Arithmetical Soundness and Completeness for $$\varvec{\Sigma }_{\varvec{2}}$$ Numerations

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \(n \ge 2\), there exists a \(\Sigma _2\) numeration \(\tau (u)\) of T such that the provability logic of the provability predicate \(\mathsf{Pr}_\tau (x)\) naturally constructed from \(\tau (u)\) is exactly \(\mathsf{K}+ \Box (\Box ^n p \rightarrow p) \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.     

Sequent Calculi for Semi-De Morgan and De Morgan Algebras

A contraction-free and cut-free sequent calculus \(\mathsf {G3SDM}\) for semi-De Morgan algebras, and a structural-rule-free and single-succedent sequent calculus \(\mathsf {G3DM}\) for De Morgan algebras are developed. The cut rule is admissible in both sequent calculi. Both calculi enjoy the decidability and Craig interpolation. The sequent calculi are applied to prove some embedding theorems: \(\mathsf {G3DM}\) is embedded into \(\mathsf {G3SDM}\) via Gödel–Gentzen translation. \(\mathsf {G3DM}\) is embedded into a sequent calculus for classical propositional logic. \(\mathsf {G3SDM}\) is embedded into the sequent calculus \(\mathsf {G3ip}\) for intuitionistic propositional logic.     

Public announcement logic with distributed knowledge: expressivity,completeness and complexity

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with distributed knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (\(\mathcal{PAL }\)) with distributed knowledge, in particular their expressivity, axiomatisations and complexity. \(\mathcal{PAL }\) extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on \(\mathcal{PACD }\), the result of adding both common and distributed knowledge to \(\mathcal{PAL }\), which is more expressive than each of its component logics. We introduce an axiomatisation of \(\mathcal{PACD }\), which is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with \(S5\) knowledge, distributed knowledge, common knowledge and public announcements at the same time. We furthermore show that \(\mathcal{PACD }\) is decidable, more precisely that it is \(\textsc {exptime}\)-complete. This result also carries over to \(\mathcal{S 5\mathcal CD }\) with common and distributed knowledge operators for all coalitions (and not only the grand coalition). Finally, we propose a notion of a trans-bisimulation to generalise certain results and give deeper insight into the proofs.     

Turing–Taylor Expansions for Arithmetic Theories

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \({\Pi_{n+1}}\) proof-theoretic ordinal \({|U|_{\Pi^0_{n+1}}}\) also denoted \({|U|_n}\). As such, to each theory U we can assign the sequence of corresponding \({\Pi_{n+1}}\) ordinals \({\langle |U|_n\rangle_{n > 0}}\). We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev’s universal model for the closed fragment of the polymodal provability logic \({\mathsf{GLP}_\omega}\). In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev’s model.     

Correia Semantics Revisited

Despite a renewed interest in Richard Angell’s logic of analytic containment (\({\mathsf{AC}}\)), the first semantics for \({\mathsf{AC}}\) introduced by Fabrice Correia has remained largely unexamined. This paper describes a reasonable approach to Correia semantics by means of a correspondence with a nine-valued semantics for \({\mathsf{AC}}\). The present inquiry employs this correspondence to provide characterizations of a number of propositional logics intermediate between \({\mathsf{AC}}\) and classical logic. In particular, we examine Correia’s purported characterization of classical logic with respect to his semantics, showing the condition Correia cites in fact characterizes the “logic of paradox” \({\mathsf{LP}}\) and provide a correct characterization. Finally, we consider some remarks on related matters, such as the applicability of the present correspondence to the analysis of the system \({\mathsf{AC}^{\ast}}\) and an intriguing relationship between Correia’s models and articular models for first degree entailment.     

Infinite Populations,Choice and Determinacy

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy (\(\mathsf {AD}\)), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \(\mathsf {AD}\) from the point of view of the research area in question is its game-theoretic character.     

Topological-Frame Products of Modal Logics

The simplest bimodal combination of unimodal logics \(\text {L} _1\) and \(\text {L} _2\) is their fusion, \(\text {L} _1 \otimes \text {L} _2\), axiomatized by the theorems of \(\text {L} _1\) for \(\square _1\) and of \(\text {L} _2\) for \(\square _2\), and the rules of modus ponens, necessitation for \(\square _1\) and for \(\square _2\), and substitution. Shehtman introduced the frame product \(\text {L} _1 \times \text {L} _2\), as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced the topological product \(\text {L} _1 \times _t \text {L} _2\), as the logic of the products of certain topological spaces. For almost all well-studies logics, we have \(\text {L} _1 \otimes \text {L} _2 \subsetneq \text {L} _1 \times \text {L} _2\), for example, \(\text {S4} \otimes \text {S4} \subsetneq \text {S4} \times \text {S4} \). Van Benthem et al. show, by contrast, that \(\text {S4} \times _t \text {S4} = \text {S4} \otimes \text {S4} \). It is straightforward to define the product of a topological space and a frame: the result is a topologized frame, i.e., a set together with a topology and a binary relation. In this paper, we introduce topological-frame products \(\text {L} _1 \times _ tf \text {L} _2\) of modal logics, providing a complete axiomatization of \(\text {S4} \times _ tf \text {L} \), whenever \(\text {L} \) is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include \(\text {T} , \text {S4} \) and \(\text {S5} \), but not \(\text {K} \) or \(\text {K4} \). We leave open the problem of axiomatizing \(\text {S4} \times _ tf \text {K} \), \(\text {S4} \times _ tf \text {K4} \), and other related logics. When \(\text {L} = \text {S4} \), our result confirms a conjecture of van Benthem et al. concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.     

