H. P. Grice's pragmatic, but not semantic, ground for rejecting Peirce's stroke

Formal systems of logic, and formal languages, have expressive resources that are symbolic. Economic use of those resources means using in the construction and implementation of the theory as few such resources as is possible without loss of any systemic powers of expression. Ideally, economy dictates that only one resource of a certain kind is to be used, if such a resource is available or definable and is effective in the construction of all the remaining expressive resources. In the case of the connective symbols of a formal language of propositional logic, this reduction to one effective symbol proves to be feasible in the case of the standard propositional logic: hence, the revelatory significance of Sheffer’s discovery (which, as seen, had already been achieved by Peirce.) For the reduction to be effective, of course, it must be the case that all other connectives (of any arity ) must be definable in terms of the single connective symbol; in this way all the other connectives can be eliminated as expressive resources without causing a loss of the ability to express what those symbols refer to. Thus, for example, instead of “”, one can write “”, and the same for all other connective symbols.

The advantages obtained from reduction of resources can be concrete in the case of implementations or applications of formal systems. For instance, in the construction of logic gates in electronic circuitry, the gate-types NAND and NOR are the electronic-theoretical interpretations of the same Boolean functions that are propositionally interpreted as the Sheffer connectives. As one ought to expect, NAND and NOR are universal gates. This means that any theoretically definable gate can be actually constructed from using just NAND gates or just NOR gates. Discoveries of this kind signal that a reduction in complexity is feasible, and this result can have economic and design advantages.

In practice, the advantage claimed from this reduction is outweighed by the fact that writing out well-formed expressions becomes prohibitively unwieldy if only one kind of connective symbol is used. For example, in the history of modern logic, Gottlob Frege’s notational variant never had a chance of being widely adopted because of the practically unmanageable demands it placed on typographical execution. One can think of this challenge as posing a trade-off between economy of resources and notational convenience. Or the trade-off is between reducing the type of resource (for instance, gate) used and needed, on the one hand, and the length or extension of the constructions that will have to be made, on the other. For example, to return to propositional logic, to express a well-formed formula like “” in terms of a single connective symbol, , one must write out the much longer equivalent well-formed formula shown below. The notational version being used in this way is significantly more unwieldy than a notational version (a grammar) that uses more, not fewer, connective symbols. Consider the formula

  

It is possible to adopt conventions that remove its complexity to some degree. For instance, stipulating that “” is to be written as “”, permits simplification of the formula above to

  

It is less obvious whether there is a deeper philosophical significance of the fact that a connective like Sheffer’s Stroke is available in a system of logic. Whitehead and Russell expressed boundless enthusiasm about Sheffer’s discovery, hinting only at an underlying significance of this while adopting the connective symbol in the second edition of Principia Mathematica. On the other hand, two other pioneering writers of logic textbooks, Hilbert and Ackermann, were unimpressed and reported on the Sheffer Stroke as if they were referring to trivia. Certainly, the Sheffer functions do not add to the logical system of standard propositional logic in any way. The simplification they make possible is an internal matter. If there are other logics for which, hypothetically, Sheffer functions are not available, this does not automatically mean that there is something wrong with those other systems insofar as they are assessed as formally constructed languages.

It was the influential thinker Ludwig Wittgenstein who attributed far-reaching significance to the fact that Sheffer functions are available. He did this in a somewhat obscure fashion in an influential logical-philosophical work.