Interpolation in 16-Valued Trilattice Logics

In a recent paper we have defined an analytic tableau calculus \({{\mathbf {\mathsf{{PL}}}}}_{\mathbf {16}}\) for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice \({SIXTEEN}_3\). This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations Open image in new window , Open image in new window , and Open image in new window that each correspond to a lattice order in \({SIXTEEN}_3\); and Open image in new window , the intersection of Open image in new window and Open image in new window . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that Open image in new window , when restricted to \(\mathcal {L}_{tf}\), the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.     

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.     

Proof Theory for Functional Modal Logic

We present some proof-theoretic results for the normal modal logic whose characteristic axiom is \(\mathord {\sim }\mathord {\Box }A\equiv \mathord {\Box }\mathord {\sim }A\). We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively. We close by proving equivalences with a Fitch-style proof system for revision theory.     

On Argumentation Logic and Propositional Logic

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.     

Arrovian Aggregation of Generalised Expected-Utility Preferences: (Im)possibility Results by Means of Model Theory

Cerreia-Vioglio et al. (Econ Theory 48(2–3):341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.     

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.     

Does Rumination Mediate the Association of Private Prayer,Stress, and Their Interaction with Depression among Christians? A Cross-Sectional Study

Mechanisms underlying significant associations between different types of prayer and depression are unknown. Based on the conceptualization of prayer as stress buffering coping style and the Response Style Theory, we proposed that rumination mediates the association of prayer types and stress with depression. This cross-sectional study (N?=?227 Christians, M_age 41.74, SD 14.82, age range 18–82 years, 76% female) used path modeling to test whether rumination mediated the associations between prayer types and prayer types by stress interactions with depression. Rumination fully mediated the associations of the colloquial and ritual prayer by stress interactions with depressive symptoms, while meditative and petitionary prayer was directly associated with depressive symptoms. Further, petitionary prayer and the ritual prayer by stress interaction were associated with more and not with less rumination and depressive symptoms, respectively. Summarized, first empirical evidence supports an integration of prayer, stress, and rumination into one model of depression. However, the exact nature of their interplay depends on the prayer type. If replicated, our findings allow the integration of prayer into cognitive-behavioral treatment of depression in religious and spiritual clients.     

Cannabis and Amphetamine Use and Socio-Ecological Proximal and Distal Factors Among School-Going Adolescents in Six Pacific Island Countries

The study aimed to assess the prevalence of cannabis and amphetamine use and socio-ecological proximal and distal factors among adolescents in six Pacific Island countries. We analyzed cross-sectional survey data from 9,849 adolescents (mean age 14.2 years, SD?=?1.2) from Cook Islands, Kiribati, Samoa, Solomon Islands, Tonga and Tuvalu that participated in the “Global School-Based Student Health Survey (GSHS)” in 2011–2013. The prevalence of past-month cannabis use was 15.5%, and lifetime amphetamine use was 14.5%, with the highest prevalence in Samoa, 37.1% and 34.6%, respectively. In adjusted mixed-effects logistic regression analysis male gender, personal attributes (anxiety, loneliness, current tobacco use, current alcohol use and school truancy) and environmental stressors (having been in a physical fight, been physically attacked, sustained an injury, hunger and being bullied) were associated with past-month cannabis and/or lifetime amphetamine use. Parental substance use was positively and parental involvement negatively associated with past-month cannabis and/or lifetime amphetamine use. Measures to prevent and control cannabis and amphetamine use in this adolescent population should include personal attributes, environmental stressors and parental support.     

Emotion Regulation and the Specific Associations with Health Anxiety

This study investigated two primary research aims surrounding health anxiety: (a) to examine what differences exist in emotion regulation strategies of health anxious and non-health anxious populations and (b) to investigate what emotion regulation strategies predict health anxiety, before and after controlling for trait cognitive and somatic anxiety. A total of 95 participants of the general population completed a series of online questionnaires. A single factor between groups MANOVA revealed that a health anxious group scored significantly higher in several emotion regulation strategies: distraction, external attribution, emotion expression and acting out; anxiety-related variables: trait cognitive and somatic anxiety; and beliefs about the negative consequences of becoming ill. Hierarchical regressions showed that several emotion regulation strategies were significantly associated with higher levels of health anxiety, before: distraction, external attribution, capitalising, positive reappraisal and emotional expression; and after: external attribution and capitalising, the control of trait anxiety. Overall, the results demonstrated that (a) emotional intelligence and behavioural attempts to understand emotional experiences may be associated with specific health anxiety symptomology, while a lack of effective cognitive strategies and worry may be responsible for the associated trait anxiety dimensions.