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 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 \(\mathsf{GLK}\).     

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.     

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.     

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).     

Epistemic modals and informational consequence

Recently, Yalcin (Epistemic modals. Mind, 116, 983–1026, 2007) put forward a novel account of epistemic modals. It is based on the observation that sentences of the form ‘\({\phi}\) &; Might \({\neg\phi}\) ’ do not embed under ‘suppose’ and ‘if’. Yalcin concludes that such sentences must be contradictory and develops a notion of informational consequence which validates this idea. I will show that informational consequence is inadequate as an account of the logic of epistemic modals: it cannot deal with reasoning from uncertain premises. Finally, I offer an alternative way of explaining the relevant linguistic data.     

Hillel and Confucius: The prescriptive formulation of the golden rule in the Jewish and Chinese Confucian ethical traditions

A prospective convert asked Hillel to teach him the entire Torahwhile standing on one foot. Hillel replied, “What is hateful to yourself, do not do to your fellow man. That isthe whole of Torah and the remainder is but commentary. Go and study it.” (Hillel:Shab. 31; emphasis added)Zigong: “Is there asingle word that can serve as a guide to conduct throughout one’s life?” Confucius said: “Perhaps the word ‘shu’, ‘reciprocity’: ‘Do not do to others what you would not want others to do to you’.” (Analects: 15.24; see alsoAnalects. 12 andZhongyong. 13.3; emphasis added)¹     

A Psychometric Evaluation of the Behavioral Inhibition Questionnaire in a Non-Clinical Sample of Israeli Children and Adolescents

This study examined the psychometric properties of a Hebrew version of the Behavioral Inhibition Questionnaire (BIQ) in a non-clinical sample of Israeli children and adolescents. We produced a Hebrew translation of the BIQ and collected 227 responses to it from parents of children aged 4–15. Some respondents in the larger sample also completed the Screen for Child Anxiety Related Emotional Disorders (SCARED) questionnaire (n?=?91) and the Conners’ Abbreviated Parent-Teacher (CONNERS) questionnaire (n?=?39), in addition to the BIQ. Lastly, 21 children of BIQ respondents (aged 8–14) completed a self-report version of the questionnaire. Confirmatory factor analysis (CFA) was performed to assess how well the established six correlated factor model of the BIQ applied to the sample data. The Hebrew BIQ demonstrated good internal consistency (Chronbach’s α?=?.94, n?=?227) and 3 month test–retest reliability, (r?=?.95, p?<?.001, n?=?21). It also showed both convergent validity, as scores on the BIQ were correlated with the SCARED (r?=?.66, p?<?.01. n?=?91), and discriminant validity, as BIQ scores were not correlated with the CONNERS (r?=?.24, n?=?39). Finally, mother reports of BI were significantly correlated to child reports of BI via the BIQ (r?=?.60, p?<?.01, n?=?21). Thus, through this preliminary study we demonstrated that the Hebrew version of the BIQ is an effective tool for screening for BI among Israeli children, making it a useful instrument for future research.     

$${\in_K}$$: 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.     

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.     

Development and Validation of the Knowledge of Parenting Strategies Scale: Measuring Effective Parenting Strategies

The current study aimed to develop and evaluate a measure of parenting knowledge, the Knowledge of Parenting Strategies Scale (KOPSS); specifically, to establish the scales internal reliability, ensure a clinically appropriate length, provide a community sample for future comparison, demonstrate adequate test–retest reliability and convergent validity, and to compare the scale to dysfunctional discipline styles. A total of n?=?865 parents were involved in the development and evaluation of the scale. In Study 1, data was collected from n?=?229 parents and Rasch analyses revealed seven items did not fit the measurement model. Study 2 involved a further sample of community families (n?=?346) and revealed the scale could be further shortened to 16 items. Study 3 revealed the scale has good test–retest reliability over a one-week period (r?=?.88, p?<?.001). Study 4 demonstrated convergent validity through a comparison to the Knowledge of Effective Parenting Scale (r?=?.583, p?=?.009). Study 5 utilised a sample of community families (n?=?190), revealing the scale was negatively correlated with hostile and lax discipline (r?=??.29, p?<?.001; r?=??.15, p?<?.05). Lastly, Study 6 showed scores on the KOPSS significantly improved following clinic-based and Internet-based Behavioural Parent Training. The KOPSS was found to be a valid and reliable measure of parenting knowledge of effective parenting strategies, which can be used to evaluate knowledge acquisition in parenting programs, and test the role of knowledge in behaviour change.     

