Metamorphoses of the Ideal

For Evald Il’enkov, philosophy is a science of the ideal. Il’enkov spent his entire life researching the logical and historical metamorphoses of the ideal. In general, he considered the ideal as a relation between at least two different things, one of which adequately represents the essence of another. At various times Il’enkov explored quite a few ideal phenomena: forms of value and forms of property, personality and talent, language, music and fine arts, not to mention numerous categories of dialectics, ethics, and aesthetics. The article also addresses the problem: which of these phenomena is the most typical and adequate form of being of the ideal?.     

Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered.     

Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLS_cw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLS_cw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.     

Contracting Intuitionistic Theories

I reformulate the AGM-account of contraction (which would yield an account also of revision). The reformulation involves using introduction and elimination rules for relational notions. Then I investigate the extent to which the two main methods of partial meet contraction and safe contraction can be employed for theories closed under intuitionistic consequence. I would like to thank the organisers, Heinrich Wansing, Sergei Odintsov and Yaroslav Shramko, of the Dresden Workshop on Constructive Negation, July 2–4, 2004, for providing the opportunity to present the ideas in this paper for the first time to a constructively critical audience. I am grateful to Sven Ove Hansson for useful comments on an earlier draft. A special note of thanks is owed also to Joongol Kim, who spotted a mistake in an earlier attempt of mine to prove a stronger form of Theorem 8.6. The results in this paper were presented to the Central Division Meeting of the American Philosophical Association in Chicago in April 2005.     

The Class of Extensions of Nelson's Paraconsistent Logic

The article is devoted to the systematic study of the lattice εN4_⊥ consisting of logics extending N4_⊥. The logic N4_⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4_⊥ and the lattice of superintuitionistic logics. Distinguish in εN4_⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate. The author acknowledges support by the Alexander von Humboldt-Stiftung and by Counsil for Grants under RF President, project NSh - 2112.2003.1.     

传统异化逻辑与主体性形而上学的共生性--论《1844年经济学哲学手稿》对启蒙辩证法的初始突破

主体性形而上学与传统异化逻辑的对峙是主体-客体对立的必然理论后果,二者之间的张力维持有赖于启蒙以来的时代精神也即抽象主体性的支撑。一旦抽象主体性被消解,主体神话被揭穿,异化就自然而然僭越为本体性存在。惟有马克思的历史辩证法才能走出主客二分的主体性与传统异化逻辑对立之悖论。在《1844年经济学哲学手稿》中,马克思就已经对传统异化逻辑做了初始性的突破。     

创伤后应激障碍患者的HPA轴功能变化的时间序列特征

已知创伤后应激障碍(posttraumatic stress disorder, PTSD)患者表现出HPA轴(hypothalamic-pituitary- adrenocortical axis)功能异常, 而皮质醇是反映PTSD患者HPA轴功能的重要生物标记。近期的研究结果提示, PTSD患者的皮质醇水平变化可能有明显的时间序列特征, 即应激事件发生后先升高, 后下降至正常水平之下。这一特征受到生物标记的时间特性, 患者病程, 应激源类型和强度, 伴生疾病等因素的干扰和掩蔽。未来研究应进一步采用长期追踪设计, 控制干扰因素的影响, 结合急性应激和慢性应激生物标记来综合验证PTSD患者皮质醇水平变化的时间序列特征|并利用该特征预测PTSD的发生、发展, 以便及时进行干预|考虑皮质醇的代谢、拮抗等机制, 结合多种生物标记综合评估、诊断PTSD患者的HPA轴功能活性。     

On Evans's Vague Object from Set Theoretic Viewpoint

Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects.     

Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.