We introduce and develop a topological semantics of conservativity logics and interpretability logics. We prove the topological compactness theorem of consistent normal extensions of the conservativity logic $\mathbf{CL}$ by extending Shehtman's ultrabouquet construction method to our framework. As a consequence, we prove that several extensions of $\mathbf{CL}$ such as $\mathbf{IL}$, $\mathbf{ILM}$, $\mathbf{ILP}$ and $\mathbf{ILW}$ are strongly complete with respect to our topological semantics.
We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform substitution and are without side conditions.
We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut-elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.
Strictly positive logics recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus RC consists of implications between formulas built up from propositional variables and constant `true' using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its…
We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and partial selection functions, independently of any strictness assumption.
In the present paper, we introduce a multi-type calculus for the logic of measurable Kleene algebras, for which we prove soundness, completeness, conservativity, cut elimination and subformula property. Our proposal imports ideas and techniques developed in formal linguistics around the notion of structural control.
De Rijke introduced a unary interpretability logic $\mathbf{il}$, and proved that $\mathbf{il}$ is the unary counterpart of the binary interpretability logic $\mathbf{IL}$. In this paper, we find the unary counterparts of the sublogics of $\mathbf{IL}$.
Frobenius Galois expansions of substructural logics:Algebraization, Kalman equivalence and positive cone semantics
School of Science, Xi’an Petroleum University（西安石油大学理学院）；School of Mathematics and Data Science, Shaanxi University of Science and Technology（陕西科技大学数学与数据科学学院）；School of Science, Xi Hang University（西航大学理学院）
In is paper we present a labelled tableau proof system that serves a wide class of interpretability logics. The system is proved sound and complete for any interpretability logic characterised by a frame condition given by a set of universal strict first order Horn sentences. As such, the current paper adds to a better proof-theoretical understanding of interpretability logics.
This paper from 2000 is a presentation of a status quæstionis at that tiime, to wit of the problem of the interpretability logic of {\em all}\/ reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
We show that the decision problem for the basic system of interpretability logic IL is PSPACE-complete. For this purpose we present an algorithm which uses polynomial space with respect to the complexity of a given formula. The existence of such algorithm, together with the previously known PSPACE hardness of the closed fragment of IL, implies PSPACE-completeness.
We obtain modal completeness of the interpretability logics ILP_0 and ILR w.r.t. generalized Veltman semantics. Our proofs are based on the notion of smart (full) labels. We also give shorter proofs of completeness w.r.t. generalized semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. generalized semantics for ILP_0 and ILR. Finally, we develop a construction that might be useful for proofs of completeness of extensions of ILW w.r.t. generalized semantics in the future, and demonstrate its usage with ILW* = ILM_0W.
This paper from 2008 is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the foundations are laid for later results. These foundations 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 ${\textbf{IL}}$. Next, the method is applied to obtain new results: the modal completeness of the logic ${\textbf{IL}}{\sf M_0}$, and modal completeness of ${\textbf{IL}…
We study modal completeness and incompleteness of several sublogics of the interpretability logic $\mathbf{IL}$. We introduce the sublogic $\mathbf{IL}^-$, and prove that $\mathbf{IL}^-$ is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between $\mathbf{IL}^-$ and $\mathbf{IL}$ with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of $\mathbf{IL}$ are modally incomplete. Finally, we prove that these incomplete logics are complete with respect to general…
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. Ho…
The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations. The logic IL(All) is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this paper we raise the previously known lower bound of IL(All) by…
The notion of a critical successor [dJV90] has been central to almost all modal completeness proofs in interpretability logics. In this paper we shall work with an alternative notion, that of an assuring successor. As we shall see, 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…
A notion of interpretation between arbitrary logics is introduced, and the poset Log of all logics ordered under interpretability is studied. It is shown that in 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 Log and the lattice of interpretability types of varieties are investigated.
The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 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 ${\textbf{IL}}({\rm All})$. In this paper we present a new principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contain…
In this paper from 2009 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 Beklemi…
In this paper from 2011 we consider $\textbf{IL}_0$, the closed fragment of the basic interpretability logic $\textbf{IL}$. We show that we can translate $\textbf{GL}_1$, the one variable fragment of Gödel-Löb's provabilty logic $\textbf{GL}$, into $\textbf{IL}_0$. Invoking a result on the PSPACE completeness of $\textbf{GL}_1$ we obtain the PSPACE hardness of $\textbf{IL}_0$.
We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic $\mathbf{IL}$. We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.