The Modal Logic of Gödel Sentences

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.     

<Emphasis Type="Italic">Wh</Emphasis>-Questions,Universal Statements and Free Choice Inferences in Child Mandarin

This study investigated 5-year-old Mandarin-speaking children’s comprehension of wh-questions, universal statements and free choice inferences. Previous research has found that Mandarin-speaking children assign a universal interpretation to sentences with a wh-word (e.g., shei ‘who’) followed by the adverbial quantifier dou ‘all’ (Zhou in Appl Psycholinguist 36:411–435, 2013). Children also compute free choice inferences in sentences that contain a modal verb in addition to a wh-word and dou (Zhou, in: Nakayama, Su, Huang (eds.) Studies in Chinese and Japanese language acquisition: in honour of Stephen Crain. John Benjamins Publishing Company, Amsterdam, pp 223–235, 2017). The present study used a Question-Statement Task to assess children’s interpretation of sentences containing shei + dou, both with and without the modal verb beiyunxu ‘was allowed to’, as well as the contrast between sentences with shei + dou, which are statements for adults, versus ones with dou + shei, which are wh-questions for adults. The 5-year-old Mandarin-speaking child participants exhibited adult-like linguistic knowledge of the semantics and pragmatics of wh-words, the adverbial quantifier dou, and the deontic modal verb beiyunxu.     

Executive Functioning Correlates of <Emphasis Type="Italic">DSM-5</Emphasis> Maladaptive Personality Traits: Initial Evidence from an Italian Sample of Consecutively Admitted Adult Outpatients

In order to evaluate the associations between computer-administered tasks of executive functioning (EF), and maladaptive personality domains and traits listed in DSM-5 Alternative Model of Personality Disorders, 53 consecutively admitted psychotherapy outpatients (female participants: n?=?27, 50.9%; male participants: n?=?26, 49.1%; participants’ mean age?=?37.28 years, SD?=?11.50 years) were administered the Psychology Experiment Building Language (PEBL) EF tasks and the Personality Inventory for DSM-5 (PID-5). According to rank-order correlation analyses, a number of non-negligible and specific associations were observed between selected PID-5 scales and indices of participants’ performance on EF tasks. MM robust regression models showed that participants’ performance on computer-administered EF tasks explained a non-negligible amount of variance in selected PID-5 scale scores (median R² value?=?.17). As a whole, our trait-level analyses of PID-5 dimensions suggest the clinical usefulness of integrating self-reports and EF laboratory tasks in routine clinical assessment.     

Scientific Realism Within Perspectivism and Perspectivism Within Scientific Realism

Perspectivism is often understood as a conception according to which subjective conditions inevitably affect our knowledge and, therefore, we are never confronted with reality and facts but only with interpretations. Hence, subjectivism and anti-realism are usually associated with perspectivism. The thesis of this paper is that, especially in the case of the sciences, perspectivism can be better understood as an appreciation of the cognitive attitude that consists in considering reality only from a certain ‘point of view’, in a way that can avoid subjectivism. Whereas the way of conceiving a notion is strictly subjective, the way of using it is open to intersubjective agreement, based on the practice of operations whose nature is neither mental nor linguistic. Therefore, intersubjectivity (that is a ‘weak’ sense of objectivity) is possible within perspectivism. Perspectivism can also help understand the notion of ‘scientific objects’ in a referential sense: they are those ‘things’ that become ‘objects’ of a certain science by being investigated from the ‘point of view’ of that science. They are ‘clipped out’ of things (and constitute the ‘domain of objects’ or the ‘regional ontology’ of that particular science) by means of standardized operations which turn out to be the same as those granting intersubjectivity. Therefore this ‘strong’ sense of objectivity, which is clearly realist, coincides with the ‘weak’ one. The notion of truth appears fully legitimate in the case of the sciences, being clearly defined for the regional ontology of each one of them and, since this truth can be extended in an analogical sense to the theories elaborated in each science, it follows that are real also the unobservable entities postulated by those theories.     

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.