Hindsight on Some Conveniently Ignored History

First, in 1903 Davis gave an 8-group interpretation to De Morgan's eight forms of a proposition. For us, this is the same subgroup (N, N, N, -) that covers both the top half of Table I and the corresponding 8-cell of logical garnets. Note especially that Davis was one of Peirce's students. By then, as noted by Ladd-Franklin (1880), Peirce was already using his own symbol for "if, then" (the claw), a symbol that could be subjected to a symmetry operation. Also by then Peirce had already devised his box-X notation (1902).

Second, fifteen years before the 1902 in Peirce, Renton (1887) was very precise about what for us covers the eight symmetries on the front face of the large 3-cube. He published a short 16-page pamphlet that left out the mate operation (N, -, N, C). Believe it or not, as early as 1887 Renton also published the 8-group table for (D4), the one that lets the eight combinations of (negation, negation, conversion) act not on (A, B) but on (subject a, predicate b). There it was, nicely made explicit. Renton even included some special vocabulary for the eight operations. The four for (N-N-) were called nonverse, adverse, reverse, converse, and the four that added conversion (- - - C) were called inverse, severse, deverse, and proverse.

A one-page review of Renton (1888) gave considerable space to a standard interpretation of the syllogism. But the review did not even acknowledge that Renton gave us an 8-group description of the "Versation of the Proposition." It is worth noting that Renton did not use a notation that had the symmetry properties built into the symbols themselves. For us the same 8-group is another subgroup, (D4) for (N, -, N, C), the one that covers the first four and the third four rows of Table I. In fact, with praise for Renton, (D4) is at its best for the logic alphabet when the eight symmetries of a rigid square act on every one of the 16 LSs on the Clock Compass.

It is worth noting that, once while at the Peirce Edition Project in Indianapolis, Max Fisch (1900-1995) emphasized that Frege and Peirce, two of the most outstanding logicians of the 1800s, both came up with dimensionalized notations. Frege had a system of branching conditionals, but, even within the Peirce community, very few people know that Peirce had TWO dimensionalized notations. The first (1896+) is the better-known Existential Graphs and the second (1902+) is the one that is like the logic alphabet, namely, his rarely-recognized box-X notation for the 16 binary connectives. The code for this one is very simple. All combinations of the four sides of a box were added to enclose (negate) the 16 combinations of the four, small subareas (TT, TF, FT, FF) that surround the center of an all-common X-frame.

Historically, look at all of the 20th century. Did we get pulled into some misplaced axiomatics, namely, into a strong formal push to isolate the smallest convenient set of rigorously complete connectives ("and", "or", "if") for the propositional calculus? If not, then what were the prevaling mind-sets that prompted two to three generations of logicians to read past the untapped potential that stands fully present and clearly exposed in the many pages of Peirce's manuscripts?

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