Naming and Diagonalization, from Cantor to Godel to Kleene

We trace self-reference phenomena to the possibility of namingfunctions by names that belong to the domain over which thefunctions are defined. A naming system is a structure of theform (D, type( ),{ }), where D is a non-empty set; for everya D, which is a name of a k-ary function, {a}: D^k D is thefunction named by a, and type(a) is the type of a, which tellsus if a is a name and, if it is, the arity of the named function.Under quite general conditions we get a fixed point theorem,whose special cases include the fixed point theorem underlyingGödel's proof, Kleene's recursion theorem and many othertheorems of this nature, including the solution to simultaneousfixed point equations. Partial functions are accommodated byincluding "undefined" values; we investigate different systemsarising out of different ways of dealing with them. Many-sortednaming systems are suggested as a natural approach to generalcomputatability with many data types over arbitrary structures.The first part of the paper is a historical reconstruction ofthe way Gödel probably derived his proof from Cantor'sdiagonalization, through the semantic version of Richard. Theincompleteness proof–including the fixed point construction–resultfrom a natural line of thought, thereby dispelling the appearanceof a "magic trick". The analysis goes on to show how Kleene'srecursion theorem is obtained along the same lines.     

周敦颐的《太极图说》、《易》象数及西方有关学说

本文从《易》象数的角度来解释周敦颐的《太极图说》与《通书》,认为这种象数的“二对生”的区别性特征是周氏学说的内在发动结构,由此而提出一些很不同于流行的“宇宙论加上伦理学”的解释模式的思路。在一些被长久争论的问题上,比如“太极与无极的关系”、“太极与两仪的关系”、“诚与太极的关系”、“中与太极和诚的关系”,都提出了新的看法。在这新的理解方式中,周敦颐的《太极图说》就不止是程朱陆王的一个含糊的先导,而是融合了儒道释的思想精华,而又有着自己独特的思想原发力与中和的精微境界的划时代的哲理学说,其蕴义(比如超出理气二分的原本发生论)并未被后来的理学家们穷尽。本文最后将周敦颐的思想与西方的毕达哥拉斯及莱布尼兹的学说进行对比,一方面揭示“象数”在伟大的哲理传统中的关键作用,另一方面又表明中西两大传统之间存在的深刻差异在结构上的原因。     

Implicit diagnosticity in an information-buying task. How do we use the information that we bring with us to a problem?

Past research suggests that people may make use of diagnosticity information when explicit data regarding P(D | H) and P(D | ? H) are given to them. However, people fall victim to pseudodiagnosticity biases and ignore P(D | ? H) when such data must be actively sought. This series of four experiments utilized judgment problems in which subjects have knowledge of P(D | ? H) but must recognize the relevance of that knowledge for the judgment at hand. It was hypothesized that subjects who genuinely understood the role of P(D | ? H) in hypothesis testing would respond to this manipulation of implicit diagnosticity by exhibiting greater confidence and lesser information buying when given evidence of relatively high diagnosticity. In the first three studies, subjects attempted to judge the guilt or innocence of suspects in several fictional crimes. In the fourth experiment, subjects attempted to judge the club membership status of students at their own university. Greater amounts of information were bought when the only available information was of low diagnosticity. Subjects also expressed greater confidence in judgments made using highly diagnostic information. However, within the legal scenario, sensitivity to diagnosticity was dependent upon the implication of the cues received. Results are discussed with respect to Bayesian probability, expected value theory, and a confidence criterion model of information purchasing.     

Sharvy’s Lucy and Benjamin Puzzle

Sharvy’s puzzle concerns a situation in which common knowledge of two parties is obtained by repeated observation each of the other, no fixed point being reached in finite time. Can a fixed point be reached?     

Common knowledge of payoff uncertainty in games

Using epistemic logic, we provide a non-probabilistic way to formalise payoff uncertainty, that is, statements such as ‘player i has approximate knowledge about the utility functions of player j.’ We show that on the basis of this formalisation common knowledge of payoff uncertainty and rationality (in the sense of excluding weakly dominated strategies, due to Dekel and Fudenberg (1990)) characterises a new solution concept we have called ‘mixed iterated strict weak dominance.’     

Democratic Answers to Complex Questions – An Epistemic Perspective

This paper addresses a problem for theories of epistemic democracy. In a decision on a complex issue which can be decomposed into several parts, a collective can use different voting procedures: Either its members vote on each sub-question and the answers that gain majority support are used as premises for the conclusion on the main issue (premise based-procedure, pbp), or the vote is conducted on the main issue itself (conclusion-based procedure, cbp). The two procedures can lead to different results. We investigate which of these procedures is better as a truth-tracker, assuming that there exists a true answer to be reached. On the basis of the Condorcet jury theorem, we show that the pbp is universally superior if the objective is to reach truth for the right reasons. If one instead is after truth for whatever reasons, right or wrong, there will be cases in which the cbp is more reliable, even though, for the most part, the pbp still is to be preferred.     

Algebraization, Parametrized Local Deduction Theorem and Interpolation for Substructural Logics over FL

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices. Dedicated to the memory of Willem Johannes Blok     

Loosely Guarded Fragment of First-Order Logic has the Finite Model Property

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.