Axiomatizing Jaśkowski’s Discussive Logic $$\mathbf {D_2}$$

We outline the rather complicated history of attempts at axiomatizing Ja?kowski’s discussive logic \(\mathbf {D_2}\) and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing \(\mathbf {D_2}\) in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of \(\mathbf {D_2}\) in the language Ja?kowski suggested in his second paper on discussive logic, by following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of \(\mathbf {D_2}\), introduced by Ciuciura, in which negation is also taken to be discussive.     

The Category of Node-and-Choice Preforms for Extensive-Form Games

It would be useful to have a category of extensive-form games whose isomorphisms specify equivalences between games. Since working with entire games is too large a project for a single paper, I begin here with preforms, where a “preform” is a rooted tree together with choices and information sets. In particular, this paper first defines the category \(\mathbf {Tree}\), whose objects are “functioned trees”, which are specially designed to be incorporated into preforms. I show that \(\mathbf {Tree}\) is isomorphic to the full subcategory of \(\mathbf {Grph}\) whose objects are converging arborescences. Then the paper defines the category \(\mathbf {NCP}\), whose objects are “node-and-choice preforms”, each of which consists of a node set, a choice set, and an operator mapping node-choice pairs to nodes. I characterize the \(\mathbf {NCP}\) isomorphisms, define a forgetful functor from \(\mathbf {NCP}\) to \(\mathbf {Tree}\), and show that \(\mathbf {Tree}\) is equivalent to the full subcategory of \(\mathbf {NCP}\) whose objects are perfect-information preforms. The paper also shows that many game-theoretic entities can be derived from preforms, and that these entities are well-behaved with respect to \(\mathbf {NCP}\) morphisms and isomorphisms.     

$${\in_K}$$: a Non-Fregean Logic of Explicit Knowledge

We present a new logic-based approach to the reasoning about knowledge which is independent of possible worlds semantics. \({\in_K}\) (Epsilon-K) is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom K _i φ → φ and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between \({\in_K}\) and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic “paradoxes” can be solved in \({\in_K}\). Every specific \({\in_K}\)-logic is defined as a certain extension of some underlying classical abstract logic.     

A Second Pretabular Classical Relevance Logic

Pretabular logics are those that lack finite characteristic matrices, although all of their normal proper extensions do have some finite characteristic matrix. Although for Anderson and Belnap’s relevance logic R, there exists an uncountable set of pretabular extensions (Swirydowicz in J Symb Log 73(4):1249–1270, 2008), for the classical relevance logic \( \hbox {KR} = \hbox {R} + \{(A\,\, \& \sim A)\rightarrow B\}\) there has been known so far a pretabular extension: \({\mathcal L}\) (Galminas and Mersch in Stud Log 100:1211–1221, 2012). In Section 1 of this paper, we introduce some history of pretabularity and some relevance logics and their algebras. In Section 2, we introduce a new pretabular logic, which we shall name \({\mathcal M}\), and which is a neighbor of \({\mathcal L}\), in that it is an extension of KR. Also in this section, an algebraic semantics, ‘\({\mathcal M}\)-algebras’, will be introduced and the characterization of \({\mathcal M}\) to the set of finite \({\mathcal M}\)-algebras will be shown. In Section 3, the pretabularity of \({\mathcal M}\) will be proved.     

Matching Topological and Frame Products of Modal Logics

The simplest combination of unimodal logics \({\mathrm{L}_1 \rm and \mathrm{L}_2}\) into a bimodal logic is their fusion, \({\mathrm{L}_1 \otimes \mathrm{L}_2}\), axiomatized by the theorems of \({\mathrm{L}_1 \rm for \square_1 \rm and of \mathrm{L}_2 \rm for \square_{2}}\). Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product\({\mathrm{L}_1 \times \mathrm{L}_2 \rm of \mathrm{L}_1 \rm and \mathrm{L}_2}\). Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product\({\mathrm{L}_1 \times_{t}\mathrm{L}_2}\), using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of \({\mathrm{S}4: \mathrm{L}_1 \times_t \mathrm{L}_2 = \mathrm{L}_1 \times \mathrm{L}_2 \rm iff \mathrm{L}_1 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_2 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_1, \mathrm{L}_2 = \mathrm{S}5}\).     

