subformulae. Then repeat this recursive deconstruction process to each of the subformulae. Eventually the formula should be reduced to its atoms, but if some subformula cannot be so reduced, then the formula is not a wff. Calculus All wffs of the form ((U | (V | W)) | ((Y | (Y | Y)) | ((X | V) | ((U | X) | (U | X))))) are axioms. Instances of are inference rules. Simplification Since the only connective of this logic is |, the symbol | could be discarded altogether, leaving only the parentheses to group the letters. A pair of parentheses must always enclose a pair of wffs. Examples of theorems in this simplified notation are (p(p(q(q((pq)(pq)))))), (p(p((qq)(pp)))). The notation can be simplified further, by letting for any U. This simplification causes the need to change some rules: More than two letters are allowed within parentheses. Letters or wffs within parentheses are allowed to commute. Repeated letters or wffs within a same set of parentheses can be eliminated. The result is a parenthetical version of the Peirce existential graphs. Another way to simplify the notation is to eliminate parentheses by using Polish Notation. For example, the earlier examples with only parentheses could be rewritten using only strokes as follows (p(p(q(q((pq)(pq)))))) becomes | p | p | q | q || pq | pq, and (p(p((qq)(pp)))) becomes, | p | p || qq | pp. This follows the same rules as the parenthesis version, with the opening parenthesis replaced with a Sheffer stroke and the (redundant) closing parenthesis removed. Or (for some formulas) one could omit both parentheses and strokes and allow the order of the arguments to determine the order of function application so that for example, applying the function from right to left (reverse Polish notation – any other unambiguous convention based on ordering would do) See also Boolean domain CMOS Gate equivalent (GE) Laws of Form Logical graph Minimal axioms for Boolean algebra NAND flash memory NAND logic Peirce's law Peirce arrow = NOR Sole sufficient operator Notes References Bocheński, Józef Maria (1960), Precis of Mathematical Logic, rev., Albert Menne, edited and translated from the French and German editions by Otto Bird, Dordrecht, South Holland: D. Reidel. Charles Sanders Peirce, 1880, "A Boolian[sic] Algebra with One Constant", in Hartshorne, C. and Weiss, P., eds., (1931–35) Collected Papers of Charles Sanders Peirce, Vol. 4: 12–20, Cambridge: Harvard University Press.