Difference between revisions of "Modus ponens"
From Encyclopedia of Mathematics
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| + | <!-- | ||
| + | m0645701.png | ||
| + | $#A+1 = 20 n = 0 | ||
| + | $#C+1 = 20 : ~/encyclopedia/old_files/data/M064/M.0604570 Modus ponens, | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| + | |||
| + | {{TEX|auto}} | ||
| + | {{TEX|done}} | ||
| + | |||
''law of detachment, rule of detachment'' | ''law of detachment, rule of detachment'' | ||
A [[Derivation rule|derivation rule]] in formal logical systems. The rule of modus ponens is written as a scheme | A [[Derivation rule|derivation rule]] in formal logical systems. The rule of modus ponens is written as a scheme | ||
| − | + | $$ | |
| − | |||
| − | |||
| − | + | \frac{A \ A \supset B }{B} | |
| + | , | ||
| + | $$ | ||
| + | where $ A $ | ||
| + | and $ B $ | ||
| + | denote formulas in a formal logical system, and $ \supset $ | ||
| + | is the logical connective of [[Implication|implication]]. Modus ponens allows one to deduce $ B $ | ||
| + | from the premise $ A $( | ||
| + | the minor premise) and $ A \supset B $( | ||
| + | the major premise). If $ A $ | ||
| + | and $ A \supset B $ | ||
| + | are true in some interpretation of the formal system, then $ B $ | ||
| + | is true. Modus ponens, together with other derivation rules and axioms of a formal system, determines the class of formulas that are derivable from a set of formulas $ M $ | ||
| + | as the least class that contains the formulas from $ M $ | ||
| + | and the axioms, and closed with respect to the derivation rules. | ||
| + | Modus ponens can be considered as an operation on the derivations of a given formal system, allowing one to form the derivation of a given formula $ B $ | ||
| + | from the derivation $ \alpha $ | ||
| + | of $ A $ | ||
| + | and the derivation $ \beta $ | ||
| + | of $ A \supset B $. | ||
====Comments==== | ====Comments==== | ||
The more precise Latin name of the law of detachment is modus ponendo ponens. In addition there is modus tollendo ponens, which is written as the scheme | The more precise Latin name of the law of detachment is modus ponendo ponens. In addition there is modus tollendo ponens, which is written as the scheme | ||
| − | + | $$ | |
| + | |||
| + | \frac{\neg B \ A \lor B }{A} | ||
| + | , | ||
| + | $$ | ||
| − | where | + | where $ \neg $ |
| + | stands for negation and $ \lor $ | ||
| + | denotes the logical "or" . | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Suppes, "Introduction to logic" , v. Nostrand (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Suppes, "Introduction to logic" , v. Nostrand (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974)</TD></TR></table> | ||
Latest revision as of 08:01, 6 June 2020
law of detachment, rule of detachment
A derivation rule in formal logical systems. The rule of modus ponens is written as a scheme
$$ \frac{A \ A \supset B }{B} , $$
where $ A $ and $ B $ denote formulas in a formal logical system, and $ \supset $ is the logical connective of implication. Modus ponens allows one to deduce $ B $ from the premise $ A $( the minor premise) and $ A \supset B $( the major premise). If $ A $ and $ A \supset B $ are true in some interpretation of the formal system, then $ B $ is true. Modus ponens, together with other derivation rules and axioms of a formal system, determines the class of formulas that are derivable from a set of formulas $ M $ as the least class that contains the formulas from $ M $ and the axioms, and closed with respect to the derivation rules.
Modus ponens can be considered as an operation on the derivations of a given formal system, allowing one to form the derivation of a given formula $ B $ from the derivation $ \alpha $ of $ A $ and the derivation $ \beta $ of $ A \supset B $.
Comments
The more precise Latin name of the law of detachment is modus ponendo ponens. In addition there is modus tollendo ponens, which is written as the scheme
$$ \frac{\neg B \ A \lor B }{A} , $$
where $ \neg $ stands for negation and $ \lor $ denotes the logical "or" .
References
| [a1] | P. Suppes, "Introduction to logic" , v. Nostrand (1957) |
| [a2] | A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974) |
How to Cite This Entry:
Modus ponens. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modus_ponens&oldid=13025
Modus ponens. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modus_ponens&oldid=13025
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article