# Sheffer stroke

2010 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:

$$(A|A)|(B|B).$$

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.

#### References

 [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