A Critical Examination of the Historical Origins of Connexive Logic

It is often assumed that Aristotle, Boethius, Chrysippus, and other ancient logicians advocated a connexive conception of implication according to which no proposition entails, or is entailed by, its own negation. Thus Aristotle claimed that the proposition ‘if B is not great, B itself is great […] is impossible’. Similarly, Boethius maintained that two implications of the type ‘If p then r’ and ‘If p then not-r’ are incompatible. Furthermore, Chrysippus proclaimed a conditional to be ‘sound when the contradictory of its consequent is incompatible with its antecedent’, a view which, in the opinion of S. McCall, entails the aforementioned theses of Aristotle and Boethius. Now a critical examination of the historical sources shows that the ancient logicians most likely meant their theses as applicable only to ‘normal’ conditionals with antecedents which are not self-contradictory. The corresponding restrictions of Aristotle’s and Boethius’ theses to such self-consistent antecedents, however, turn out to be theorems of ordinary modal logic and thus don’t give rise to any non-classical system of genuinely connexive logic.     

Stoic Sequent Logic and Proof Theory

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.     

Embedding Friendly First-Order Paradefinite and Connexive Logics

Journal of Philosophical Logic - First-order intuitionistic and classical Nelson–Wansing and Arieli–Avron–Zamansky logics, which are regarded as paradefinite and connexive logics,...     

Variable Sharing in Connexive Logic

Journal of Philosophical Logic - However broad or vague the notion of connexivity may be, it seems to be similar to the notion of relevance even when relevance and connexive logics have been shown...     

Aristotle's thesis in consistent and inconsistent logics

A typical theorem of conaexive logics is Aristotle's Thesis(A), (AA).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In addition, while inconsistent but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made.     

A Nelsonian Response to ‘the Most Embarrassing of All Twelfth-century Arguments’

Alberic of Paris put forward an argument, ‘the most embarrassing of all twelfth-century arguments’ according to Christopher Martin, which shows that the connexive principles contradict some other logical principles that have become deeply entrenched in our most widely accepted logical theories. Building upon some of Everett Nelson’s ideas, we will show that the steps in Alberic of Paris’ argument that should be rejected are precisely the ones that presuppose the validity of schemas that are nowadays taken as some of the most trivial logical truths: (A∧B) →_AB A and (A∧B) →_AB B, i.e. Simplification.     

逻辑现代化:今天的理解与实践——王宪钧先生诞辰100周年纪念大会逻辑现代化专题研讨侧记

<正>王宪钧先生毕生致力于提高中国的逻辑教学与研究的水平,三十余年前率先提出"逻辑课程现代化"的口号,对当时的逻辑界有振聋发聩之功。此后"逻辑现代化"成为我国逻辑学发展的主旋律。时至今日,三十年的"逻辑现代化"实现或完成了什么,在新的形势下,"逻辑现代化"有什么新的内涵或重点,或简言之,我们今后该如何发展?这些问题需要逻辑界的总结与共识。为此,王宪钧先生诞辰     

Expressing Second-order Sentences in Intuitionistic Dependence Logic

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.     

Arthur Prior and medieval logic

Though Arthur Prior is now best known for his founding of modern temporal logic and hybrid logic, much of his early philosophical career was devoted to history of logic and historical logic. This interest laid the foundations for both of his ground-breaking innovations in the 1950s and 1960s. Because of the important r?le played by Prior??s research in ancient and medieval logic in his development of temporal and hybrid logic, any student of Prior, temporal logic, or hybrid logic should be familiar with the medieval logicians and their work. In this article we give an overview of Prior??s work in ancient and medieval logic.     

Special Column for 2018 National Workshop of Modern Logic(Part Ⅱ):Editorial IntroductionCSSCI

In late October last year,the workshop of national modern logic was held in Xiamen organized by Professional Committee of Modem Logic,the Logic Association of China and Philosophy department of Xiamen University.More than 100 scholars or graduate students attended the meeting.The topics of the workshop contain philosophical logic,mathematical logic,philosophy of logic,logic and language,and so on.There are several     

“当代中国逻辑学史”学术研讨会综述

2009年9月19日至20日,由中山大学逻辑与认知研究所主办、贵州大学哲学系承办的“当代中国逻辑学史”学术研讨会在贵州大学召开,来自中山大学、中国社会科学院、中国人民大学、南开大学、南京大学、浙江大学、复旦大学、华南师范大学、西南大学、贵州大学等单位10多位专家学者参加了会议。本次会议主题是研讨《当代中国逻辑学史》著作的内容。     

Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory

What has been the historical relationship between set theory and logic? On the óne hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand, set theory influenced logic by suggesting to Schröder, Löwenheim and others the use of infinitely long expressions. The question of which logic was appropriate for set theory — first-order logic, second-order logic, or an infinitary logic — culminated in a vigorous exchange between Zermelo and Gödel around 1930.     

