Light Affine Set Theory: A Naive Set Theory of Polynomial Time

In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL.In this paper, we consider a naive set theory based on Intuitionistic Light Affine Logic (ILAL), a simplification of LLL introduced by [1], and call it Light Affine Set Theory (LAST). The simplicity of LAST allows us to rigorously verify its polytime character. In particular, we prove that a function over {0, 1}* is computable in polynomial time if and only if it is provably total in LAST.     

Extensionality and Restriction in Naive Set Theory

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.     

The Axioms of Set Theory

In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that this concept is given to us with a certain sense as the objective focus of a ”phenomenologically reduced“ intentional experience. The concept of set that ZF describes, I claim, is that of a multiplicity of coexisting elements that can, as a consequence, be a member of another multiplicity. A set is conceived as a quantitatively determined collection of objects that is, by necessity, ontologically dependent on its elements, which, on the other hand, must exist independently of it. A close scrutiny of the essential characters of this conception seems to be sufficient to ground the set-theoretic hierarchy and the axioms of ZF. This revised version was published online in August 2006 with corrections to the Cover Date.     

幼儿朴素理论发展的研究现状

朴素物理学、朴素生物学和朴素心理学是幼儿朴素理论发展的三个核心领域,本体集合、因果原则集合和内部一致的知识体系则是朴素理论的三个重要成分。该文以幼儿对三个核心领域的因果认知发展为中心,以朴素理论三个重要组成成分为线索,对近期幼儿朴素理论发展的研究成果进行了简要的介绍和讨论,并提出进一步研究所需要解决的问题。     

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.     

The Consistency of The Naïve Theory of Properties

If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naïve set theory? Probably not, but limiting naïve comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics.     

学前儿童朴素情绪理论的发展

对情绪的心理层面的理解是幼儿心理理论发展中出现得相对较迟的能力, 他们如何通过愿望和信念等不同的心理状态对人们的情绪做出判断？随着年龄的增长, 他们采用不同策略推测情绪的能力又发生了什么变化？本研究结合主人公愿望是否得到满足的状态, 通过利用图片故事个别测查的方法探查了90名3~5岁学前儿童在“内容错误信念”任务中对他人情绪的推测。结果显示, 年幼儿童、尤其是3岁幼儿的高兴情绪理解受其对情境是否满足愿望的认知的影响, 呈现出显著的“积极情绪偏差效应”; 学前儿童对情绪的认知表现出从不能采用信念—愿望策略到错误地采用信念—愿望策略、到正确采用信念—愿望策略的发展模式, 且其基于信念的情绪理解在一定程度上呈现出落后于其错误信念理解的“滞后效应”。结果还显示幼儿的惊奇情绪判断不仅由信念成分决定, 也受愿望理解的影响。这些结果突显出心理状态在儿童情绪理解中的重要作用, 揭示出学前儿童相对完整的朴素情绪理论发展模式。     

The Open-Endedness of the Set Concept and the Semantics of Set Theory

Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after all. More generally, this article constitutes a case study in whether the need to account for conceptual progress can ever motivate a revision of semantics or logic. I end by expressing skepticism about the prospects of a so-called non-proof-based justification for this kind of revisionism about set theory.     

Aggregate Theory Versus Set Theory

Maddy's (1990) arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light of this, and the recent search for `New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for Science. Either way, the Continuum Hypothesis has no physical interest.     

Arithmetical Set Theory

It is well known that number theory can be interpreted in the usual set theories, e.g. ZF, NF and their extensions. The problem I posed for myself was to see if, conversely, a reasonably strong set theory could be interpreted in number theory. The reason I am interested in this problem is, simply, that number theory is more basic or more concrete than set theory, and hence a more concrete foundation for mathematics. A partial solution to the problem was accomplished by WTN in [2], where it was shown that a predicative set theory could be interpreted in a natural extension of pure number theory, PN, (i.e. classical first-order Peano Arithmetic). In this paper, we go a step further by showing that a reasonably strong fragment of predicative set theory can be interpreted in PN itself. We then make an attempt to show how to develop predicative fragments of mathematics in PN.If one wishes to know what is meant by reasonably strong and fragment please read on.     

Ethical Theory,Completeness &; Consistency

This paper argues that many leading ethical theories are incomplete, in that they fail to account for both right and wrong. It also argues that some leading ethical theories are inconsistent, in that they allow that an act can be both right and wrong. The paper also considers responses on behalf of the target theories.     

Ideal Objects for Set Theory

Journal of Philosophical Logic - In this paper, we argue for an instrumental form of existence, inspired by Hilbert’s method of ideal elements. As a case study, we consider the existence of...     

Consistency Theory for the General Nonparametric Classification Method

Psychometrika - Parametric likelihood estimation is the prevailing method for fitting cognitive diagnosis models—also called diagnostic classification models (DCMs). Nonparametric concepts...     

The Naive Topology of the Conscious Subject

What does our naïve conception of a conscious subject demand of the nature of conscious beings? In a series of recent papers David Barnett has argued that a range of powerful intuitions in the philosophy of mind are best explained by the hypothesis that our naïve conception imposes a requirement of mereological simplicity on the nature of conscious beings. It is argued here that there is a much more plausible explanation of the intuitions in question. Our naïve conception of a conscious subject imposes a requirement of topological integrity.