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 \(\mathsf{GLK}\).     

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.     

Making sense of (in)determinate truth: the semantics of free variables

It is argued that truth value of a sentence containing free variables in a context of use (or the truth value of the proposition it expresses in a context of use), just as the reference of the free variables concerned, depends on the assumptions and posits given by the context. However, context may under-determine the reference of a free variable and the truth value of sentences in which it occurs. It is argued that in such cases a free variable has indeterminate reference and a sentence in which it occurs may have indeterminate truth value. On letting, say, x be such that \(x^2=4\), the sentence ‘Either \(x=2\) or \(x=-2\)’ is true but the sentence ‘\(x=2\)’ has an indeterminate truth value: it is determinate that the variable x refers to either 2 or \(-2\), but it is indeterminate which of the two it refers to, as a result ‘\(x=2\)’ has a truth value but its truth value is indeterminate. The semantic indeterminacy is analysed in a ‘radically’ supervaluational (or plurivaluational) semantic framework closely analogous to the treatment of vagueness in McGee and McLaughlin (South J Philos 33:203–251, 1994, Linguist Philos 27:123–136, 2004) and Smith (Vagueness and degrees of truth, Oxford University Press, Oxford, 2008), which saves bivalence, the T-schema and the truth-functional analysis of the boolean connectives. It is shown that on such an analysis the modality ‘determinately’ is quite clearly not an epistemic modality, avoiding a potential objection raised by Williamson (Vagueness, Routledge, London, 1994) against such ‘radically’ supervaluational treatments of vagueness, and that determinate truth (rather than truth simpliciter) is the semantic value preserved in classically valid arguments. The analysis is contrasted with the epistemicist proposal of Breckenridge and Magidor (Philos Stud 158:377–400, 2012) which implies that (in the given context) ‘\(x=2\)’ has a determinate but unknowable truth value.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Foundationalism with infinite regresses of probabilistic support

There is a long-standing debate in epistemology on the structure of justification. Some recent work in formal epistemology promises to shed some new light on that debate. I have in mind here some recent work by David Atkinson and Jeanne Peijnenburg, hereafter “A&P”, on infinite regresses of probabilistic support. A&P show that there are probability distributions defined over an infinite set of propositions {\(p_{1}, p_{2}, p_{3}, {\ldots }, p_{n}, {\ldots }\}\) such that (i) \(p_{i}\) is probabilistically supported by \(p_{i+1}\) for all i and (ii) \(p_{1}\) has a high probability. Let this result be “APR” (short for “A&P’s Result”). A&P oftentimes write as though they believe that APR runs counter to foundationalism. This makes sense, since there is some prima facie plausibility in the idea that APR runs counter to foundationalism, and since some prominent foundationalists argue for theses inconsistent with APR. I argue, though, that in fact APR does not run counter to foundationalism. I further argue that there is a place in foundationalism for infinite regresses of probabilistic support.     

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.     

Semantic Distances in Depression: Relations Between ME and PAST,FUTURE, JOY,SADNESS, HAPPINESS

Using the Semantic Distance Task, we investigated the semantic distances between ME and five metaphorically conceptualized notions: PAST, FUTURE, JOY, SADNESS, and HAPPINESS. Three Polish-speaking groups participated in the study: depressive subjects (\(n = 30\)), patients in remission (\(n = 12\)), and non-depressed individuals (\(n = 30\)). T-test and the Kruskal–Wallis nonparametric equivalent of ANOVA analyses showed that subjects in remission placed ME significantly farther away from PAST than non-depressed individuals and depressed patients. Data mining algorithms indicated the distances ME–SADNESS, ME–PAST, and ME–FUTURE as the three strongest predictors of group membership. We interpret the findings in the light of a contrast effect and defense mechanisms. We propose that intergroup differences are especially prominent in tasks requiring creation of semantic associative relations, that is, in the first stage of conceptual processing. We suggest treating the results as confirmation that Beck’s theory of depression applies at the level of notion comprehension, proving that processing of key concepts in depression symptoms (particularly PAST) runs differently in all three groups under consideration.     

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.     

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.