Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

  title={Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth},
  author={Carlo Nicolai},
  journal={Studia Logica},
We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘$$\fancyscript{PKF}$$PKF-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules (in Basic De Morgan Logic) are all valid in fixed-point models; the second by ‘$$\fancyscript{KF}$$KF-like’ theories first introduced by Solomon Feferman, that lose this… 
A novel version of possible worlds semantics featuring both classical and nonclassical worlds is introduced and the completeness of a family of noncongruent modal logics whose internal logic is non classical with respect to this semantics is established.
In 'Some Remarks on Extending an Interpreting Theories with a Partial Truth Predicate' Reinhardt famously proposed an instrumentalist interpretation of the truth theory Kripke-Feferman (KF) in
Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE
By formulating the theory PKF over HYPE one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF, thus matching the strength of predicative analysis.
On the Costs of Nonclassical Logic
The reasons for this asymmetry are assessed by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true.
On the Costs of Classical Logic
This article compares classical (or -like) and nonclassical (or -like) axiomatisations of the fixed-point semantics developed by Kripke (J Philos 72(19): 690–716, 1975). Following the line of
Comparing Axiomatic Theories of Truth
S U M M A R Y : This paper is a direct continuation of (Łełyk, Wcisło, 2017), where one particular way of comparing axiomatic theories of truth was discussed at length in the context of extensions of
Systems for non-reflexive consequence
. Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive
Iterated reflection over full disquotational truth
This article shows how in the context of a weaker logic, which is called Basic De Morgan Logic, a collection of Tarski-biconditionals can coherently start with such a fully disquotational truth theory and arrive at a strong compositional truth theory by applying a natural uniform reflection principle a finite number of times.
Is the HYPE about strength warranted?
In comparing classical and non-classical solutions to the semantic paradoxes arguments relying on strength have been influential. In this paper I argue that non-classical solutions should preserve
Abstract Nonclassical theories of truth that take truth to be transparent have some obvious advantages over any classical theory of truth (which must take it as nontransparent on pain of


Axiomatizing Kripke's theory of truth
It is argued that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system ramified analysis or a system of Tarskian ramified truth up to ωω.
Some remarks on extending and interpreting theories with a partial predicate for truth
It is argued that there is a way of reading Kripke's theory as an implementation of G6del's suggestion regarding the possibility of resolving the paradoxes using the notion of meaningful applicability, and essentially Burge's argument can be given, but only in a stronger formal system.
Toward Useful Type-Free Theories. I
The informal argument that the paradoxes are blocked in ZF is that its axioms are true in the cumulative hierarchy of sets where (i) unlike the theory of types, a set may have members of various (ordinal) levels, but (ii) the level of a set is greater than that of each of its members.
On the Costs of Nonclassical Logic
The reasons for this asymmetry are assessed by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true.
Saving Truth From Paradox
Preface Introduction PART ONE: A SELECTIVE BACKGROUND 1. Chapter 1: Self-Reference and Tarski>'s Theorem 2. Validity and the Unprovability of Soundness 3. Kripke>'s Theory of Truth (Strong Kleene
Reflecting on Incompleteness
To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of
Iterated reflection principles and the ω-rule
The ω-rule, with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is
Proof Theory: Some Applications of Cut-Elimination
The interplay between the axiomatic and the semantic approach to truth is discussed and the criterion of ℕ-categoricity is focused on and its usefulness and limits are discussed.
Proof Theory
Proof theory began in the 1920’s as a part of Hilbert’s program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving