A contemporary interpretation of Marx’s thoughts on modernity

Unlike some western scholars who limit their interpretation of modernity and its source to conceptual, cultural, value, and psychological dimensions, Marx pointed out that modernity came mainly from modern production system. Starting from the historical context of his time, Marx explored various aspects of modernity and pointed out that modernity was inherent in the logic of capital, resided in the process of historical evolution, arose in social conflicts and segmentation, and presented itself in a global horizon. The logic of capital, the historical viewpoint, the theory of contradiction and a global perspective are fundamental in Marx’s analysis of the problems of modernity. Marx’s ideas of modernity are methodologically significant to the construction of modernity in contemporary Chinese society. __________ Translated from Zhongguo Shehui Kexue 中国社会科学 (China Social Sciences), 2005 (4) by Xing Lijun     

Kant on Complete Determination and Infinite Judgement

In the Transcendental Ideal Kant discusses the principle of complete determination: for every object and every predicate A, the object is either determinately A or not-A. He claims this principle is synthetic, but it appears to follow from the principle of excluded middle, which is analytic. He also makes a puzzling claim in support of its syntheticity: that it represents individual objects as deriving their possibility from the whole of possibility. This raises a puzzle about why Kant regarded it as synthetic, and what his explanatory claim means. I argue that the principle of complete determination does not follow from the principle of excluded middle because the externally negated or ‘negative’ judgement ‘Not (S is P)’ does not entail the internally negated or ‘infinite’ judgement ‘S is not-P.’ Kant's puzzling explanatory claim means that empirical objects are determined by the content of the totality of experience. This entails that empirical objects are completely determinate if and only if the totality of experience has a completely determinate content. I argue that it is not a priori whether experience has such a completely determinate content and thus not analytic that objects obey the principle of complete determination.     

Varieties of Reflection in Kant's Logic

For Kant, ‘reflection’ (Überlegung, Reflexion) is a technical term with a range of senses. I focus here on the senses of reflection that come to light in Kant's account of logic, and then bring the results to bear on the distinction between ‘logical’ and ‘transcendental’ reflection that surfaces in the Amphiboly chapter of the Critique of Pure Reason. Although recent commentary has followed similar cues, I suggest that it labours under a blind spot, as it neglects Kant's distinction between ‘pure’ and ‘applied’ general logic. The foundational text of existing interpretations is a passage in Logik Jäsche that appears to attribute to Kant the view that reflection is a mental operation involved in the generation of concepts from non-conceptual materials. I argue against the received view by attending to Kant's division between ‘pure’ and ‘applied’ general logic, identifying senses of reflection proper to each, and showing that none accords well with the received view. Finally, to take account of Kant's notion of transcendental reflection I show that we need to be attentive to the concerns of applied logic and how they inform the domain-relative transcendental logic that Kant presents in the first Critique.     

Louis Couturat,modern logic,and the international auxiliary language

ABSTRACT

To some extent, the early twentieth century revival of universal languages was the work of logicians and mathematicians. Pioneers of modern logic such as Frege, Russell and Peano wanted to overcome the diversity and deficiencies of natural languages. Through the rigour of formal logic, they aimed at providing scientific thinking with a reliable medium free from the ambiguity and inconsistencies of ordinary language. This article shows some interconnections between modern logic and the search for a common tongue that would unite scientists and people of all nations. The French mathematician and philosopher Louis Couturat is a key figure in understanding the interplay between these two movements. Through his work in composing the Ido language as an alternative to Esperanto, Couturat gave a new life to the Leibnizian idea of a universal characteristics. In addition, his multifaceted work provides a valuable insight into some political implications of early analytic philosophy.     

Possibilities,models, and intuitionistic logic: Ian Rumfitt’s The boundary stones of thought

AIan Rumfitt's new book presents a distinctive and intriguing philosophy of logic, one that ultimately settles on classical logic as the uniquely correct one–or at least rebuts some prominent arguments against classical logic. The purpose of this note is to evaluate Rumfitt's perspective by focusing on some themes that have occupied me for some time: (i) the role and importance of model theory and, in particular, the place of counter-arguments in establishing invalidity, (ii) higher-order logic, and (iii) the logical pluralism/relativism articulated in my own recent *Varieties of logic*.     

Mathematical concepts for analyzing ambiguous situations

Abstract

We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.     

A General Setting for Dedekind's Axiomatization of the Positive Integers

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ?₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable.     

Future Contradictions

A common and much-explored thought is ?ukasiewicz's idea that the future is ‘indeterminate’—i.e., ‘gappy’ with respect to some claims—and that such indeterminacy bleeds back into the present in the form of gappy ‘future contingent’ claims. What is uncommon, and to my knowledge unexplored, is the dual idea of an overdeterminate future—one which is ‘glutty’ with respect to some claims. While the direct dual, with future gluts bleeding back into the present, is worth noting, my central aim is simply to sketch and briefly explore an alternative glutty-future view, one that is conservative—indeed, entirely classical—with respect to the present.

The structure of the paper runs as follows. §1 briefly sketches the target gap picture of an indeterminate future yielding gappy claims at the present. §2 presents the direct dual idea—a glut picture of an overdeterminate future yielding glutty claims at present. §3 sketches the central idea, a more interesting glut picture in which the future contains contradictory states but the present remains entirely classical. §4 contains a general defence of the idea, leaving it open as to whether the gappy-future view enjoys substantive virtues over the proposed glutty-future view of §3.     

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.