Propositional Epistemic Logics with Quantification Over Agents of Knowledge

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal (epistemic) operators or over agents of knowledge and extended by predicate symbols that take modal (epistemic) operators (or agents) as arguments. Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There exist epistemic logics whose languages have the above mentioned properties (see, for example Corsi and Orlandelli in Stud Log 101:1159–1183, 2013; Fitting et al. in Stud Log 69:133–169, 2001; Grove in Artif Intell 74(2):311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science (LNCS), vol 1193, pp 71–85, 1996). But these logics are obtained from first-order modal logics, while a logic of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal (epistemic) operators and predicate symbols that take modal (epistemic) operators as arguments. Among the logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re (between ‘knowing that’ and ‘knowing of’). We show the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) with the help of the loosely guarded fragment (LGF) of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal (epistemic) operators. The family of this logics coincides with \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols (see Grove and Halpern in J Log Comput 3(4):345–378, 1993). Some logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as counterparts of logics defined in Grove and Halpern (J Log Comput 3(4):345–378, 1993). We prove that the satisfiability problem for these logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) is Pspace-complete using their counterparts in Grove and Halpern (J Log Comput 3(4):345–378, 1993).     

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.     

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.     

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.     

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 \(\mathsf{GLK}\).     

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.     

$${\in_K}$$: 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.     

On All Strong Kleene Generalizations of Classical Logic

By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the \({{\bf SK}^\mathcal{P}}\) and \({\bf SK}^{\mathcal{N}}\) calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, we also hint at its philosophical significance.     

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.     

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.     

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 \({\mathrm{S}4: \mathrm{L}_1 \times_t \mathrm{L}_2 = \mathrm{L}_1 \times \mathrm{L}_2 \rm iff \mathrm{L}_1 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_2 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_1, \mathrm{L}_2 = \mathrm{S}5}\).     

Bayes factor design analysis: Planning for compelling evidence

A sizeable literature exists on the use of frequentist power analysis in the null-hypothesis significance testing (NHST) paradigm to facilitate the design of informative experiments. In contrast, there is almost no literature that discusses the design of experiments when Bayes factors (BFs) are used as a measure of evidence. Here we explore Bayes Factor Design Analysis (BFDA) as a useful tool to design studies for maximum efficiency and informativeness. We elaborate on three possible BF designs, (a) a fixed-n design, (b) an open-ended Sequential Bayes Factor (SBF) design, where researchers can test after each participant and can stop data collection whenever there is strong evidence for either \(\mathcal {H}_{1}\) or \(\mathcal {H}_{0}\), and (c) a modified SBF design that defines a maximal sample size where data collection is stopped regardless of the current state of evidence. We demonstrate how the properties of each design (i.e., expected strength of evidence, expected sample size, expected probability of misleading evidence, expected probability of weak evidence) can be evaluated using Monte Carlo simulations and equip researchers with the necessary information to compute their own Bayesian design analyses.     

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.     

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.     

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.     

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.