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''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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645701.png" /></td> </tr></table>
+
$$
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645703.png" /> denote formulas in a formal logical system, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645704.png" /> is the logical connective of [[Implication|implication]]. Modus ponens allows one to deduce <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645705.png" /> from the premise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645706.png" /> (the minor premise) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645707.png" /> (the major premise). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m0645709.png" /> are true in some interpretation of the formal system, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457011.png" /> as the least class that contains the formulas from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457012.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457013.png" /> from the derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457015.png" /> and the derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457017.png" />.
+
\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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457018.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\neg B \  A \lor B }{A}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457019.png" /> stands for negation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064570/m06457020.png" /> denotes the logical  "or" .
+
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
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article