Modal Deduction in Second-Order Logic and Set Theory - II

In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor such a translation to work with specific cases of extended modal logics.     

Abstract argumentation and explanation applied to scientific debates

Abstract argumentation has been shown to be a powerful tool within many fields such as artificial intelligence, logic and legal reasoning. In this paper we enhance Dung’s well-known abstract argumentation framework with explanatory capabilities. We show that an explanatory argumentation framework (EAF) obtained in this way is a useful tool for the modeling of scientific debates. On the one hand, EAFs allow for the representation of explanatory and justificatory arguments constituting rivaling scientific views. On the other hand, different procedures for selecting arguments, corresponding to different methodological and epistemic requirements of theory evaluation, can be formulated in view of our framework.     

A General Framework for $$ {FDE}$$ FDE -Based Modal Logics

We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the literature can be expressed in our framework.

     

A neural cognitive model of argumentation with application to legal inference and decision making

Formal models of argumentation have been investigated in several areas, from multi-agent systems and artificial intelligence (AI) to decision making, philosophy and law. In artificial intelligence, logic-based models have been the standard for the representation of argumentative reasoning. More recently, the standard logic-based models have been shown equivalent to standard connectionist models. This has created a new line of research where (i) neural networks can be used as a parallel computational model for argumentation and (ii) neural networks can be used to combine argumentation, quantitative reasoning and statistical learning. At the same time, non-standard logic models of argumentation started to emerge. In this paper, we propose a connectionist cognitive model of argumentation that accounts for both standard and non-standard forms of argumentation. The model is shown to be an adequate framework for dealing with standard and non-standard argumentation, including joint-attacks, argument support, ordered attacks, disjunctive attacks, meta-level attacks, self-defeating attacks, argument accrual and uncertainty. We show that the neural cognitive approach offers an adequate way of modelling all of these different aspects of argumentation. We have applied the framework to the modelling of a public prosecution charging decision as part of a real legal decision making case study containing many of the above aspects of argumentation. The results show that the model can be a useful tool in the analysis of legal decision making, including the analysis of what-if questions and the analysis of alternative conclusions. The approach opens up two new perspectives in the short-term: the use of neural networks for computing prevailing arguments efficiently through the propagation in parallel of neuronal activations, and the use of the same networks to evolve the structure of the argumentation network through learning (e.g. to learn the strength of arguments from data).     

Fibring Argumentation Frames

This paper is part of a research program centered around argumentation networks and offering several research directions for argumentation networks, with a view of using such networks for integrating logics and network reasoning. In Section 1 we introduce our program manifesto. In Section 2 we motivate and show how to substitute one argumentation network as a node in another argumentation network. Substitution is a purely logical operation and doing it for networks, besides developing their theory further, also helps us see how to bring logic and networks closer together. Section 3 develops the formal properties of the new kind of network and Section 4 offers general discussion and comparison with the literature.     

Coinductive models and normal forms for modal logics (or how we learned to stop worrying and love coinduction)

We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language—but whose proofs were known to be mere routine—now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem.We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms—e.g., the conjunctive normal form of propositional logic—the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.     

广义论证视域下朱熹论辩逻辑体系的考察

近年来,广义论证视域下的中国古代逻辑史研究成为学界关注的热点,在丰硕的研究成果中,仍存在“怎样深化”与“如何拓展”等问题。在广义论证视域下对朱熹论辩逻辑体系进行考察,一方面可以打破以往研究主要集中在先秦时期的囿限,进一步拓展其研究范围;另一方面也可以在逻辑体系的构建中深化中国古代逻辑研究。从广义论证角度看,朱熹的逻辑是“名义界分”“知言穷理”等说理论证规则的集合,基于上述逻辑,朱熹同陆九渊、张栻等人围绕“太极”“仁说”等进行论辩实践,逐步建立起以名、辞理论为基础的“名、辞–辩”的逻辑体系。上述体系对后世影响深远。从广义论证角度考察朱熹论辩逻辑体系,对新时期回应如何做“中国古代逻辑”具有启发意义。     

