This paper introduces sequent systems for Visser's two propositional logics: Basic Propositional Logic (BPL) and Formal Propositional Logic (FPL). It is shown through semantical completeness that the cut rule is admissible in each system. The relationships with Hilbert-style axiomatizations and with other sequent formulations are discussed. The cut-elimination theorems are also demonstrated by syntactical methods.

We present a version of Herbelin’s image-calculus in the call-by-name setting to study the precise correspondence between normalization and cut-elimination in classical logic. Our translation of λμ-terms into a set of terms in the calculus does not involve any administrative redexes, in particular η-expansion on μ-abstraction. The isomorphism preserves β,μ-reduction, which is simulated by a local-step cut-elimination procedure in the typed case, where the reduction system strictly follows the “ cut=redex” paradigm. We show that the underlying untyped calculus is confluent (...) and enjoys the PSN property for the isomorphic image of λμ-calculus, which in turn yields a confluent and strongly normalizing local-step cut-elimination procedure for classical logic. (shrink)

We introduce a dual-context style sequent calculus which is complete with respectto Kripke semantics where implication is interpreted as strict implication in the modal logic K. The cut-elimination theorem for this calculus is proved by a variant of Gentzen's method.

We introduce a Gentzen style formulation of Basic Propositional Calculus(BPC), the logic that is interpreted in Kripke models similarly tointuitionistic logic except that the accessibility relation of eachmodel is not necessarily reflexive. The formulation is presented as adual-context style system, in which the left hand side of a sequent isdivided into two parts. Giving an interpretation of the sequents inKripke models, we show the soundness and completeness of the system withrespect to the class of Kripke models. The cut-elimination theorem isproved (...) in a syntactic way by modifying Gentzen's method. Thisdual-context style system exemplifies the effectiveness of dual-contextformulation in formalizing various non-classical logics. (shrink)

Within a weak subsystem of second-order arithmetic , that is -conservative over , we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.

We introduce and analyze a new axiomatic theory $\mathsf {CD}$ of truth. The primitive truth predicate can be applied to sentences containing the truth predicate. The theory is thoroughly classical in the sense that $\mathsf {CD}$ is not only formulated in classical logic, but that the axiomatized notion of truth itself is classical: The truth predicate commutes with all quantifiers and connectives, and thus the theory proves that there are no truth value gaps or gluts. To avoid inconsistency, the instances (...) of the T-schema are restricted to determinate sentences. Determinateness is introduced as a further primitive predicate and axiomatized. The semantics and proof theory of $\mathsf {CD}$ are analyzed. (shrink)

専門的な予備知識は仮定せずに完全性定理や計算可能性から論じ、第一および第二不完全性定理、Rosserの定理、Hilbertのプログラム、G ̈odelの加速定理、算術の超準モデル、Kolmogorov複雑性などを紹介して、不完全性定理の数学的意義と、その根源にある哲学的問題を説く。.

明治時代、人々は狐憑きやコックリさんなどの怪現象に右往左往していた。若き哲学者の井上圓了は、それらに合理的な道筋をつけることこそが哲学普及につながると信じて奔走。柳田國男からは見地の違いから「井上圓了 君には徹頭徹尾反対」と言われながらも、開学した「哲学館」(現東洋大学)で「妖怪学」の講義を行い、日本各地で膨大な怪異談を収集した。妖怪学者であり、哲学者、宗教改革者であった隠れた偉人、初の評伝。.

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; (...) this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy. (shrink)

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as the given (...) totality. (shrink)

In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to (...) constructive investigations of variants of determinacy statements for those games in which the players make choices from only finitely many alternatives. As a result, three formulations of open determinacy, that are all classically equivalent with each other, are equivalent to three different semi-classical principles, namely Markov’s Principle, Lesser Limited Principle of Omniscience and Limited Principle of Omniscience, over a suitable constructive base theory that proves clopen determinacy. Open and clopen determinacies for these games are thus separated. Some basic results on variants of inductive definitions for \ positive operators will also be given. (shrink)

This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms of the reflection principle, (...) are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 , Theorem 107 and Theorem 108. (shrink)

Truth is often considered to be a logico-linguistic tool for expressing indirect endorsements and infinite conjunctions. In this article, I will point out another logico-linguistic function of truth: to enable and validate what I call a blind argument, namely, an argument that involves indirectly endorsed statements. Admitting this function among the logico-linguistic functions of truth has some interesting consequences. In particular, it yields a new type of so-called conservativeness argument, which poses a new type of threat to deflationism about truth.

