On Interpreting Chaitin's Incompleteness Theorem

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory.I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.     

The Incompleteness of RGL

RGLis a version of the modal logic GLbased on the relevant logic R. It is shown that the class of RKframes that verify all theorems of RGLalso verify a scheme that we call (!). If RGLhas (!) as a theorem, however, it is not a relevant logic. I go on to show that not all instances of (!) are theorems of RGL, hence this logic is not complete over any class of RKframes.     

The Incompleteness of Ideal Theory

Can one give an account of a perfectly just society without invoking principles governing our responses to injustice? My claim is that addressing this question puts us in a position to reveal ambiguities and problems with the way in which Rawls draws the ideal/nonideal theory distinction that have so far gone unnoticed. In the first part of my paper, I demonstrate that Rawls’s original definition of the ideal/nonideal theory distinction is ambiguous as it is composed of two different conceptual distinctions, before clarifying the distinctions involved, paying particular attention to the unfamiliar distinction between primary and secondary principles. I then show that we can best account for what Rawls is actually doing at the level of ideal and nonideal theory by invoking this distinction between primary and secondary principles. This result sets the stage for my argument in the second part. I first explain why Rawls does not have access to an understanding of the strict compliance condition that can account for the irrelevance of secondary principles for a complete account of the principles regulating a perfectly just basic structure. I then point out that there is a tension between what Rawls claims to be doing at the level of ideal theory and what he is actually doing at the level of ideal theory. On this basis, I argue that Rawls’s ideal (domestic and international) conceptions of justice are incomplete because they do not encompass secondary principles. The Conclusion unpacks the contributions this article makes to the ideal/nonideal theory debate.     

Fictionalism and Incompleteness

The modal fictionalist faces a problem due to the fact that her chosen story seems to be incomplete—certain things are neither fictionally true nor fictionally false. The significance of this problem is not localized to modal fictionalism, however, since many fictionalists will face it too. By examining how the fictionalist should analyze the notion of truth according to her story, and, in particular, the role that conditionals play for the fictionalist, I develop a novel and elegant solution to the incompleteness problem.     

The Alleged Incompleteness of Public Reason

According to John Rawls's ideal of liberal public reason, comprehensive moral, religious and philosophical doctrines should play no more than an auxiliary or marginal role in the political life of constitutional democracies. David Reidy has recently claimed that since liberal public reason is incomplete, comprehensive doctrines, and non-public reasons, must play a wider role than Rawls admits. In response, I argue that Reidy's arguments do not establish that liberal public reason is incomplete. Furthermore, even if the substantive values embodied in liberal public reason were insufficient to determine certain fundamental decisions, such indeterminacy need not be eliminated by recourse to comprehensive doctrines. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Incompleteness of Theories of Games

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.     

Incompleteness and the Halting Problem

Studia Logica - We present an abstract framework in which we give simple proofs for Gödel’s First and Second Incompleteness Theorems and obtain, as consequences, Davis’,...     

Incompleteness and the Barcan formula

A (normal) system of prepositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systemswith the Barcan Formula which are incomplete, while those without are complete. In the second group it is thosewithout the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence.     

Liar-type Paradoxes and the Incompleteness Phenomena

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.     

Expressive Power and Incompleteness of Propositional Logics

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this.     

亚里士多德逻辑的现代意义

亚里士多德是逻辑的创始人,也是形而上学的开拓者,他的逻辑为西方哲学提供了一种工具和眼界,一直促进和影响西方哲学的发展.因此,理解亚里士多德逻辑,不仅有助于理解他本人的逻辑观和哲学思想,也有助于理解西方人的逻辑观念和哲学.对照亚里士多德逻辑和现代逻辑,则有助于深入地理解逻辑的本质,有助于清楚地认识西方传统哲学和现代哲学的同异,有助于揭示为什么说西方哲学的主要特征是逻辑分析的,从而有助于更加深刻地理解哲学的本质.     

Incompleteness and harm avoidance in OCD symptom dimensions

Incompleteness (INC) and harm avoidance (HA) are motivational core dimensions of OCD. While HA-related concepts (e.g., inflated responsibility, overestimation of threat) are a main focus of current cognitive-behavioural OCD research, there is also a renewed interest in INC feelings and "not just right experiences" with an inability to achieve "closure" concerning actions/perceptions. This study systematically examines the association of OCD symptom dimensions with INC and HA in a large clinical OCD sample (n=202). Hierarchical linear multiple regression analyses controlling for anxiety, depression and symptom severity demonstrated a unique association of symmetry/ordering and checking (but not of contamination/washing and obsessional thoughts) with INC, and of obsessional thoughts and checking with HA. Thus, in contrast with symmetry/ordering (predominantly INC-related) and obsessional thoughts (predominantly HA-related), checking was motivationally heterogeneous, i.e., associated with INC and HA to a comparable and substantial degree. Contamination/washing failed to show a unique association with HA in two of three analyses, and with INC in all three analyses. Symptom severity uniquely contributed to INC in two of three analyses, but not to HA. Clinically, our results indicate that a conceptualization of OCD as an anxiety disorder is too narrow.     

"仁"的多元伦理阐释

在"仁学"的创立发展过程中,形成了几种比较系统的理论倾向,主要包括:心理主义的解释倾向;形而上学方向、人与人的伦理关系阐释.这些阐释方向基本上规定了对"仁"进行现代理论构建和实践阐释的空间.     

Incompleteness and Disgust Predict Treatment Outcome in Pediatric Obsessive-Compulsive Disorder

Increasing evidence suggests that pediatric obsessive-compulsive disorder (OCD) is motivated not only by fear but also by feelings of incompleteness and disgust. However, it is currently unclear whether emotion involvement in OCD symptoms is associated with treatment response in youth with OCD. The present study examined whether treatment outcome for youth with OCD was predicted by the degree to which fear, disgust, and incompleteness were involved in baseline OCD symptoms. Children and adolescents with OCD entering treatment for this condition (N = 111) were administered standardized OCD symptom measures and an interview designed to assess the degree of fear, incompleteness, and disgust experienced during current OCD symptoms. Follow-up assessments occurred on average 13 months after baseline with each participant coded for outcome according to internationally acknowledged change criteria for pediatric OCD. Higher levels of incompleteness and disgust as part of baseline OCD symptoms predicted poorer outcome. The degree of fear during baseline OCD symptoms did not predict outcome. If replicated under controlled conditions, these results suggest that incompleteness and disgust may act as barriers to improvement in pediatric OCD and that treatment modifications that target these emotion-related motivators may improve outcome for a subset of youth.