Edoardo Rivello

Consider the following sentence: (1) is not true. It has long been known that the sentence, (1), produces a paradox, the socalled liar’s paradox: it seems impossible consistently to maintain that (1) is true, and impossible consistently to maintain that (1) is not true: if (1) is true, then (1) says, truly, that (1) is not true so that (1) is not true; on the other hand, if (1) is not true, then what (1) says is the case, i.e., (1) is true. Given such a paradox, one might be sceptical of the notion of truth, or at least of the prospects of giving a scientifically respectable account of truth. Alfred Tarski’s great accomplishment was to show how to give — contra this scepticism — a formal definition of truth for a wide class of formalized languages. Tarski did not, however, show how to give a definition of truth for languages (such as English) that contain their own truth predicates. He thought that this could not be done, precisely because of the liar’s paradox. More generally, Tarski reckoned that any language with its own truth predicate would be inconsistent, as long as it obeyed the rules of standard classical logic, and had the ability to refer to its own sentences. As we will see in our remarks on Theorem 2.1 in Section 2.3, Tarski was not quite right: there are consistent classical interpreted languages that refer to their own sentences and have their own truth predicates. (This point originates in Gupta 1982 and is strengthened in Gupta and Belnap 1993.) Given the close connection between meaning and truth, it is widely held that any semantics for a language L, i.e., any theory of meaning for L, will be closely related to a theory of truth for : indeed, it is commonly held that something like a Tarskian theory of truth for will be a central part of a semantics for L. Thus, the impossibility of giving a Tarskian theory of truth for languages with their own truth predicates threatens the project of giving a semantics for languages with their own truth predicates. We had to wait until the work of Kripke 1975 and of Martin & Woodruff 1975 for a systematic formal proposal of a semantics for languages with their own truth predicates. The basic thought is simple: take the offending sentences, such as (1), to be neither true nor false. Kripke, in particular, shows how to implement this thought for a wide variety of languages, in effect employing a semantics with three values, true, false and neither. It is safe to say that Kripkean approaches have replaced Tarskian pessimism as the new orthodoxy concerning languages with their own truth predicates. One of the main rivals to the three-valued semantics is the Revision Theory of Truth, or RTT, independently conceived by Hans Herzberger and Anil Gupta, and first presented in publication in 1982. The first monographs on the topic are by Yaqūb and the locus classicus, Gupta & Belnap, both in 1993. The Revision Theory of Truth (RTT) is designed to model the kind of reasoning that the liar sentence leads to, within a two-valued context. The central idea is the idea of a revision process: a process by which we revise hypotheses about the truth-value of one or more sentences. The present article’s purpose is to outline the Revision Theory of Truth.

The classical theory of definitions bans so-called circular definitions, namely, definitions of a unary predicate P, based on stipulations of the form $$Px =_{\mathsf {Df}} \phi,$$where ϕ is a formula of a fixed first-order language and the definiendumP occurs into the definiensϕ. In their seminal book The Revision Theory of Truth, Gupta and Belnap claim that “General theories of definitions are possible within which circular definitions [...] make logical and semantic sense” [p. IX]. In order to sustain their claim, they develop in this book one general theory of definitions based on revision sequences, namely, ordinal-length iterations of the operator which is induced by the definition of the predicate. Gupta-Belnap’s approach to circular definitions has been criticised, among others, by D. Martin and V. McGee. Their criticisms point to the logical complexity of revision sequences, to their relations with ordinary mathematical practice, and to their merits relative to alternative approaches. I will present an alternative general theory of definitions, based on a combination of supervaluation and ω-length revision, which aims to address some criticisms raised against revision sequences, while preserving the philosophical and mathematical core of revision.

The model of self-referential truth presented in this paper, named Revision-theoretic supervaluation, aims to incorporate the philosophical insights of Gupta and Belnap’s Revision Theory of Truth into the formal framework of Kripkean fixed-point semantics. In Kripke-style theories the final set of grounded true sentences can be reached from below along a strictly increasing sequence of sets of grounded true sentences: in this sense, each stage of the construction can be viewed as an improvement on the previous ones. I want to do something similar replacing the Kripkean sets of grounded true sentences with revision-theoretic sets of stable true sentences. This can be done by defining a monotone operator through a variant of van Fraassen’s supervaluation scheme which is simply based on ω-length iterations of the Tarskian operator. Clearly, all virtues of Kripke-style theories are preserved, and we can also prove that the resulting set of “grounded” true sentences shares some nice features with the sets of stable true sentences which are provided by the usual ways of formalising revision. What is expected is that a clearer philosophical content could be associated to this way of doing revision; hopefully, a content directly linked with the insights underlying finite revision processes.

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination which characterises this move.

A paper by Beneš, published in 1954, was an attempt to prove the consistency of $\mathsf{NF}$ via a partial model of Hailperin’s finite axiomatization of $\mathsf{NF}$. Here, I offer an analysis of Beneš’s proof in a De Giorgi-style setting for set theory. This approach leads to an abstract version of Beneš’s theorem that emphasizes the monotone and invariant content of the axioms proved to be consistent, in a sense of monotony and invariance that this paper intends to state rigorously and to help clarify. Moreover, some tentative speculation will be made about possible developments of the topic in the following two directions: which set theories can be proved to be consistent via Beneš-like constructions, and how can we elaborate on Beneš’s model to get a consistency proof for full $\mathsf{NF}$?

Revision sequences were introduced in 1982 by Herzberger and Gupta as a mathematical tool in formalising their respective theories of truth. Since then, revision has developed in a method of analysis of theoretical concepts with several applications in other areas of logic and philosophy. Revision sequences are usually formalised as ordinal-length sequences of objects of some sort. A common idea of revision process is shared by all revision theories but specific proposals can differ in the so-called limit rule, namely the way they handle the limit stages of the process. The limit rules proposed by Herzberger and by Belnap show different mathematical properties, called periodicity and reflexivity, respectively. In this paper we isolate a notion of cofinally dependent limit rule, encompassing both Herzberger’s and Belnap’s ones, to study periodicity and reflexivity in a common framework and to contrast them both from a philosophical and from a mathematical point of view. We establish the equivalence of weak versions of these properties with the revision-theoretic notion of recurring hypothesis and draw from this fact some observations about the problem of choosing the “right” limit rule when performing a revision-theoretic analysis.