Two arguments favoring propositionalist accounts of attitude sentences are being revisited: the Church-Langford translation argument and Thomason's argument against quotational theories of indirect discourse. None of them proves to be decisive, thus leaving the option of searching for a developed quotational alternative. Such an alternative is found in an interpreted logical form theory of attitude ascription. The theory differentiates elegantly among different attitudes but it fails to account for logical dependencies among them. It is argued, however, that the concept of (...) logical consequence does not well apply to dependencies among belief sentences and that the requirement to account for logical relations among such sentences should be relaxed. (shrink)

In this paper we will be concerned with the interpretability logic of PA and in particular with the fact that this logic, which is denoted by ILM, does not have the interpolation property. An example for this fact seems to emerge from the fact that ILM cannot express Σ₁-ness. This suggests a way to extend the expressive power of interpretability logic, namely, by an additional operator for Σ₁-ness, which might give us a logic with (...) the interpolation property. We will formulate this extension, give an axiomatization which is modally complete and arithmetically complete (although for proofs of these theorems we refer to an earlier paper), and investigate interpolation. We show that this logic still does not have the interpolation property. (shrink)

This paper is a presentation of astatus quæstionis, to wit of the problemof the interpretability logic of all reasonablearithmetical theories.We present both the arithmetical side and themodal side of the question.Dedicated to Dick de Jongh on the occasion of his 60th birthday.

PA is Peano arithmetic. The formula $\operatorname{Interp}_\mathrm{PA}(\alpha, \beta)$ is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, Pr PA (x), to the case of the formalized interpretability relation $\operatorname{Interp}_\mathrm{PA}(x, y)$ . The relevant modal logic, in addition to the usual provability operator `□', (...) has a binary operator ` $\triangleright$ ' to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem. (shrink)

We obtain modal completeness of the interpretability logics IL $\!\!\textsf {P}_{\textsf {0}}$ and ILR w.r.t. generalised Veltman semantics. Our proofs are based on the notion of full labels [2]. We also give shorter proofs of completeness w.r.t. the generalised semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. the generalised semantics for IL $\textsf {P}_{\textsf {0}}$ and ILR. Finally, we develop a construction that might be useful for proofs of completeness of extensions of (...) ILW w.r.t. the generalised semantics in the future, and demonstrate its usage with $\textbf {IL}\textsf {W}^\ast = \textbf {IL}\textsf {WM}_{\textsf {0}}$. (shrink)

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 $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, 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 I$\Sigma_1$. We then study interpretability logics, obtaining upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely related to linear frames. The technique is also applied to yield the nontrivial result that IL(PRA) $\subset$ ILM. (shrink)

Interpreted Logical Forms are objects composed of a syntactic structure annotated with the semantic values of each node of the structure. We criticize the view that ILFs are the objects of propositional attitude verbs such as believe, as this is developed by Larson and Ludlow. Our critique arises from a tension in the way that sen-.

De Rijke introduced a unary interpretability logic $$\textbf{il}$$, and proved that $$\textbf{il}$$ is the unary counterpart of the binary interpretability logic $$\textbf{IL}$$. In this paper, we find the unary counterparts of the sublogics of $$\textbf{IL}$$.

One major obstacle in providing a compositional semantics for natural languages is that it is not clear how we should deal with propositional attitude contexts. In this paper I will discuss the Interpreted Logical Form proposal , focusing on the case of belief. This proposal has been developed in different ways by authors such as Harman (1972), Higginbotham (1986,1991), Segal (1989) and Larson and Ludlow (1993). On this approach, the that-clause of a belief report is treated as a singular term, (...) referring to the interpreted logical form (ILF) of its embedded sentence. The ILF of a sentence is made up of two parts : a syntactic representation of the sentence at the level of logical form, and an assignment of semantic values to parts of the representation. Thus, given a belief report such as. (shrink)

Jim Edwards; VIII*—Interpreted Logical Forms and Knowing Your Own Mind, Proceedings of the Aristotelian Society, Volume 99, Issue 1, 1 June 1999, Pages 169–190.

