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Languages in which self reference is possible1

  • Raymond M. Smullyan (a1)
Abstract

This paper treats of semantical systems S of sufficient strength so that for any set W definable in S (in a sense which will be made precise), there must exist a sentence X which is true in S if and only if it is an element of W. We call such an X a Tarski sentence for W. It is the sentence which (in a purely extensional sense) says of itself that it is in W. If W is the set of all expressions not provable in some syntactical system C, then X is the Gödel sentence which is true (in S) if and only if it is not provable (in C). We provide a novel method for the construction of these sentences, which yields sentences particularly simple in structure. The method is applicable to a variety of systems, including a form of elementary arithmetic, and some systems of protosyntax self applied. In application to the former, we obtain an extremely simple and direct proof of a theorem, which is essentially Tarski's theorem that the truth set of elementary arithmetic is not arithmetically definable.

The crux of our method is in the use of a certain function, the ‘norm’ function, which replaces the classical use of the diagonal function. To give a heuristic idea of the norm function, let us define the norm of an expression E (of informal English) as E followed by its own quotation.

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1

I wish to express my deepest thanks to Professor Rudolph Carnap, of the University of California at Los Angeles, and to Professor John Kemeny and Dr. Edward J. Cogan, of Dartmouth College, for some valuable suggestions. I also wish to thank the referee for some very helpful revisions.

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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 Lp which formalizes the diagonal function in the same way as Sp 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 Lp 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 inline-graphic. 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)10x − 1, rather than x. 10x.

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 inline-graphic (where we take n any number ≠ 1). Then we define inline-graphic [Add (α, 1, γ) & Add (β, 1, γ · 10α)] (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.

Footnotes

1 I wish to express my deepest thanks to Professor Rudolph Carnap, of the University of California at Los Angeles, and to Professor John Kemeny and Dr. Edward J. Cogan, of Dartmouth College, for some valuable suggestions. I also wish to thank the referee for some very helpful revisions.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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