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。     

奎因对模态逻辑的批评

奎因对模态逻辑的批评包括：(i)因为模态语境是指称不透明的,所以对其量化纳入没有意义;(ii)一些关于必然真理的经验知识与关于必然性的语言学理论不相容;(iii)模态语境中的量化纳入承诺了“亚里士多德式的本质主义”。丘奇,克里普克和福雷斯达尔对奎因提出了批评。他们提出,模态构造应该是外延上不透明但指称上透明的。奎因式诉诸于模态环境指称不透明的批评,在不采用单称词项的模态逻辑中不成立。他对本质主义的批评引入了一种必然性,但这种必然性没有办法用语言陈述词项进行解释。通过对语言中名称使用的讨论,我们可以看出,有关包含名称的同一性陈述的真值的经验知识,和卡苏洛称之为“一般模态地位”的先验知识是相容的。既然奎因论证了真同一性陈述的必然真是逻辑真,而且所有逻辑真都遵循戴维森式的意义理论（另文讨论）,那么我们可以给出一个关于包含名称的同一性的真陈述的模态地位的先验知识的语言学理论。     

Thick,Thin, and Becoming a Virtuous Arguer

A virtue account is focused on the character of those who argue. It is frequently assumed, however, that virtues are not action guiding, since they describe how to be and so fail to give us specific actions to take in a sticky situation. In terms of argumentation, we might say that being a charitable arguer is virtuous, but knowing so provides no details about how to argue successfully. To close this gap, I develop a parallel with the thick-thin distinction from ethics and use Hursthouse’s notion of “v-rules”. I also draw heavily from the work in argumentation by Daniel Cohen to develop Wayne Brockriede’s notion of arguing lovingly. But “argue lovingly” has a delicious ambiguity. For Brockriede it describes how we engage with others arguers. It can also mean, however, a loving attachment to knowledge, understanding, and truth. Applying the thick-thin distinction to argumentation in general and loving argumentation in particular shows that a virtue theoretic approach to argumentation is valuable for two reasons: it can provide one articulation of what it means to be a virtuous arguer and provide some insights into how to become one.     

Liar-type Paradoxes and the Incompleteness Phenomena

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.     

A general tableau method for propositional interval temporal logics: Theory and implementation

In this paper, we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema's CDT logic interpreted over partial orders (BCDT⁺ for short). It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented.     

Vagueness-Adaptive Logic: A Pragmatical Approach to Sorites Paradoxes

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley-Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.     

走向模型论的模态逻辑

增加特定的基数量词,扩张一阶语言,就可以导致实质性地增强语言的表达能力,这样许多超出一阶逻辑范围的数学概念就能得到处理。由于在模型的层次上基本模态逻辑可以看作一阶逻辑的互模拟不变片断,显然它不能处理这些数学概念。因此,增加说明后继状态类上基数概念的模态词,原则上我们就能以模态的方式处理所有基数。我们把讨论各种模型论逻辑的方式转移到模态方面。     

