Sheffer stroke

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2020 Mathematics Subject Classification: Primary: 03B05 [MSN][ZBL]

Sheffer bar

A logical operation, usually denoted by $|$, given by the following truth table:

$A$ $B$ $A|B$
$T$ $T$ $F$
$T$ $F$ $T$
$F$ $T$ $T$
$F$ $F$ $T$

Thus, the assertion $A|B$ means that $A$ and $B$ are incompatible, i.e. are not true simultaneously. All other logical operations can be expressed by the Sheffer stroke. For example, the assertion $\neg A$ (the negation of $A$) is equivalent to the assertion $A|A$; the disjunction $A\lor B$ of two assertions $A$ and $B$ is expressed as:


The conjunction $A\&B$ and the implication $A\to B$ are expressed as $(A|B)|(A|B)$ and $A|(B|B)$, respectively. Sheffer's stroke was first considered by H. Sheffer.


[1] H.M. Sheffer, "A set of five independent postulates for Boolean algebras, with applications to logical constants" Trans. Amer. Math. Soc. , 14 (1913) pp. 481–488


The Sheffer stroke operation is also called alternative denial.


[a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1950) pp. 139
[a2] W. Marek, J. Onyszkiewicz, "Elements of logic and the foundations of mathematics in problems" , Reidel & PWN (1982) pp. 4
How to Cite This Entry:
Sheffer stroke. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article