Jon AS, I noticed that I hadn't answered the question about incomplete propositions. In the 1930s, the logician Alonzo Church introduced lambda expressions as a notation for deriving a predicate (rheme) from a proposition. In effect, Peirce invented "lambda expressions" about 40 years before Church: every proposition with N-blanks (for Peirce) is equivalent to an N-adic lambda expression for Church and all logicians following Church.CSP: In the first place, I say that every relationship concerns some definite number of correlates ... We may express this as saying that every relation has a definite number of blanks to be filled by indices, or otherwise ...In a complete proposition there are no blanks. (CP 3.464-465, 1897)CSP: By a rheme, or predicate, will here be meant a blank form of proposition which might have resulted by striking out certain parts of a proposition, and leaving a blank in the place of each, the parts stricken out being such that if each blank were filled with a proper name, a proposition (however nonsensical) would thereby be recomposed. (CP 4.560, 1906)JAS> Similar passages include CP 2.379 (1902), CP 2.272 (1903), and CP 4.454 (1903). Rhemes as incomplete propositions are monads, dyads, triads, etc. based on the number of blanks; but a complete proposition is a medad, because it has no blanks.fA proposition p: "2 + 2 = 4"A rheme derived from p: "_ + _ = _"A lambda expression derived from p: "(λ x, y, z) (x + y = z)"Peirce was ahead of his time, and his pioneering terminology may be confusing to a modern reader. The developments in logic during the 20th c and 21st c reinvented and built on many ideas that were introduced by Peirce. In some cases, they went beyond Peirce. But in other cases, Peirce still has a great deal to teach the 21st c logicians.In any case, studying 21st c logic is extremely helpful for understanding what Peirce meant and the importance of his insights for developments today.John
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