A cirquent calculus system with clustering and ranking

Cirquent calculus is a new proof-theoretic and semantic approach introduced by G. Japaridze for the needs of his theory of computability logic (CoL). The earlier article “From formulas to cirquents in computability logic” by Japaridze generalized formulas in CoL to circuit-style structures termed cirquents. It showed that, through cirquents with what are termed clustering and ranking, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present paper syntactically constructs a cirquent calculus system with clustering and ranking, sound and complete w.r.t. the propositional fragment of cirquent-based semantics. Such a system captures the propositional version of what is called extended IF logic, thus being an axiomatization of a nontrivial fragment of that logic.     

墨家逻辑推理模式的新解释

墨家是先秦诸多学派之一,墨家逻辑也是中国古代本土逻辑思想的典范之一。墨子及其后学创立了中国思想史上第一个＂以名举实,以辞舒意,以说出故＂的墨家逻辑体系,成为中国古代逻辑思想发展的优秀代表。墨家逻辑的主要推理模式包括：＂辟＂、＂侔＂、＂援＂、＂推＂等。墨家逻辑思想的研究开启了中国逻辑思想研究的先河,墨家逻辑思想研究是中国逻辑思想研究的核心内容之一。国际逻辑学界对作为非印—欧语言系统的中国逻辑的关注,显示了中国逻辑独立存在的价值。今天的中国逻辑思想研究处于现代逻辑发展与中国现代文化发展的交汇点上,需要我们从逻辑和中国文化的角度来研究中国逻辑思想。用逻辑的一般特性来分析墨家逻辑,依据工具性、形式性和有效性这三个方面,是解释墨家逻辑的一个新角度。     

Logic and Mathematics in the Seventeenth Century

According to the received view (Bocheński, Kneale), from the end of the fourteenth to the second half of nineteenth century, logic enters a period of decadence. If one looks at this period, the richness of the topics and the complexity of the discussions that characterized medieval logic seem to belong to a completely different world: a simplified theory of the syllogism is the only surviving relic of a glorious past. Even though this negative appraisal is grounded on good reasons, it overlooks, however, a remarkable innovation that imposes itself at the beginning of the sixteenth century: the attempt to connect the two previously separated disciplines of logic and mathematics. This happens along two opposite directions: the one aiming to base mathematical proofs on traditional (Aristotelian) logic; the other attempting to reduce logic to a mathematical (algebraical) calculus. This second trend was reinforced by the claim, mainly propagated by Hobbes, that the activity of thinking was the same as that of performing an arithmetical calculus. Thus, in the period of what Bocheński characterizes as ‘classical logic’, one may find the seeds of a process which was completed by Boole and Frege and opened the door to the contemporary, mathematical form of logic.     

Johnson and the Soundness Doctrine

Why informal logic? Informal logic is a group of proposals meant to contrast with, replace, and reject formal logic, at least for the analysis and evaluation of everyday arguments. Why reject formal logic? Formal logic is criticized and claimed to be inadequate because of its commitment to the soundness doctrine. In this paper I will examine and try to respond to some of these criticisms. It is not my aim to examine every argument ever given against formal logic; I am limiting myself to those that, as a matter of historical fact, were instrumental in the replacement of formal logic by informal logic and initially established informal logic as a separate discipline (in particular, Toulmin’s attacks on what he calls the “analytic ideal” will not form part of the discussion and were not instrumental in this way, only becoming appreciated later). If the criticism of the soundness doctrine is defective, then the move from formal logic to informal logic was not theoretically well-motivated. It is this motivation that I wish to bring into question, rather than the adequacy or inadequacy of formal or informal logic as such. While I will tend to the view that formal logic is as adequate as it is reasonable to expect, the real issue is whether it is inadequate for the reasons that, as a matter of historical fact, were used to motivate its rejection.     

Skolem's Discovery of Gödel-Dummett Logic

Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.     

Dynamic Deontic Logic and its Paradoxes

In Meyer’s promising account [7] deontic logic is reduced to a dynamic logic. Meyer claims that with his account “we get rid of most (if not all) of the nasty paradoxes that have plagued traditional deontic logic.” But as was shown by van der Meyden in [4], Meyer’s logic also contains a paradoxical formula. In this paper we will show that another paradox can be proven, one which also effects Meyer’s “solution” to contrary to duty obligations and his logic in general. Presented by Hannes Leitgeb     

多值逻辑与语义赋值博弈

文章在扩展博弈上,给出了多值逻辑的语义赋值博弈的一般框架,避免了博弈者在多值逻辑的语义博弈中声明无穷对象的问题;然后通过Eloise赢的策略定义博弈的语义概念——赋值,证明了多值逻辑的博弈语义与Tarski语义是等价的;最后,根据语义赋值博弈框架对经典逻辑进行了博弈化。