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.     

Two Kinds of Consequential Implication

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by \(\rightarrow \) and \(\Rightarrow \)). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for \(\Rightarrow \), CI\(\Rightarrow \), is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for \(\rightarrow \), CI. The main drawback ot the weaker operator \(\Rightarrow \) is that it lacks unrestricted contraposition, but the final section of the paper argues that \(\Rightarrow \) has some properties which make it a valuable alternative to \(\rightarrow \), turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.     

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.     

Praise,blame, and demandingness

Consequentialism has been challenged on the grounds that it is too demanding. I will respond to the problem of demandingness differently from previous accounts. In the first part of the paper, I argue that consequentialism requires us to distinguish the justification of an act \(\varphi\) from the justification of an act \(\psi\), where \(\psi\) is an act of praise or blame. In the second part of the paper, I confront the problem of demandingness. I do not attempt to rule out the objection; instead, I argue that if certain plausible empirical claims about moral motivation are true, we morally ought not to blame people for failing to meet certain very demanding obligations. With this theory, we create a space in consequentialism for intuitions questioning the plausibility of demanding obligations. I conclude the paper by showing that separate justifications for \(\varphi\) and \(\psi\) may also give us a theoretical niche for intuitions about supererogation.     

Arithmetical Completeness Theorem for Modal Logic $$\mathsf{}$$

We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \(\Sigma _2\) provability predicate of T whose provability logic is precisely the modal logic \(\mathsf{K}\). For this purpose, we introduce a new bimodal logic \(\mathsf{GLK}\), and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \(\mathsf{GLK}\).     

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.     

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.     

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.     

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.     

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.     

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.     

Can probability theory explain why closure is both intuitive and prone to counterexamples?

Epistemic closure under known implication is the principle that knowledge of \(\varphi\) and knowledge of \(\varphi \rightarrow \psi\), together, imply knowledge of \(\psi\). This principle is intuitive, yet several putative counterexamples have been formulated against it. This paper addresses the question, why is epistemic closure both intuitive and prone to counterexamples? In particular, the paper examines whether probability theory can offer an answer to this question based on four strategies. The first probability-based strategy rests on the accumulation of risks. The problem with this strategy is that risk accumulation cannot accommodate certain counterexamples to epistemic closure. The second strategy is based on the idea of evidential support, that is, a piece of evidence supports a proposition whenever it increases the probability of the proposition. This strategy makes progress and can accommodate certain putative counterexamples to closure. However, this strategy also gives rise to a number of counterintuitive results. Finally, there are two broadly probabilistic strategies, one based on the idea of resilient probability and the other on the idea of assumptions that are taken for granted. These strategies are promising but are prone to some of the shortcomings of the second strategy. All in all, I conclude that each strategy fails. Probability theory, then, is unlikely to offer the account we need.     

Truth,Partial Logic and Infinitary Proof Systems

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an \(\omega \)-rule.     

Asymptotically Correct Standardization of Person-Fit Statistics Beyond Dichotomous Items

The \(l_z\) statistic (Drasgow et al. in Br J Math Stat Psychol 38:67–86, 1985) is one of the most popular person-fit statistics (Armstrong et al. in Pract Assess Res Eval 12(16):1–10, 2007). Snijders (Psychometrika 66:331–342, 2001) derived the asymptotic null distribution of \(l_z\) when the examinee ability parameter is estimated. He also suggested the \(l^*_z\) statistic, which is the asymptotically correct standardized version of \(l_z\). However, Snijders (Psychometrika 66:331–342, 2001) only considered tests with dichotomous items. In this paper, the asymptotic null distribution of \(l_z\) is derived for mixed-format tests (those that include both dichotomous and polytomous items). The asymptotically correct standardized version of \(l_z\), which can be considered as the extension of \(l^*_z\) to such tests, is suggested. The Type I error rate and power of the suggested statistic are examined from several simulated datasets. The suggested statistic is computed using a real dataset. The suggested statistic appears to be a satisfactory tool for assessing person fit for mixed-format tests.     

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.     

Meaningful Words and Non-Words Repetitive Articulatory Rate (Oral Diadochokinesis) in Persian Speaking Children

Repetitive articulatory rate or Oral Diadochokinesis (oral-DDK) shows a guideline for appraisal and diagnosis of subjects with oral-motor disorder. Traditionally, meaningless words repetition has been utilized in this task and preschool children have challenges with them. Therefore, we aimed to determine some meaningful words in order to test oral-DDK in Persian speaking preschool children. Participants were 142 normally developing children, (age range 4–6 years), who were asked to produce /motæka, golabi/ as two meaningful Persian words and /pa-ta-ka/ as non-word in conventional oral-DDK task. We compared the time taken for 10-times fast repetitions of two meaningful Persian words and the tri-syllabic nonsense word /pa-ta-ka/. Praat software was used to calculate the average time that subjects took to produce the target items. In 4–5 year old children, \(\hbox {mean}\pm \hbox {SD}\) of time taken for 10-times repetitions of /pa-ta-ka, motæka, golabi/ were \(7.72\pm 1.02, 6.58\pm 1.68\), and \(6.65\pm 1.13\) seconds respectively, and in 5–6 year old children were \(7.57\pm 0.95, 6.35\pm 1.38\), and \(6.30\pm 1.01\) seconds respectively. Findings showed that the main effect of type of words on oral diadochokinesis was significant (\({p}\,{<}\,0.001\)). Children repeated meaningful words /motæka, golabi/ faster than the non-word /pa-ta-ka/. Sex and age factors had no effect on time taken for repetition of oral-DDK test. It is suggested that Speech Therapists can use meaningful words to facilitate oral-DDK test for children.     

Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures

Every transformation monoid comes equipped with a canonical topology, the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever \({\mathbf {A}}\) is a countable \(\aleph _0\)-categorical G-finite structure whose automorphism group has a trivial center and if \({\mathbf {B}}\) is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism.     

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.