H. P. Grice recognises C. S. Peirce's geniality in aiming at REDUCTION -- but notes that, when reconstructing "not," "and," "or," and "if," reduction may not be the best policy -- especially if communicators are abiding by less philosophical expectations, like a 'principle of cooperative helpfulness' subsuming a 'desideratum of conversational clarity'

No doubt, one reason is that a system that would have only the Sheffer Stroke as its connective would require use of unwieldy formulaic expressions.