Difference between revisions of "Peirce arrow"
From Encyclopedia of Mathematics
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| − | A two-place [[Logical operation|logical operation]], usually denoted by | + | {{TEX|done}} |
| + | A two-place [[Logical operation|logical operation]], usually denoted by $\downarrow$, specified by the following [[Truth table|truth table]]: | ||
| − | < | + | <center> |
| + | {| border="1" class="wikitable" style="text-align:center; width:300px;" | ||
| + | |$A$||$B$||$A\downarrow B$ | ||
| + | |- | ||
| + | |$T$||$T$||$F$ | ||
| + | |- | ||
| + | |$T$||$F$||$F$ | ||
| + | |- | ||
| + | |$F$||$T$||$F$ | ||
| + | |- | ||
| + | |$F$||$F$||$T$ | ||
| + | |} | ||
| + | </center> | ||
| − | Therefore, the statement | + | Therefore, the statement $A\downarrow B$ denotes "neither A nor B". Peirce's arrow has the property that all logical operations can be expressed in terms of it. For example, the statement $\neg A$ (the [[Negation|negation]] of $A$) is equivalent to the statement $A\downarrow A$, while the [[Conjunction|conjunction]] $A\&B$ of two statements $A$ and $B$ is expressed as $(A\downarrow A)\downarrow(B\downarrow B)$ and the disjunction $A\lor B$ is equivalent to $(A\downarrow B)\downarrow(A\downarrow B)$. This arrow was introduced by C. Peirce. |
====Comments==== | ====Comments==== | ||
| − | A more familiar two-place logical operation in terms of which all others can be expressed is the so-called [[Sheffer stroke|Sheffer stroke]] "AB" : either | + | A more familiar two-place logical operation in terms of which all others can be expressed is the so-called [[Sheffer stroke|Sheffer stroke]] "AB": either $A$ or $B$ but not both. The Peirce arrow and the Sheffer stroke are each other negations. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139</TD></TR></table> | ||
Revision as of 12:22, 12 August 2014
A two-place logical operation, usually denoted by $\downarrow$, specified by the following truth table:
| $A$ | $B$ | $A\downarrow B$ |
| $T$ | $T$ | $F$ |
| $T$ | $F$ | $F$ |
| $F$ | $T$ | $F$ |
| $F$ | $F$ | $T$ |
Therefore, the statement $A\downarrow B$ denotes "neither A nor B". Peirce's arrow has the property that all logical operations can be expressed in terms of it. For example, the statement $\neg A$ (the negation of $A$) is equivalent to the statement $A\downarrow A$, while the conjunction $A\&B$ of two statements $A$ and $B$ is expressed as $(A\downarrow A)\downarrow(B\downarrow B)$ and the disjunction $A\lor B$ is equivalent to $(A\downarrow B)\downarrow(A\downarrow B)$. This arrow was introduced by C. Peirce.
Comments
A more familiar two-place logical operation in terms of which all others can be expressed is the so-called Sheffer stroke "AB": either $A$ or $B$ but not both. The Peirce arrow and the Sheffer stroke are each other negations.
References
| [a1] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139 |
How to Cite This Entry:
Peirce arrow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peirce_arrow&oldid=32867
Peirce arrow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peirce_arrow&oldid=32867
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article