By an unfounded chain for a function f:X→X we mean a sequence nω of elements of X s.t. fxn+1=xn for every n. Unfounded chains can be regarded as a generalization of fixed points, but on the other hand are linked with concepts concerning non-well-founded situations, as ungrounded sentences and the hypergame. In this paper, among other things, we prove a lemma in general topology, we exhibit an extensional recursive function from the set of sentences of PA into itself without an (...) unfounded chain, and we prove that every term in a Magari algebra has an unfounded chain. (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)

In researching presuppositions dealing with logic and dynamic of belief we distinguish two related parts. The first part refers to presuppositions and logic, which is not necessarily involved with intentional operators. We are primarily concerned with classical, free and presuppositonal logic. Here, we practice a well known Strawson’s approach to the problem of presupposition in relation to classical logic. Further on in this work, free logic is used, especially Van Fraassen’s research of the role of presupposition in supervaluations logical systems. (...) At the end of the first part, presuppositional logic, advocated by S.K. Thomason, is taken into consideration. The second part refers to the presuppositions in relation to the logic of the dynamics of belief. Here the logic of belief change is taken into consideration and other epistemic notions with immanent mechanism for the presentation of the dynamics. Three representative and dominant approaches are evaluated. First, we deal with new, less classical, situation semantics. Besides Strawson’s theory, the second theory is the theory of the belief change, developed by Alchourron, Gärdenfors, and Makinson (AGM theory). At the end, the oldest, universal, and dominant approach is used, recognized as Hintikka’s approach to the analysis of epistemic notions. (shrink)

This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...) to weak Heyting algebras and implicative semilattices. The book also dives into the Blok-Esakia theorem and provides an outline of the intuitionistic modal logic KM which is closely related to the Gödel-Löb provability logic GL. One chapter scrutinizes Esakia’s work interpreting modal diamond as the derivative of a topological space within the setting of point-free topology. The final chapter in the volume is dedicated to the derivational semantics of modal logic and other related issues. (shrink)

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic (...) manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem.Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developedwithout referencetoelements.By Gödel's generalized incompleteness theorem for consistent arithmetical systemTwe mean any statement of the following sort: if every recursive set is definable inT, thenTis essentially undecidable [41]; or if all recursive functions are definable inT, thenTis essentially undecidable [41]; or if every recursive set is definable inT, thenT0andR0 are recursively inseparable [39]; or if all re sets are representable inT, thenT0is creative [28], [39]; or ifTis a Rosser theory, thenT0andR0are effectively inseparable [39]. (shrink)

The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered – definability by formulas that do not (...) preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems. (shrink)

-/- Provability logic is a modal logic that is used to investigate what arithmetical theories can express in a restricted language about their provability predicates. The logic has been inspired by developments in meta-mathematics such as Gödel’s incompleteness theorems of 1931 and Löb’s theorem of 1953. As a modal logic, provability logic has been studied since the early seventies, and has had important applications in the foundations of mathematics. -/- From a philosophical point of view, provability logic is interesting because (...) the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic. Furthermore, provability logic provides tools to study the notion of self-reference. (shrink)

This paper treats a kind of a modal logic based on the intuitionistic propositional logic which arose from the provability predicate in the first order arithmetic. The semantics of this calculus is presented in both a relational and an algebraic way.Completeness theorems, existence of a characteristic model and of a characteristic frame, properties of FMP and FFP and decidability are proved.

This paper investigates the algebraic structure of the Medvedev lattice M. We prove that M is not a Heyting algebra. We point out some relations between M and the Dyment lattice and the Mucnik lattice. Some properties of the degrees of enumerability are considered. We give also a result on embedding countable distributive lattices in the Medvedev lattice.

For every sequence |p n } n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula 0 A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, 0 A is a theorem ofPA.

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR.

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical completeness theorem with respect to PA is obtained for LR.

We make use of a Theorem of Burris-McKenzie to prove that the only decidable variety of diagonalizable algebras is that defined by 0=1. Any variety containing an algebra in which 01 is hereditarily undecidable. Moreover, any variety of intuitionistic diagonalizable algebras is undecidable.

The proof of the Second Incompleteness Theorem consists essentially of proving the uniqueness and explicit definability of the sentence asserting its own unprovability. This turns out to be a rather general phenomenon: Every instance of self-reference describable in the modal logic of the standard proof predicate obeys a similar uniqueness and explicit definability law. The efficient determination of the explicit definitions of formulae satisfying a given instance of self-reference reduces to a simple algebraic problem-that of solving the corresponding fixed-point equation (...) in the modal logic. We survey techniques for the efficient calculation of such fixed-points. (shrink)

This paper is devoted to the algebraization of an arithmetical predicate introduced by S. Feferman. To this purpose we investigate the equational class of Boolean algebras enriched with an operation (g=rtail), which translates such predicate, and an operation τ, which translates the usual predicate Theor. We deduce from the identities of this equational class some properties of (g=rtail) and some ties between (g=rtail) and τ; among these properties, let us point out a fixed-point theorem for a sufficiently large class of (...) (g=rtail)-τ polynomials. The last part of this paper concerns the duality theory for (g=rtail)-τ algebras. (shrink)

In Honour of John Crossley’s 85th Birthday.

We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic \(\textbf{IL}\). We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. (...) However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Gödel sentences, using a purely proof-theoretical method. (shrink)

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into (...) K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

A class of toposes is introduced and studied, suitable for semantical analysis of an extension of the Heyting predicate calculus admitting Gödel's provability interpretation.