² By a semantical system S, we mean a set E of expressions (strings of signs), together with a subset S of expressions called sentences of S (determined by a set of rules of formation), together with a subset T of S, of elements called true sentences of S (determined by a set of ‘rules of truth’ for S).

³ (At the referee's suggestion) – In an extensional system, the only way we can translate the meta-linguistic phrase ‘X says that X _∈W is by the phrase ’X is true if and only if X _∈W. Thus the requirement for X to be a Tarski sentence for W, is exceedingly weak; any sentence X which is either both true and in W, or false and not in W, will serve. However, this is as much as we need of a Tarski sentence (for undecidability results). If we were considering an intensional system, then we would define a Tarski sentence for W as a sentence X which not only is true if and only if X _∈W, but which actually expresses the proposition that X _∈W.

⁴ The former will be carried out in this paper and the latter in a forthcoming paper, Systems of protosyntax self applied.

⁵ In contrast with this construction, let us define the diagonalization of E as the result of substituting the quotation of E for all occurrences of the variable ‘x’ in E. Then the following Tarski sentence for W (when formalized) is the classical construction: W contains the diagonalization of ‘W contains the diagonalization of x’. This latter construction involves substitution (inherent in diagonalization), whereas the norm function involves concatenation (the norm of E being E followed by its quotation), which is far easier to formalize (cf. Section 5).

⁶ ‘Yields falsehood when appended to its quotation’ yields falsehood when appended to its quotation. This is Quine's version of the famous semantical paradox.

⁷ If we wished to construct a miniature system L_p which formalizes the diagonal function in the same way as S_p does the norm function, we take four signs, viz., ‘φ’, ‘*’, ‘D’, ‘x’, and the rules R ₁ (same as S _p), R ₂: If E₁ designates E₂, then ⌜DE₁⌝ designates the diagonalization of E₂ (i.e., the result of replacing each occurrence of ‘x’ in E₂ by the quotation of E₂), R ₃: If E₁ designates E₂, then ⌜φE₂⌝ is a sentence and is true in L_p if and only if E₂ has the property P. Then the expression of Theorem 2.1, which designates itself, is ‘D*Dx*’, and the Tarski sentence (of Theorem 2.2) for P is ‘φD*φDx*’.

⁸ The word ‘norm’ was suggested by the following usage: In Mathematics, when we have a function f(x, y) of two arguments, the entity f(x, x) is sometimes referred to as the norm of x — e.g., in Algebra the norm of a vector ∊ is f(∊, ∊), where f is the function which assigns to any pair of vectors the square root of their inner product. In this paper, the crucial function f (which allows us to introduce a notion of representability) is the function which assigns, to each pair (E₁, E₂) of expressions, the expression . Thus for this f, f(E, E) is our norm of E.

⁹ Professor Quine has kindly suggested that I remark that the norm of a predicate is a sentence which says in effect that the predicate is autological, in the sense of Greiling (i.e., that the predicate applies to itself.)

¹⁰ We could have used the single primitive ‘⊂’ [class inclusion] in place of the joint use of ‘↓’ and ‘ = ’ (as occurring between abstracts). We would then have a system formulated in a logic based on inclusion and abstraction (in the sense of Quine). All results of this paper would still go through.

¹¹ Had we used g ₀, rather than g for our Gödel correspondence, then, if x were the g.n. of E, the g.n. of the norm of E would have been (x + 1)10^x − 1, rather than x. 10^x.

¹² As indicated at the beginning of Section 4, quantification is defined, using a formula which employs the identity sign between class abstracts.

¹³ E.g., the correspondence g ₀. To show that, relative to g ₀, the norm function is weakly definable, we must construct a formula ⌜φ(α, β)⌝ such that, for any two numbers n and m, φ(n, m) is true ⇔ m + 1 = (n+1). 10ⁿ (cf. (9)). We first define addition as follows: Add (where we take n any number ≠ 1). Then we define [Add (α, 1, γ) & Add (β, 1, γ · 10^α)] (this construction can be simplified by introducing descriptors). Thus the tricky correspondence g ₀ + 1, which we used, was introduced only for purposes of simplicity, and is certainly not necessary for the success of our program.

¹⁴ We can profitably avoid repetition of analogous arguments for S and L by regarding both as special cases of a more general structure. This approach will be presented in a forthcoming paper. Abstract structure of unsaturated theories, in which we study, in considerable generality, the deeper properties of undecidable systems, uncovered by Gödel and Rosser.