An attractive semantic theory presented by Richard K. Larson and Peter Ludlow takes a report of propositional attitudes, e.g 'Tom believes Judy Garland sang', to report a believing relation between Tom and an interpreted logical form constructed from 'Judy Garland sang'. We briefly outline the semantic theory and indicate its attractions. However, the definition of interpreted logical forms given by Larson and Ludlow is shown to be faulty, and an alternative definition is offered which matches their intentions. This definition is (...) then shown to imply that Tom does not know his own mind, a result without intuitive support. A third definition is offered to deal with this problem. (shrink)

The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryński.

Jean Nicod (1893–1924) is a French philosopher and logician who worked with Russell during the First World War. His PhD, with a preface from Russell, was published under the titleLa géométrie dans le monde sensiblein 1924, the year of his untimely death. The book did not have the impact he deserved. In this paper, I discuss the methodological aspect of Nicod’s approach. My aim is twofold. I would first like to show that Nicod’s definition of various notions of equivalence between (...) theories anticipates, in many respects, the (syntactic and semantic) model-theoretic notion of interpretation of a theory into another. I would secondly like to present the philosophical agenda that led Nicod to elaborate his logical framework: the defense of rationalism against Bergson’s attacks. (shrink)

This paper is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the fundaments are laid for later results. These fundaments consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic IL. Next, the method is applied to obtain new results: (...) the modal completeness of the logic ILMo, and modal completeness of ILW*. (shrink)

The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryski.

A Kripke-style semantics developed by de Jongh and Veltman is used to investigate relations between several extensions of interpretability logic, IL.

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

Interpretability logic is a modal logic that can be used to describe relative interpretability between extensions of a given first-order arithmetical theory. Verbrugge semantics is a generalization of the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. The Van Benthem Correspondence Theorem establishes modal logic as the bisimulation invariant fragment of first-order logic. In this paper we show that a special type of bisimulations, the (...) so-called w-bisimulations, enable an analogue of the Van Benthem Theorem for interpretability logic w.r.t. Verbrugge semantics. (shrink)

The interpretability logic ILP is the interpretability logic of all sufficiently strong |$\varSigma _1$|-sound finitely axiomatised theories, such as the Gödel-Bernays set theory. The interpretability logic IL is a strict subset of the intersection of the interpretability logics of all so-called reasonable theories, IL(All). It is known that both ILP and ILW are decidable, however their complexity has not been resolved previously. In [10] it was shown that the basic interpretability logic (...) IL is PSPACE-complete. Here we prove the same for ILP and ILW. (shrink)

An elementary theory of concatenation,QT+, is introduced and used to establish mutual interpretability of Robinson arithmetic, Minimal Predicative Set Theory, quantifier-free part of Kirby’s finitary set theory, and Adjunctive Set Theory, with or without extensionality. The most basic arithmetic and simplest set theory thus turn out to be variants of string theory.

In a recent paper, Barrett & Halvorson (2016) define a notion of equiva- lence for first-order theories, which they call “Morita Equivalence.” To argue that Morita equivalence is a reasonable measure of “theoretical equivalence,” they make use of the claim that Morita extensions “say no more” than the theories they are extending. The goal of this paper is to challenge this central claim by raising objections to their argument for it and by showing why there is good reason to think (...) that the claim itself is false. In light of these criticisms, this paper develops a natural way for the advocate of Morita equivalence to respond. However, this response makes her criterion equivalent to mutual faithful interpretability, an already well-known and philosophically contentious barometer of theoretical equivalence. (shrink)

Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that (...) are modally equivalent but not bisimilar, and showing how these GL-models can be converted to Veltman models with the same properties. In the process we develop some useful constructions: games on Veltman models, chains, and general method of transformation from GL-models/frames to Veltman ones. (shrink)

We prove that a variant of Robinson arithmetic $\mathsf{Q}$ with nontotal operations is interpretable in the theory of concatenation $\mathsf{TC}$ introduced by A. Grzegorczyk. Since $\mathsf{Q}$ is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether $\mathsf{Q}$ is interpretable in $\mathsf{TC}$. An immediate consequence is essential undecidability of $\mathsf{TC}$.

