² I take ‘∼’ and ‘⊃’ to be the only primitive connectives of QC (and hence of SC), and ‘∀’ to be the only primitive quantifier letter of QC.

³ For the sake of definiteness the individual variables of QC may be understood to run in the following alphabetical order: ‘x’, ‘y’, ‘z’, ‘x’, ‘x́’, ‘ý’, etc.

⁴ Like conventions will hold mutatis mutandis when X or Y both are individual constants rather than variables. See §8.

⁵ I owe the reference to Professor Bas C. van Fraassen. What Beth showed is that a formula A of QC is valid in the standard sense if and only if it is satisfied by every assignment of truth-values to the atomic formulas of QC. But the truth-values Beth assigns to the atomic formulas of QC that do not figure among the subformulus of A are clearly idle, and hence may be discarded. The reader may wish to verify that in clause (d) of our definition of satisfaction B(Y/X) cannot do duty for B[Y/X].

⁶ This because every individual variable of QC may, for example, be assigned the same member d of a given domain D and ‘ƒ’ be assigned some superset of {d} short of D, in which case each one of ‘ƒ(x)’, ‘ƒ(y)’, ‘ƒ(z)’, etc., comes out true in D, but ‘(∀x)ƒ(x)’ does not. I owe the example to Professor Richmond H. Thomason.

⁷ I.e., where Χ₁^′ is the member of Σ′ correlated under M with X_1˙ That A_1–1(Χ₁^′/Χ₁) cannot do duty here for A_1–1[Χ₁^′/Χ₁], was brought to my attention by Professor van Fraassen.

⁸ A like convention will hold mutatis mutandis when Χ or Υ or both are individual constants rather than variables. See §8.

⁹ I follow here Hailperin's handling of vacuous quantifications (see Quantification theory and empty individual domains, this Journal, vol. 18 (1953), pp. 197–200).

¹⁰ As axioms and rules of inference for QC=, when the individual constants of QC= have leave to designate nothing or designate something not a value of ‘x’, ‘y’, ‘z’, etc., use those of QC¹ = (which do not acknowledge as a domain) or those of QC² = (which do) in H. Leblanc and R. H. Thomason's Completeness theorems for some presupposition-free logics, forthcoming in Fundamenta mathematicae.

¹¹ I owe the above phrasing of (iii) to a referee.