The present paper suggests relative truth definability as a tool for comparing conceptual aspects of axiomatic theories of truth and gives an overview of recent developments of axiomatic theories of truth in the light of it. We also show several new proof-theoretic results via relative truth definability including a complete answer to the conjecture raised by Feferman in [13].

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA0, those in second order set theory extending NBG are. In this article, we establish the equivalence between Δ01-LFP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-LFP}}$$\end{document} and Δ01-FP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-FP}}$$\end{document}, which assert the existence of a least and of a fixed point, respectively, for positive elementary operators. Our proof also shows (...) the equivalence between ID1 and ID^1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\it ID}_1}$$\end{document}, both of which are defined in the standard way but with the starting theory PA replaced by ZFC. (shrink)

We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies have been extensively studied, we point out that well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting α⊢wfwhereα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${_\alpha \vdash \mathsf{wf}\, {\rm where}\, _\alpha}$$\end{document} stands for the iteration of Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} comprehension along some ordinal α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha}$$\end{document} and wf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{wf}}$$\end{document} stands for the well-foundedness of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha}$$\end{document}. (shrink)

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: fan theorem for decidable fans but arbitrary bars; continuity principle (...) and the axiom of choice both for arbitrary formulae; and $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO ; but that limited principle of omniscience LPO makes the situation completely classical. (shrink)

We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti-Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε .Analogously to the usual forcing and the usual generic (...) extension for FA-models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA-set theory can be developed in a similar way to that for FA-set theory.Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent. (shrink)

We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for \ formulae are provable in \. It is shown to be larger than the proof-theoretic ordinal \ by power of base 2. (...) We also show a similar result for the structural transfinite induction, defined with fundamental sequences. (shrink)

The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over (...) subject matters ‘richer’ than arithmetic, such as set theory. The move to this new formal setting provides deflationists with better defence and brings a broader perspective to the debate. (shrink)

Recently, Edwards and Liaschenko questioned the validity of an argument put forward by Dr Søren Holm and Joseph Dunne concerning the impossibility of a theory of nursing. Taking into consideration the premises of the argument, I describe how Maritain's conception of philosophy allows for the possibility of a theory of nursing conceived as a philosophy of nursing art that is both practical and propositional in nature. As well, I identify how the philosophy of nursing art guides nursing art in developing (...) and applying other kinds of artistic practical nursing knowledge: artistic nursing rules and artistic prudential nursing judgements. Finally, the relationship of ethical knowledge to artistic practical nursing knowledge is set down. (shrink)

Although dual-process or divided-mind models of self-control dominate the literature, they suffer from empirical and conceptual challenges. We propose an alternative approach, suggesting that self-control can be characterized by a fragmented part versus integrated whole dynamic. Whereas responses to events derived from fragmented parts of the mind undermine self-control, responses to events derived from integrated wholes enhance self-control. We review empirical evidence from psychology and related disciplines that support this model. We, moreover, discuss the implications of this work for psychology, (...) neuroscience, economics, and philosophy. In particular, we highlight how this model addresses many of the conceptual and empirical short-comings of divided-mind models. We suggest that understanding self-control as the interplay between fragmented parts versus integrated wholes, moreover, provides novel insights and testable new hypotheses. (shrink)

Recently, a concern has been raised about a particular kind of behaviour that is adversely affecting the quality of nursing theoretical discourse. With the behaviour being attributed to nurses’ tendency to think in binary terms, it has been proposed that nurses replace their binary way of thinking with thinking that is inclusive and expansive and is based on an epistemology of contradiction. While agreeing that the behaviour of concern is indeed unscholarly, I disagree that the culprit is the binary. In (...) this commentary, I argue that the culprit is sophistical argumentation involving a binary (real or not) and that, by not making room for legitimate binaries, we run the danger of ‘throwing out the baby with the bathwater’. Further, I argue that the proffered solution is illusory. Its epistemological basis prohibits the deliverance of that which is promised. (shrink)

