Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes $$\textrm{E}_k$$ and $$\textrm{U}_k$$ introduced in [1] are exactly the classes induced by $$\Sigma _k$$ and $$\Pi _k$$ respectively via the transformation procedure in (...) any first-order theory. (shrink)

Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves (...) to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic. (shrink)

We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.

We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$. Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$. In addition, we show that this conservation theorem is optimal in the (...) sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$. In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles. (shrink)

Monism is a metaphysical view according to which there is only one fundamental object. This paper will explore monism within the context of truthmaker theory, or Truthmaker Monism, a view rarely discussed in literature. Although few truthmaker theorists defend monism, at least explicitly, some theories seem to share the spirit of monism to some extent. Interestingly, they are proposed as solutions for the same problem, called the problem of negative truth. A close examination will show that while each of these (...) solutions can account for the problem, they are unsatisfactory for different reasons. This paper will suggest a novel solution to the problem of negative truth which preserves the advantages of existing solutions while avoiding the difficulties with which they are faced. (shrink)

We systematically study several versions of the disjunction and the existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm {B}$, $\Delta (\mathrm {B})$, and $\Sigma (\mathrm {B})$ of formulas of modal arithmetic and study basic properties of them. Then, we prove several implications between the properties. In particular, among other things, we prove that for any consistent recursively enumerable extension T of $\mathbf {PA}(\mathbf {K})$ with $T \nvdash \Box \bot $, the $\Sigma (\mathrm {B})$ -disjunction property, the (...) $\Sigma (\mathrm {B})$ -existence property, and the $\mathrm {B}$ -existence property are pairwise equivalent. Moreover, we introduce the notion of the $\Sigma (\mathrm {B})$ -soundness of theories and prove that for any consistent recursively enumerable extension of $\mathbf {PA}(\mathbf {K4})$, the modal disjunction property is equivalent to the $\Sigma (\mathrm {B})$ -soundness. (shrink)

Erdelyi distinguishes between cognitive and emotional forms of repression, but argues that they use the same general mechanism. His discussion of experimental memory findings, on the one hand, and clinical examples, on the other, does indeed indicate considerable overlap. As an in-between level of evidence, research findings on emotion in neuroscience, as well as experimental and social/personality psychology, further support his argument.

This book tracks the emergence of the modern Japanese nation in the nineteenth century through the history of some of its local aspirants. It explores how kokugaku (Japan studies) scholars envisioned their place within Japan and the globe, while living in a castle town and domain far north of the political capital.

Prior studies suggest that psychological difficulties arise from higher trait rejection sensitivity (RS)—heightened vigilance and differential detection of social rejection cues and defensive response to. On the other hand, from an evolutionary perspective, rapid and efficient detection of social rejection cues can be considered beneficial. We conducted a survey and an electrophysiological experiment to reconcile this seeming contradiction. We compared the effects of RS and rejection detection capability (RDC) on perceived interpersonal experiences (Study 1) and on neurocognitive processes in response (...) to cues of social rejection (disgusted faces; Study 2). We found that RS and RDC were not significantly related, although RS was positively related to perceived social rejection experiences and RDC was positively related to perceived social inclusion experiences. Event-related brain potentials (ERPs) revealed that higher RS was related to cognitive avoidance (i.e., P1) and heightened motivated attention (i.e., late positive potential: LPP), but not to facial expression encoding (i.e., N170) toward disgusted faces. On the other hand, higher RDC was related to heightened N170 amplitude, but not to P1 and LPP amplitudes. These findings imply that sensitivity to rejection is apparently distinct from the ability to detect social rejection cues and instead reflects intense vigilance and defensive response to those cues. We discussed an alternative explanation of the relationship between RS and RDC from a signal detection perspective. (shrink)

This volume investigates how the phenomenology of religion was accepted and developed in different national contexts. It consists of interviews with senior scholars, who are experts on the development of the phenomenology of religion in their countries, along with commentary and analysis.

We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions. First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert–Bernays’ consistency statement. Secondly, we improve Buchholz’s schematic proof of provable $\Sigma_1$ -completeness. Then among other things, we show that Hilbert–Bernays’ conditions and Löb’s conditions are (...) mutually incomparable. We also show that neither Hilbert–Bernays’ conditions nor Löb’s conditions accomplish Gödel’s original statement of the second incompleteness theorem. (shrink)

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

We study modal completeness and incompleteness of several sublogics of the interpretability logic. We introduce the sublogic, and prove that is sound and complete with respect to Veltman prestructures which are introduced by Visser. Moreover, we prove the modal completeness of twelve logics between and with respect to Veltman prestructures. On the other hand, we prove that eight natural sublogics of are modally incomplete. Finally, we prove that these incomplete logics are complete with respect to generalized Veltman prestructures. As a (...) consequence of these investigations, we obtain that the twenty logics studied in this paper are all decidable. (shrink)

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \, there exists a \ numeration \\) of T such that the provability logic of the provability predicate \\) naturally constructed from \\) is exactly \ \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.

It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of Rosser-type formalizations of (...) Yablo’s paradox for each consistent but not Σ1-sound theory is dependent on the choice of a standard proof predicate. (shrink)

We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation between quantified provability logics with respect to \ arithmetical interpretations.

