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Self-reference and modal logic. (English) Zbl 0596.03001
Universitext. New York etc.: Springer-Verlag. XII, 333 p. DM 88.00 (1985).
Self-reference, as a general concept, is very complex. It penetrates deeply into the nature of language, as does the concept of reference, at least when conceived in general terms not excluding reference also to the fact of referring. By restricting the concept, however, both with respect to ”reference” and to ”self”, sufficient simplicity may be obtained to allow smooth formalization.
The book under review is a well written, instructive text on selected aspects of self-reference that lend themselves to a uniform treatment in terms of modal logic. Its uniformity is obtained, first by restricting self-reference to sentential self-reference, where a sentence may refer to certain properties of itself, like in ”this subsentence contains exactly forty seven letters”. The sentence is partially referring to itself through an external act of interpretation. Reference through description, which is also natural and known to generate other types of self-reference, is not considered in the book. Next, a restriction is made to a Gödel-context where the fixed point lemma is obtained: With any formula W x of a formal language, there is a fixed point sentence $$\phi$$ of the language such that $$\vdash \phi \equiv W \ulcorner \phi \urcorner$$. Here $$\ulcorner \phi \urcorner$$ is the Gödel number of the sentence $$\phi$$, which thus says of itself that it has the property expressed by W x - however with no assurance that $$\phi$$ is provable or true. By way of example, a Henkin sentence h is the first point to Pr x, the provability predicate, and a Gödel sentence g the fixed point to $$\neg \Pr x$$. The Gödel sentence g thus says of itself that it is not provable. Gödel showed in 1931, on the basis of certain soundness conditions, that g as well as $$\neg g$$ are in fact unprovable (Gödel’s first incompleteness theorem; Gödel’s second incompleteness theorem concerns the nonprovability in a consistent theory T of the consistency of T).
Not until the 1970s, however, did the subject of partial sentential self- reference itself become a subject of mathematical study. From several directions it was then recognized that the proofs of Gödel’s theorems and Löb’s (about the truth of h) were propositional in character with an additional modal operator for provability, i.e., themselves subject to mathematical study. This is what the book is about, namely to provide a modal analysis of fixed points to Pr x-related properties which lend themselves to such a study.
In a modal provability logic, PRL, the modal operator $$\square$$ (for provability) is provided with axioms that correspond to the conditions of Löb on Pr x that allow derivation of Gödel’s second incompleteness theorem, as well as to the formalized Löb’s theorem. As shown by detailed syntactical and semantical studies, PRL suffices for modal analysis of self-referential sentences arising from this particular context. Here, in part I of the book, smooth theory is obtained with extremely well-behaved self-reference: the fixed points are unique; they are explicitly definable; and they have common explicit definitions. Likewise for part II, which is devoted to multi-modal generalizations of the modal analysis in part I. In part III, more involved provability predicates are considered where the fixed points need be neither unique nor explicitly definable. Like for Rosser sentences, which are demonstrably nonprovable and nonrefutable on a weaker soundness, namely consistency, than that behind Gödel’s first incompleteness theorem. A Rosser sentence asserts, not its outright unprovability, but rather that its negation must be provable before it is. A modal analysis of Rosser sentences is given by incorporating the ordering of derivations into the modal logic as so called witness comparisons. In the final chapter, witness comparisons are introduced to the multi-modal schemes, whereby a uniform treatment is obtained, covering many basic applications of self- reference in the 1970s. A goal of this final chapter is said to be a shift from the study of self-reference as an object of study towards laying a foundation of self-reference as a tool for application.
Surely a well aimed goal (almost of a complementaristic nature) for a well written book. A book that may be recommended not only for metamathematicians, but also for scientists trying to develop languages for studies of language. That is, language (and self-reference) in a more general sense, including not only formal languages, but programming languages and even genetic languages [cf. R. Solovay, ”Explicit Henkin sentences”, J. Symb. Logic. 50, 91-93 (1985) for reference to such contexts]. Not that the book, with its strict mathematical nature obtained through fragmentation of self-reference, can allow a self- reference to include reference also to its applicability (a problem with complementaristic resolutions). But it provides the reader with a wealth of well understood specific instances of partial sentential self- reference, that may well come to function as building stones in a developing knowledge of self-reference in a more general sense, also with the partiality of self-reference as an object of study. (Hopefully, later editions will be supplied with an index!)
Reviewer: L.Löfgren

##### MSC:
 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations 03A05 Philosophical and critical aspects of logic and foundations 03B45 Modal logic (including the logic of norms) 03B30 Foundations of classical theories (including reverse mathematics)