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

Classes are the objects of the second sort of second-order set theory. They have sets as their members and behave like sets, but paradoxes tell us that many classes cannot be sets. Then, what are classes? Predicativism about classes suggests that classes are predicates of sets, and this article investigates the question from the predicativist point of view in light of recent developments in the use of classes in set theory. Predicativism has been considered too restrictive and unable to accommodate (...) the use of classes in set theory. This diagnosis, however, is only true of a certain specific type of predicativism. In this article, we propose a new type of predicativism, which we call liberal predicativism, and argue that predicativism is still a highly viable option, and our liberal version provides a sufficiently versatile and workable nominalist concept of classes for set theory. (shrink)

We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg . We first introduce a weak weak set theory as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals:1. and ,2. and . We also show (...) that neither Reg nor affects the proof-theoretic strength, i.e., where T is Basic plus any combination of Ext and. (shrink)

We study filters in residuated structures that are associated with congruence relations (which we call -filters), and develop a semantical theory for general substructural logics based on the notion of primeness for those filters. We first generalize Stone’s sheaf representation theorem to general substructural logics and then define the primeness of -filters as being “points” (or stalkers) of the space, the spectrum, on which the representing sheaf is defined. Prime FL-filters will turn out to coincide with truth sets under various (...) well known semantics for certain substructural logics. We also investigate which structural rules are needed to interpret each connective in terms of prime -filters in the same way as in Kripke or Routley-Meyer semantics. We may consider that the set of the structural rules that each connective needs in this sense reflects the difficulty of giving the meaning of the connective. A surprising discovery is that connectives , ⅋ of linear logic are linearly ordered in terms of the difficulty in this sense. (shrink)

By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 234–268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse (...) mathematics–Friedman–Simpson’s “orthodox” reverse mathematics, Cook’s bounded reverse mathematics and large cardinal theory–can be reformulated within one language so that we can compare them more directly. (shrink)

Distracted attention is considered responsible for most car accidents, and many functional magnetic resonance imaging researchers have addressed its neural correlates using a car-driving simulator. Previous studies, however, have not directly addressed safe driving performance and did not place pedestrians in the simulator environment. In this fMRI study, we simulated a pedestrian-rich environment to explore the neural correlates of three types of safe driving performance: accurate lane-keeping during driving, the braking response to a preceding car, and the braking response to (...) a crossing pedestrian. Activation of the bilateral frontoparietal control network predicted high driving accuracy. On the other hand, activation of the left posterior and right anterior superior temporal sulci preceding a sudden pedestrian crossing predicted a slow braking response. The results suggest the involvement of different cognitive processes in different components of driving safety: the facilitatory effect of maintained attention on driving accuracy and the distracting effect of social–cognitive processes on the braking response to pedestrians. (shrink)

Prediction is essential for the efficiency of many cognitive processes; however, this process is not always perfect. Predictive coding theory suggests that the brain generates and updates a prediction to respond to an upcoming event. Although an electrophysiological index of prediction, the stimulus preceding negativity, has been reported, it remains unknown whether the SPN reflects the prediction accuracy, or whether it is associated with the prediction error, which corresponds to a mismatch between a prediction and an actual input. Thus, the (...) present study aimed to investigate this question using electroencephalography. Participants were asked to predict the original pictures from pictures that had undergone different levels of pixelation. The SPN amplitude was affected by the level of pixelation and correlated with the subjective evaluation of the prediction accuracy. Furthermore, late positive components were negatively correlated with SPN. These results suggest that the amplitude of SPN reflects the prediction accuracy; more accurate prediction increases the SPN and reduces the prediction error, resulting in reduced LPC amplitudes. (shrink)

We investigate the role of coalgebraic predicate logic, a logic for neighborhood frames first proposed by Chang, in the study of monotonic modal logics. We prove analogues of the Goldblatt–Thomason theorem and Fine’s canonicity theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic. The elementary equivalence here can be relativized to the classes of monotonic, quasi-filter, augmented quasi-filter, filter, or augmented filter neighborhood frames, respectively. The original, Kripke-semantic versions of the theorems follow as a (...) special case concerning the classes of augmented filter neighborhood frames. (shrink)

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $, of Kripke–Feferman truth (...) theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators. (shrink)

This paper studies several systems of the transfinite iteration and autonomous progression of self-applicable truth and determines their proof-theoretic strength.