Let \ be a class of formulas. We say that a theory T in classical logic has the \-disjunction property if for any \ sentences \ and \, either \ or \ whenever \. First, we characterize the \-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of (...) formalized partial disjunction properties. (shrink)

Previous studies have found that serum lipid levels independently associate with mental health problems in adulthood. However, little is known about the association between serum lipid levels and positive aspects of mental health such as resilience and self-esteem, which develop in adolescence. The aim of this study is to examine the association between serum lipid levels and resilience and self-esteem in Japanese adolescents. Data were pooled data from the Adachi Child Health Impact of Living Difficulty study in 2016 and 2018, (...) a school-based, cross-sectional study in Adachi City, Tokyo, Japan. Resilience of the child was assessed by caregivers, and self-esteem was self-identified via questionnaires. Serum lipid levels [total cholesterol, low-density lipoproteins, and high-density lipoproteins ] were assessed in school health checkup, in addition to height and weight measurements. Multiple linear regression was applied to investigate the association between standardized serum lipid levels and resilience and self-esteem. LDL showed inverse association with resilience [β = −1.26, 95% confidence interval = −2.39 to −0.14] after adjusting for child’s BMI, month of birth, sex, absence of parent, household income, caregiver’s mental health, and lifestyle. We also found an inverse association of total cholesterol and higher LDL cholesterol with self-esteem. HDL cholesterol was not associated with resilience and self-esteem. Among Japanese adolescent, total and LDL cholesterol may be biomarkers of resilience and self-esteem. (shrink)

Solovay proved the arithmetical completeness theorem for the system GL of propositional modal logic of provability. Montagna proved that this completeness does not hold for a natural extension QGL of GL to the predicate modal logic. Let Th(QGL) be the set of all theorems of QGL, Fr(QGL) be the set of all formulas valid in all transitive and conversely well-founded Kripke frames, and let PL(T) be the set of all predicate modal formulas provable in Tfor any arithmetical interpretation. Montagna’s results (...) are described as Th(QGL) ⊊ (Fr(QGL), PL(PA) ⊈ Fr(QGL), and Th(QGL) ⊊ PL(PA). In this paper, we prove the following three theorems: (1) Fr(QGL) ⊈ PL(T) for any Σ1-sound recursively enumerable extension T of I Σ1, (2) PL(T) ⊈ Fr(QGL) for any recursively enumerable S1755020312000275_inline1 A -theory T extending I Σ1, and (3) Th(QGL) ⊊ Fr(QGL) ∩ PL(T) for any recursively enumerable S1755020312000275_inline2 A -theory T extending I Σ2. To prove these theorems, we use iterated consistency assertions and nonstandard models of arithmetic, and we improve Artemov’s lemma which is used to prove Vardanyan’s theorem on the Π0 2-completeness of PL(T). (shrink)

In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic \. Secondly, we introduce a new normal modal logic \ which is a proper extension of \, and prove that there exists a Rosser provability predicate whose provability logic includes \.

Solovay proved (Israel J Math 25(3–4):287–304, 1976) that the propositional provability logic of any ∑2-sound recursively enumerable extension of PA is characterized by the propositional modal logic GL. By contrast, Montagna proved in (Notre Dame J Form Log 25(2):179–189, 1984) that predicate provability logics of Peano arithmetic and Bernays–Gödel set theory are different. Moreover, Artemov proved in (Doklady Akademii Nauk SSSR 290(6):1289–1292, 1986) that the predicate provability logic of a theory essentially depends on the choice of a binumeration of the (...) theory which is used to construct the provability predicate. In this paper, we compare predicate provability logics of I∑ n ’s. For a binumeration α(x) of a recursive theory T, let PL α(T) be the predicate provability logic of T defined by α(x). We prove that for any natural numbers i, j such that 0 < i < j, there exists a ∑1 binumeration α(x) of some recursive axiomatization of I∑ i such that ${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsupseteq \bigcap_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$ and ${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsubseteq \bigcup_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$ , where β(x) ranges over all ∑1 binumerations of recursive axiomatizations of I∑ j. (shrink)

We provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan theorem is not (...) equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation. (shrink)

We investigate the Friedman–Goldfarb–Harrington theorem from two perspectives. Firstly, in the frameworks of classical and modal propositional logics, we study the forms of sentences whose existence is guaranteed by the FGH theorem. Secondly, we prove some variations of the FGH theorem with respect to Rosser provability predicates.

We introduce and investigate the notion of uniform Lyndon interpolation property which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \, \, \ and \ enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property using layered bisimulations Gödel’96, logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Springer, Berlin, 1996). Also we give a new upper bound on the complexity of uniform interpolants for \ (...) and \. (shrink)

We systematically study several principles and give a principle which is weaker than disjunctive Markov’s principle. We also show that the principle is underivable and strictly weaker than MP∨ in certain extensions of the system EL of elementary analysis.

Child poverty leads to various negative consequences, including low self-esteem, which is a risk factor for mental illness, suicide, or poor academic achievement. However, little is known about why child poverty leads to low self-esteem. We aimed to elucidate the association of child poverty and low self-esteem based on the ecological model, which includes family-level, school-level, and community-level factors. Data were obtained from the Adachi Child Health Impact of Living Difficulty (A-CHILD) study in 2016, and participants included 1,652 children in (...) fourth grade (534 pairs), sixth grade (530 pairs), and eighth grade (588 pairs) living in Adachi City, Tokyo. A questionnaire survey was implemented to assess child poverty, parental mental health, parental involvement with children, parental social capital by caregivers, and self-esteem and school social capital by children. The structural equation model was applied to elucidate the association between child poverty and low self-esteem, using family-level (parental mental health and parental involvement with children), school-level (school social capital), and community-level (parental social capital) factors. Child poverty was associated with low self-esteem. Child poverty leads to poor parental involvement, which can be indirectly associated with poor parental mental health and poor parental social capital, and poor parental involvement was directly or indirectly associated with low self-esteem through poor school social capital. To mitigate the impact of child poverty on low self-esteem, comprehensive health policies targeting family-level (parental mental health and parental involvement with children), school-level (school social capital), and community-level (parental social capital) factors may be effective. (shrink)

By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...) also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem. (shrink)