After section 8 the Sheffer stroke sees no usage.

Hence any formal system including the Sheffer stroke must also include a means of indicating grouping.

The Sheffer stroke commutes but does not associate.

The expressive adequacy of the Sheffer stroke points to the pa also being a algebra of type .

To add to the complexity of the treatment, 8 introduces the notion of substituting a "matrix", and the Sheffer stroke:

The following is an example of a formal system based entirely on the Sheffer stroke, yet having the functional expressiveness of the propositional logic:

W. V. Quine's Mathematical Logic also made much of the Sheffer stroke.

For Quine, there is but one connective, the Sheffer stroke, and one quantifier, the universal quantifier.

In his best known work, he showed that the classical propositional calculus could be derived from one axiom and one rule, both expressed using the Sheffer stroke.

This follows the same rules as the parenthesis version, with opening parenthesis replaced with a Sheffer stroke and the (redundant) closing parenthesis removed.

Three bases for Boolean algebra are in common use, the lattice basis, the ring basis, and the Sheffer stroke or NAND basis.

It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND).

Sheffer stroke: Is the contemporary logical NAND (NOT-AND), i.e., "incompatibility", meaning:

Similarly, it is sufficient to have only and as logical connectives, or to have only the Sheffer stroke (NAND) or the Peirce arrow (NOR) operator.

Sheffer introduced what is now known as the Sheffer stroke in 1913; it became well known only after its use in the 1925 edition of Whitehead and Russell's Principia Mathematica.

Researchers have known for some time that single equational axioms (i.e., 1-bases) exist for Boolean algebra, including representation in terms of disjunction and negation and in terms of the Sheffer stroke.

Wittgenstein's N-operator is however an infinitary analogue of the Sheffer stroke, which applied to a set of propositions produces a proposition that is equivalent to the denial of every member of that set.

With the dot representing the Nand logical operation (also known as the Sheffer stroke), with the following meaning: p Nand q is true if and only if not both p and q are true.

The first published proof was by Henry M. Sheffer in 1913, so the NAND logical operation is sometimes called Sheffer stroke; the logical NOR is sometimes called Peirce's arrow.

It may be called by various other names including the polon, pipe (by the Unix community, referring to the I/O pipeline construct), Sheffer stroke (by computer or mathematical logicians), verti-bar, vbar, stick, vertical line, vertical slash, or bar, think colon, poley, or divider line.

Likewise, the propositional calculus could be formulated using a single connective, having the truth table either of the logical NAND, usually symbolized with a vertical line called the Sheffer stroke, or its dual logical NOR (usually symbolized with a vertical arrow or with a dagger symbol).