Interpretability Logic

Archive for Mathematical Logic, 2014

In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory-like Elementary Arithmetic EA, IΣ 1 , or the Gödel-Bernays theory of sets and classes GBhave suprema. This partially answers a question posed by VítěslavŠvejdar in his paper [Šve78]. The partial solution ofŠvejdar's problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result. Dedicated to Dirk van Dalen on the occasion of his 80th birthday.

Annals of Pure and Applied Logic, 2009

In this paper we study IL(PRA), the interpretability logic of PRA. As PRA is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to PRA: IL(PRA) is not ILM or ILP. IL(PRA) does of course contain all the principles known to be part of IL(All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of PRA and see what their consequences in the modal logic IL(PRA) are. These properties are reflected in the so-called Beklemishev Principle B, and Zambella's Principle Z, neither of which is a part of IL(All). Both principles and their interrelation are submitted to a modal study. In particular, we prove a frame condition for B. Moreover, we prove that Z follows from a restricted form of B. Finally, we give an overview of the known relationships of IL(PRA) to important other interpetability principles. We thank Lev Beklemishev for his help and suggestions. Evan Goris did a thorough proofread of an early draft and suggested a simplification of the notion of Bsimulation. We thank Albert Visser for fruitful discussions and challenges. We also thank Franco Montagna for his many contributions to the subject. Two unknown referees improved our paper considerably with their remarks and suggestions.

arXiv (Cornell University), 2020

The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL(All). In this paper we present a new principle R in IL(All). We show that R does not follow from the logic ILP0W * that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W * : showing that R follows semantically but not syntactically from ILP0W *. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. 1 Technically speaking the property of so-called essential reflexivity is sufficient. A theory is

Erkenntnis, 2000

This paper is a presentation of a status qu stionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.

arXiv (Cornell University), 2020

The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set Γ. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set Γ, where each sentence in Γ has a well understood (meta)-mathematical content in T , the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and IΣ 1. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) ⊂ ILM. 2 Σ 1-sound is sufficient here. 3 There is a paper by de Jongh, Jumelet and Montagna [22] where an alternative proof of Solovay's theorem is given. In that proof, using the diagonal lemma, one finds some sentences with the required properties rather than defining the sentences and then proving the necessary properties.

1-sound, nitely axiomatized theories containing I 0 + SUPEXP. Examples are: I 0 + SUPEXP, I n (n > 0), ACA 0 , GB. Theories in this class are sound and complete for the logic ILP. See 55]. The second class is that of sequential, locally essentially re exive theories containing I 1. Examples are PA and ZF. Theories in this class satisfy are sound and complete for the logic ILM. This result was proved independently by Alessandro Berarducci and Volodya S h a vrukov. See 5] and 39]. Outside of these major classes we k n o w very little. See section 9 and appendix B.

Studia Logica, 1991

Logic Group Preprint Series, 2012

This paper is a presentation of a status qu stionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.

Bulletin of the Section of Logic, 2023

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic IL. We will introduce the logics BIL and BIL + in the propositional language with a modal operator □ and a binary operator ⇒ such that BIL ⊆ BIL + ⊆ IL. The logic BIL is generated by the relational structures ⟨X, R, N ⟩, called basic frames, where ⟨X, R⟩ is a Kripke frame and ⟨X, N ⟩ is a neighborhood frame. We will prove that the logic BIL + is generated by the basic frames where the binary relation R is definable by the neighborhood relation N and, therefore, the neighborhood semantics is suitable to study the logic BIL + and its extensions. We shall also study some axiomatic extensions of BIL and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic BIL + and some of its extensions are complete respect with the class of neighborhood frames.

The Journal of Symbolic Logic

A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.