochen Burghardt: keep consistent with text before general rule

2014-12-28T20:07:14Z

keep consistent with text before general rule

← Older revision		Revision as of 22:07, 28 December 2014
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123.17.142.123: Add some spaces between mathematic symbols

2014-03-04T15:16:47Z

Add some spaces between mathematic symbols

Show changes

en>Markhurd: clean up, replaced: ’ → ' (2) using AWB

2014-01-22T15:02:40Z

clean up, replaced: ’ → ' (2) using AWB

← Older revision		Revision as of 17:02, 22 January 2014
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			{{Transformation rules}}

			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>

			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>

			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:

			: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:

			:<math>\frac{P \to Q,\; P}{\therefore Q}</math>

			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.

			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."

			== Formal notation ==

			The ''modus ponens'' rule may be written in [[sequent]] notation:
			:<math>P \to Q,\; P\;\; \vdash\;\; Q</math>

			where ⊦ is a [[metalogic]]al symbol meaning that ''Q'' is a [[syntactic consequence]] of ''P'' → ''Q'' and ''P'' in some [[formal system\|logical system]];

			or as the statement of a truth-functional [[Tautology (logic)\|tautology]] or [[theorem]] of propositional logic:

			:<math>((P \to Q) \land P) \to Q</math>

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

			== Explanation ==

			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]].

			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 [[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]].

			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]].

			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''.

			== Relation to Modus Tollens ==
			Any Modus Ponens rule can be proved using a Modus Tollens rule and transposition.
			:The proof is as follows.
			:1. P → Q
			:2. P /∴ Q
			:3.~Q → ~P 1 Transposition
			:4.~~P 2 Double Negation
			:5.~~Q 3,4 Modus Tollens
			:6. 5 Double Negation

			==Justification via truth table==
			The validity of ''modus ponens'' in classical two-valued logic can be clearly demonstrated by use of a [[truth table]].
			{\| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:45%"
			\|+ ''' '''
			\|-
			! style="width:15%" \| ''p''
			! style="width:15%" \| ''q''
			! style="width:15%" \| ''p'' → ''q''
			\|-
			\| bgcolor=#FFFF00 \| T
			\| bgcolor=#00FF00 \| T
			\| bgcolor=#00FFFF \| T
			\|- style="color:gray"
			\| bgcolor=#FFFF00 \| T \|\| F \|\| F
			\|- style="color:gray"
			\| F \|\| T
			\| bgcolor=#00FFFF \| T
			\|- style="color:gray"
			\| F \|\| F
			\| bgcolor=#00FFFF \| T
			\|}
			<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.

			==See also==
			*[[What the Tortoise Said to Achilles]]
			*[[Condensed detachment]]

			== References ==
			{{Reflist}}

			== Sources ==
			*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.
			*Herbert B. Enderton, 2001, ''A Mathematical Introduction to Logic Second Edition'', Harcourt Academic Press, Burlington MA, ISBN 978-0-12-238452-3.

			== External links ==
			*{{springer\|title=Modus ponens\|id=p/m064570}}
			*{{PhilPapers\|search\|modus_ponens}}
			* ''[http://mathworld.wolfram.com/ModusPonens.html Modus ponens]'' at Wolfram MathWorld

			{{DEFAULTSORT:Modus Ponens}}
			[[Category:Rules of inference]]
			[[Category:Latin logical phrases]]
			[[Category:Theorems in propositional logic]]
			[[Category:Classical logic]]

en>EmausBot: Robot: eo:Matematika indukto is a good article

2012-08-25T10:16:52Z

Robot: eo:Matematika indukto is a good article

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