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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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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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