Languages with self-reference. I: Foundations (or: We can have everything in first-order logic!)
The domain of artificial intelligence has on several occasions stimulated to unconventional thoughts on language, notably on natural language with its possibilities for uttering freely self-referring sentences, let be at the price of inexplicable meanings. The present paper provides one such outlook with the explicitly stated hope of having provided a language, even a first order language, that is able to express its own syntax and semantics. In particular, thoughts on quotation and unquotation are believed to improve on Kripke's partial truth-predicate approach in preserving the excluded middle for the predicate. \{In the reviewer's opinion, some of the exposed ideas are contradictory to established results of mathematical logic. By way of example, the alleged language L, that can express its own syntax and semantics, implies a consistent theory T in L, expressing L's syntax and semantics. The statement that T is consistent, i.e., that it has a model (the intended language L), is an example of a statement concerning L's semantics, which cannot be proved in the alleged theory T for L (Gödel's second incompleteness theorem). There are semantic properties of L that require a metalanguage of a strictly higher order than that of L, thus contradicting the author's capital statement that ''we can have everything in first-order logic''.\}
Kripke's partial truth-predicate approach
scientific article; zbMATH DE number 3928954