Argument Structure and Disciplinary Perspective

Many in the informal logic tradition distinguish convergent from linked argument structure. The pragma-dialectical tradition distinguishes multiple from co-ordinatively compound argumentation. Although these two distinctions may appear to coincide, constituting only a terminological difference, we argue that they are distinct, indeed expressing different disciplinary perspectives on argumentation. From a logical point of view, where the primary evaluative issue concerns sufficient strength of support, the unit of analysis is the individual argument, the particular premises put forward to support a given conclusion. Structure is internal to this unit. From a dialectical point of view, where the focus concerns how well a critical discussion comes to a reasoned conclusion of some disputed question, the argumentation need not constitute a single unit of argument. The unit of dialectical analysis will be the entire argumentation made up of these several arguments. The multiple/co-ordinatively compound distinction is dialectical, while the linked/convergent distinction is logical. Keeping these two pairs of distinctions separate allows us to see certain attempts to characterize convergent versus linked arguments as rather characterizing multiple versus co-ordinatively compound arguments, in particular attempts of Thomas, Nolt, and Yanal, and to resolve straightforwardly conflicts, tensions, or anomalies in their accounts. Walton's preferred Suspension/Insufficient Proof test to identify linked argument structure correctly identifies co-ordinatively compound structure. His objection to using the concept of relevance to explicate the distinction between linked and convergent structure within co-ordinatively compound argumentation can be met through explicating relevance in terms of inference licenses. His counterexample to the Suspension/No Support test for identifying linked structure which this approach supports can itself be straightforwardly dealt with when the test is explicated through inference licenses.     

Representation of Argumentation in Text with Rhetorical Structure Theory

Various argumentation analysis tools permit the analyst to represent functional components of an argument (e.g., data, claim, warrant, backing), how arguments are composed of subarguments and defenses against potential counterarguments, and argumentation schemes. In order to facilitate a study of argument presentation in a biomedical corpus, we have developed a hybrid scheme that enables an analyst to encode argumentation analysis within the framework of Rhetorical Structure Theory (RST), which can be used to represent the discourse structure of a text. This paper describes the hybrid representation scheme and illustrates its use for investigation of contexts that license omission of elements of an argument. The analyses given in the paper involve reconstruction of enthymemes. Defeasible argumentation schemes serve as a constraint on reconstruction. In addition, the examples illustrate several other types of contextual constraints on reconstruction of enthymemes.     

论证的基于标记的论据状态(英文)

本文针对抽象论证体系中的论据状态给出了一种基于标记的定义方法。与基于扩张的传统方法相比,此定义能够更加细致准确地描述论据状态。同时,本文对论据状态的等级进行了定义和区分,并把该等级作为接受或驳斥论据的依据。由于该方法基于标准论证体系定义,在实现上使用了已有的基于论证的证明方法,因此,该定义与以往的抽象论证体系是完全兼容的。     

Abstract argument games via modal logic

Inspired by some logical considerations, the paper proposes a novel perspective on the use of two-players zero-sum games in abstract argumentation. The paper first introduces a second-order modal logic, within which all main Dung-style semantics are shown to be formalizable, and then studies the model checking game of this logic. The model checking game is then used to provide a systematic game theoretic proof procedure to test membership with respect to all those semantics formalizable in the logic. The paper discusses this idea in detail and illustrates it by providing a game for the so-called skeptical preferred and skeptical semi-stable semantics.     

Psychological laws and non-monotonic logic

In this essay I enter into a recently published debate between Stephen Schiffer and Jerry Fodor concerning whether adequate sense can be made of the ceteris paribus conditions in special science laws, much of their focus being on the case of putative psychological laws. Schiffer argues that adequate sense cannot be made of ceteris paribus clauses, while Fodor attempts to overcome Schiffer's arguments, in defense of special science laws. More recently, Peter Mott has attempted to show that Fodor's response to Schiffer fails, and furthermore that further study shows that the logical framework in which Schiffer and Fodor address their issue is susceptible to inconsistency.In this essay I argue that adequate sense can be made of ceteris paribus conditions. Against Mott, I argue that recent work in the model theory of non-monotonic logic indicates how his problem involving logical inconsistencies can be overcome. Against Schiffer, I argue that the claims that he makes against ceteris paribus clauses would lead to a fatal skepticism concerning indefinitely many of the claims we make about the world (and indeed that his claims would be destructive of the view of the special sciences that Schiffer himself presents in his paper), and that the semantical considerations from non-monotonic logic that I present provide a suitable framework for dealing with his complaints. Thus I come out on the whole on Fodor's side of this debate, although for my own reasons, as I argue against much of Fodor's own argumentation.