The notion of a critical successor [5] in relational semantics has been central to most classic modal completeness proofs in interpretability logics. In this paper we shall work with a more general notion, that of an assuring successor. This will enable more concisely formulated completeness proofs, both with respect to ordinary and generalised Veltman semantics. Due to their interesting theoretical properties, we will devote some space to the study of a particular kind of assuring labels, the so‐called full labels (...) and maximal labels. After a general treatment of assuringness, we shall apply it to obtain a completeness result for the modal logic w.r.t. generalised semantics for a restricted class of frames. (shrink)

In this paper we define the notion of -linearity for and discuss interpretability (and noninterpretability) of -linear orders in structures and theories.

In [6] Albert Visser shows that ILP completely axiomatizes all schemata about provability and relative interpretability that are provable in finitely axiomatized theories. In this paper we introduce a system called $\text{ILP}^{\omega}$ that completely axiomatizes the arithmetically valid principles of provability in and interpretability over such theories. To prove the arithmetical completeness of $\text{ILP}^{\omega}$ we use a suitable kind of tail models; as a byproduct we obtain a somewhat modified proof of Visser's completeness result.

In [6] Albert Visser shows that ILP completely axiomatizes all schemata about provability and relative interpretability that are provable in finitely axiomatized theories. In this paper we introduce a system called ILP that completely axiomatizes the arithmetically valid principles of provability in and interpretability over such theories. To prove the arithmetical completeness of ILP we use a suitable kind of tail models; as a byproduct we obtain a somewhat modified proof of Visser's completeness result.

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.In this paper we present a new principle R in IL. 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. (shrink)

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic $\mathbf{IL}$. We will introduce the logics $\mathbf{BIL}$ and $\mathbf{BIL}^{+}$ in the propositional language with a modal operator $\square$ and a binary operator $\Rightarrow$ such that $\mathbf{BIL}\subseteq\mathbf{BIL}^{+}\subseteq\mathbf{IL}$. The logic $\mathbf{BIL}$ is generated by the relational structures \(\left \), called basic frames, where \(\left \) is a Kripke frame and \(\left \) is a neighborhood frame. We will prove that the (...) class='Hi'>logic $\mathbf{BIL}^{+}$ 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 $\mathbf{BIL}^{+}$ and its extensions. We shall also study some axiomatic extensions of $\mathsf{\mathbf{BIL}}$ 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. (shrink)

A generalized Veltman semantics developed by de Jongh is used to investigate correspondences between several extensions of intepretability logic . In this paper we present some new results on independences.

In this paper we carry out a comparative study of $\mathrm{I}\Sigma_1$ and PRA. We will in a sense fully determine what these theories have to say about each other in terms of provability and interpretability. Our study will result in two arithmetically complete modal logics with simple universal models.

In this paper we study the modal behavior of Σ-preservativity, an extension of provability which is equivalent to interpretability for classical superarithmetical theories. We explain the connection between the principles of this logic and some well-known properties of HA, like the disjunction property and its admissible rules. We show that the intuitionistic modal logic given by the preservativity principles of HA known so far, is complete with respect to a certain class of frames.

We study modal completeness and incompleteness of several sublogics of the interpretability logic. We introduce the sublogic, and prove that is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between and with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of are modally incomplete. Finally, we prove that these incomplete logics are complete with respect to generalized Veltman prestructures. (...) As a consequence of these investigations, we obtain that the twenty logics studied in this paper are all decidable. (shrink)

In this paper we will investigate different axiomatic theories of truth that are minimal in some sense. One criterion for minimality will be conservativity over Peano Arithmetic. We will then give a more fine-grained characterization by investigating some interpretability relations. We will show that disquotational theories of truth, as well as compositional theories of truth with restricted induction are relatively interpretable in Peano Arithmetic. Furthermore, we will give an example of a theory of truth that is a conservative extension (...) of Peano Arithmetic but not interpretable in it. We will then use stricter versions of interpretations to compare weak theories of truth to subsystems of second-order arithmetic. (shrink)