Factor analysis models via I-divergence optimization

Given a positive definite covariance matrix \(\widehat{\Sigma }\) of dimension n, we approximate it with a covariance of the form \(HH^\top +D\), where H has a prescribed number \(k<n\) of columns and \(D>0\) is diagonal. The quality of the approximation is gauged by the I-divergence between the zero mean normal laws with covariances \(\widehat{\Sigma }\) and \(HH^\top +D\), respectively. To determine a pair (H, D) that minimizes the I-divergence we construct, by lifting the minimization into a larger space, an iterative alternating minimization algorithm (AML) à la Csiszár–Tusnády. As it turns out, the proper choice of the enlarged space is crucial for optimization. The convergence of the algorithm is studied, with special attention given to the case where D is singular. The theoretical properties of the AML are compared to those of the popular EM algorithm for exploratory factor analysis. Inspired by the ECME (a Newton–Raphson variation on EM), we develop a similar variant of AML, called ACML, and in a few numerical experiments, we compare the performances of the four algorithms.     

Resampling-Based Inference Methods for Comparing Two Coefficients Alpha

The two-sample problem for Cronbach’s coefficient \(\alpha _C\), as an estimate of test or composite score reliability, has attracted little attention compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistical procedures of comparing two coefficients \(\alpha _{C,1}\) and \(\alpha _{C,2}\). The null hypothesis of interest is \(H_0 : \alpha _{C,1} = \alpha _{C,2}\), which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed for two-group multivariate non-normal models under the general asymptotically distribution-free (ADF) setting. These statistical tests ensure a better control of the type-I error, in finite or very small sample sizes, when the state-of-affairs ADF large-sample test may fail to properly attain the nominal significance level. By proper choice of a studentized test statistic, the resampling tests are modified in order to be valid asymptotically even in non-exchangeable data frameworks. Moreover, extensions of this approach to other designs and reliability measures are discussed as well. Finally, the usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.     

Human Psychophysical Functions,an Update: Methods for Identifying their form; Estimating their Parameters; and Evaluating the Effects of Important Predictors

Stevens’ power law for the judgments of sensation has a long history in psychology and is used in many psychophysical investigations of the effects of predictors such as group or condition. Stevens’ formulation \(\varPsi = {aP}^{n}\), where \(\varPsi \) is the psychological judgment, P is the physical intensity, and \(n\) is the power law exponent, is usually tested by plotting log \((\varPsi )\) against log (P). In some, but by no means all, studies, effects on the scale parameter, \(a\), are also investigated. This two-parameter model is simple but known to be flawed, for at least some modalities. Specifically, three-parameter functions that include a threshold parameter produce a better fit for many data sets. In addition, direct non-linear computation of power laws often fit better than regressions of log-transformed variables. However, such potentially flawed methods continue to be used because of assumptions that the approximations are “close enough” as to not to make any difference to the conclusions drawn (or possibly through ignorance the errors in these assumptions). We investigate two modalities in detail: duration and roughness. We show that a three-parameter power law is the best fitting of several plausible models. Comparison between this model and the prevalent two parameter version of Stevens’ power law shows significant differences for the parameter estimates with at least medium effect sizes for duration.     

Contextual-Hierarchical Reconstructions of the Strengthened Liar Problem

In this paper we shall introduce two types of contextual-hierarchical (from now on abbreviated by ‘ch’) approaches to the strengthened liar problem. These approaches, which we call the ‘standard’ and the ‘alternative’ ch-reconstructions of the strengthened liar problem, differ in their philosophical view regarding the nature of truth and the relation between the truth predicates T r ⁿ and T r ⁿ⁺¹ of different hierarchy-levels. The basic idea of the standard ch-reconstruction is that the T r ⁿ⁺¹-schema should hold for all sentences of \(\mathcal {L}^{n}\). In contrast, the alternative ch-reconstruction, for which we shall argue in section four, is motivated by the idea that T r ⁿ and T r ⁿ⁺¹ are coherent in the sense that the same sentences of \(\mathcal {L}^{n}\) should be true according to T r ⁿ and T r ⁿ⁺¹. We show that instances of the standard ch-reconstruction can be obtained by iterating Kripke’s strong Kleene jump operator. Furthermore, we will demonstrate how instances of the alternative ch-reconstruction can be obtained by a slight modification of the iterated axiom system KF and of the iterated strong Kleene jump operator.     

Private Announcements on Topological Spaces

In this work, we present a multi-agent logic of knowledge and change of knowledge interpreted on topological structures. Our dynamics are of the so-called semi-private character where a group G of agents is informed of some piece of information \(\varphi \), while all the other agents observe that group G is informed, but are uncertain whether the information provided is \(\varphi \) or \(\lnot \varphi \). This article follows up on our prior work (van Ditmarsch et al. in Proceedings of the 15th TARK. pp 95-102, 2015) where the dynamics were public events. We provide a complete axiomatization of our logic, and give two detailed examples of situations with agents learning information through semi-private announcements.     

A Categorical Equivalence Motivated by Kalman’s Construction

An equivalence between the category of MV-algebras and the category \({{\rm MV^{\bullet}}}\) is given in Castiglioni et al. (Studia Logica 102(1):67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations \({a = \neg \neg a, (a \rightarrow b) \vee (b\rightarrow a) = 1}\) and \({a \odot (a\rightarrow b) = a \wedge b}\). An object of \({{\rm MV^{\bullet}}}\) is a residuated lattice which in particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs (A, I), where A is an MV-algebra and I is an ideal of A.