SUO: Re: Zeroth Order Ontology
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
ZOO. Discussion Note 13
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Historically speaking, the entry point for this discussion is located here,
with Peirce's discovery of a dual pair of logical operators that he called
the "amphecks", from the Greek for "cutting both ways", or that later came
to be called Sheffer's "strokes", or to computer scientists, NAND and NNOR.
| [A Boolian Algebra With One Constant]
|
| Every logical notation hitherto proposed has an unnecessary number of signs.
| Is is by means of this excess that the calculus is rendered easy to use and
| that a symmetrical development of the subject is rendered possible; at the
| same time, the number of primary formulae is thus greatly multiplied, those
| signifying facts of logic being very few in comparison with those which
| merely define the notation. I have thought that it might be curious to
| see the notation in which the number of signs should be reduced to a
| minimum; and with this view I have constructed the following. The
| apparatus of the Boolian calculus consists of the signs, =, > (not
| used by Boole, but necessary to express particular propositions),
| +, -, x [·], 1, 0. In place of these seven signs, I propose to
| use a single one.
|
| I begin with the description of the notation for conditional
| or "secondary" propositions. The different letters signify
| propositions. Any one proposition written down by itself
| is considered to be asserted. Thus,
|
| A
|
| means that the proposition A is true.
| Two propositions written in a pair are
| considered to be both denied. Thus,
|
| A B
|
| means that the propositions A and B
| are both false; and
|
| A A
|
| means that A is false. We may have pairs of pairs of propositions
| and higher complications. In this case we shall make use of commas,
| semicolons, colons, periods, and parentheses, just as [in] chemical
| notation, to separate pairs which are themselves paired. These
| punctuation marks can no more count for distinct signs of
| algebra, than the parentheses of the ordinary notation.
|
| To express the proposition: "If S then P",
| first write:
|
| A
|
| for this proposition. But the proposition
| is that a certain conceivable state of things
| is absent from the universe of possibility.
| Hence instead of A we write:
|
| B B
|
| Then B expresses the possibility of S being true and
| P false. Since, therefore, SS denies S, it follows
| that (SS, P) expresses B. Hence we write:
|
| SS, P; SS, P.
|
| C.S. Peirce, 'Collected Papers', CP 4.12-14, circa 1880.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o