² See Carnap, R., Logical foundations of probability, Chicago, 1950, pp. 295–296Google Scholar, and Leblanc, H., On logically false evidence statements, this Journal, vol. 22 (1957), no. 4, pp. 345–349.Google Scholar A1–A4 come from Rosser, J. B., Logic for mathematicians, New York, 1953, pp. 54–59Google Scholar; A5–A9 are equivalent, as the reader may check, to Al, A2, A3, A4′, and A5 in Leblanc, loc. cit., p. 345.

³ See Popper, K. R., The logic of scientific discovery, New York, 1959Google Scholar, Appendix ^*iv and Appendix ^*v. B1–B6 come from Popper, loc. cit., pp. 332–333 (where the axiom B1' mentioned in footnote 6 but nowhere given is, in Popper's notation, p(ab, c) =p(ba,c)); B7 comes from p. 357. Popper's system S for which B1–B6 (plus an extra requirement to the effect that Pr(A, B) ≠ Pr(C, D) for at least two C and D of S) are intended, differs from PC in several respects: (1) S is a Boolean algebra of sorts, equipped with variables ranging over arbitrary objects, the two Boolean functors ‘—’ and ‘̑’, the standard connectives, and the standard quantifiers, whereas PC is a prepositional calculus; (2) the functor ‘Pr’, which we treat as a primitive sign of the metalanguage of PC, is treated by Popper as a primitive sign of S; (3) Tarski's elementary algebra, which we presume to be part of the metalanguage of PC, is presumed by Popper to be part of S. As for the ‘Pr(A, ∼A) = 1' of B7, which serves with us as a definiens for the metalogical phrase ‘├’, it serves in Popper as a definiens for a phrase of S, ‘A is necessary’. Credit for method B should nonetheless go to Popper.

⁴ Popper undoubtedly meant B1–B7 to be equivalent to such requirements as A1–A9. He did not, however, establish (1) that B1–B7 yield A1–A7 (and vice-versa), nor, in particular, (2) that B1–B7 yield A1–A4. He proved instead (3) that with the four letters ‘A’, ‘B’, ‘C’, and ‘D’ serving as variables of S and the phrase ‘A = B’ of S defined as ‘(C)(Pr(A, C) = Pr(B, C))’, B1–B6 yield Huntington's, E. V. fourth set of postulates for Boolean algebra in Transactions of the American Mathematical Society, vol. 35 (1933), pp. 274–304.Google Scholar That (3), the result we shall use in section 3 to obtain (1), is weaker than (2) should be clear from footnote 7.

⁵ The above proof of All, considerably shorter than the author's original one, is due to the referee, who should be thanked for his kindly advice and suggestions.

⁶ The above proof of B11 is modeled after Popper's proof of formula (70), loc. cit., page 353.

⁷ The outcomes in question yield as theorems only wffs of the form A ≡ B and hence do not constitute a complete set of postulates for PC. This point was brought to the author's attention by Professor Robert McNaughton. Popper's requirements for Pr in Appendix ^*v differ slightly from B1–B6. In footnote 6, page 333, Popper points out, however, that his requirements are equivalent to B1–B6.

⁸ It follows from the above that the outcomes of substituting ‘∼’ for ‘—’, ‘V’ for ‘̆’ (or ‘&’ for ‘̑’), and ‘≡’ for ‘=’ in a set of postulates for Boolean algebra of the Huntington type yield as theorems all tautologies so-called of PC once they are supplemented with B18 and B23. The same result holds true with the one-premise rule

B23′. If ├(A&B) = A, then ├A⊃B,

in place of B23. The two-premise rule B23 was suggested by the referee as an alternative to B23′, which the author had used in a previous version of this paper.