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.     

Maybe There Are No Subject-Predicate Sentences in Chinese

In this essay, I argue for the conclusion that the Chinese sentences that are regularly translated into subject-predicate sentences in English may be understood as all non-subject-predicate sentences. My argument is based on the premise that some grammatical features are crucial to yield the sense of contrast between the completeness of subject and the incompleteness of predicate. The absence of such grammatical features in Chinese makes it impossible to establish any criterion for the distinction between subject and predicate in Chinese.     

汉语主动句、被动句的命题表征项目顺序特点

以中国大学生为被试,采用句子—图画验证任务(sentence-picture verification task )探讨了中国学生理解汉语主动句、被动句所形成的命题表征中项目顺序特点。结果表明,无论是语义水平较高的不可逆句,还是语义水平较低的可逆句,中国学生理解汉语主动句、被动句所建构的命题表征中项目顺序均为“施事→受事”。本研究结果初步表明,中国学生理解汉语主动句和被动句时,能根据它们不同的表层结构（汉语主动句的语言表达顺序为“施事→受事”、汉语被动句的语言表达顺序为“受事→施事”）建构相同的深层结构（语义内容的命题表征项目顺序“施事→受事”）,可能是一个按照“施事→受事”固定方向进行的系列认知心理加工的过程     

Modal Logics of Succession for 2-Dimensional Integral Spacetime

We consider the problem of axiomatizing various natural successor logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages, and prove completeness theorems. We also establish that the irreflexive successor logic in the standard modal language (i.e. the language containing and ) is not finitely axiomatizable.     

“Four-Valued” Semantics for the Relevant Logic R

This paper sets out two semantics for the relevant logic R based on Dunn's four-valued semantics for first-degree entailments. Unlike Routley's semantics for weak relevant logics, they do not use two ternary accessibility relations. Unlike Restall's semantics, they capture all of R. But there is a catch. Both of the present semantics are neighbourhood semantics, that is, they include sets of propositions in the specification of their frames.     

CE is Not a Conservative Extension of E

The logic CE (for Classical E) results from adding Boolean negation to Anderson and Belnap"s logic E. This paper shows that CE is not a conservative extension of E.     

Supererogation in Deontic Logic: Metatheory for DWE and Some Close Neighbours

In "Doing Well Enough: Toward a Logic for Common Sense Morality", Paul McNamara sets out a semantics for a deontic logic which contains the operator It is supererogatory that. As well as having a binary accessibility relation on worlds, that semantics contains a relative ordering relation, . For worlds u, v and w, we say that u w v when v is at least as good as u according to the standards of w. In this paper we axiomatize logics complete over three versions of the semantics. We call the strongest of these logics DWE for Doing Well Enough.     

Plural Action Sentences and Logical Form: Reply to Himmelreich

This paper replies to Himmelreich's ‘The Paraphrase Argument Against Collective Actions’ [2017], which presents three putative counterexamples to the multiple agents analysis of plural action sentences. The paper shows that the argument from the first example, the discursive dilemma, fails because it relies crucially on a simplification of the target analysis, and that the others don't bear on the question because they turn out on examination to be about individual rather than group action sentences.     

Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers p, p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most , the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel–Dummett logics with quantifiers over propositions.     

A Note on Graded Modal Logic

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.     

汉语句子的标记因素对外国留学生理解的影响

以不同标记程度汉语句子为材料,母语是英语、日语和韩语的留学生为被试,母语为汉语的大学生作参照,研究汉语句子的标记因素对留学生理解的影响。结果发现:(1)有标记结构规则的句子是比较难理解的;(2)受母语的影响,有时标记程度相同的汉语句子对不同母语类型的留学生难易有差异,甚至表现相反;(3)使用频率多可能会使有标记的句子理解起来比较容易。研究结果为在对外汉语教学中有针对性教学提供参考。     

Predicate Logics on Display

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.     

工作记忆容量的差异对第二语言句法歧义句加工的影响

采用眼动技术, 以句子阅读过程中, 句子不同区段的首次注视时间作为句子即时性加工的指标, 回扫次数和总的注视时间作为句子非即时加工的指标, 考察了非熟练的汉-英双语者, 工作记忆容量的差异对第二语言(英语)暂时句法歧义句加工的影响。结果显示: 高工作记忆容量的被试, 在句子不同区段的首次注视时间和总的注视时间均短于低工作记忆容量的被试; 高、低工作记忆容量被试的回扫模式相似。研究结果表明, 对于非熟练的汉-英双语者, 工作记忆容量的差异既影响英语句法歧义句即时性的加工, 也影响非即时性的加工, 即工作记忆容量的差异既影响英语句法歧义句初始的句法加工过程, 也影响句法的重新分析过程。研究结果在一定程度上支持工作记忆容量限制的理论。     

A Note on the Interpolation Property in Tense Logic

It is proved that all bimodal tense logics which contain the logic of the weak orderings and have unbounded depth do not have the interpolation property.     

Extended Quantum Logic

The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.     

Modal Logic,Truth, and the Master Modality

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a so-called master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic.     

Constructing Finite Least Kripke Models for Positive Logic Programs in Serial Regular Grammar Logics

A serial context-free grammar logic is a normal multimodal logicL characterized by the seriality axioms and a set of inclusionaxioms of the form _ts1..._sk. Such an inclusion axiom correspondsto the grammar rule t s₁... s_k. Thus the inclusion axioms ofL capture a context-free grammar . If for every modal index t, the set of words derivable fromt using is a regular language, then L is a serial regular grammar logic. In this paper, we present an algorithm that, given a positivemultimodal logic program P and a set of finite automata specifyinga serial regular grammar logic L, constructs a finite leastL-model of P. (A model M is less than or equal to model M' iffor every positive formula , if M then M' .) A least L-modelM of P has the property that for every positive formula , P iff M . The algorithm runs in exponential time and returnsa model with size 2^O(n³). We give examples of P and L, for bothof the case when L is fixed or P is fixed, such that every finiteleast L-model of P must have size 2⁽ⁿ⁾. We also prove that ifG is a context-free grammar and L is the serial grammar logiccorresponding to G then there exists a finite least L-modelof _s p iff the set of words derivable from s using G is a regularlanguage.     

状态不确定独立否定句的加工机制

采用自定步速阅读方法与图片核证范式(picture verification paradigm)探讨状态不确定独立否定句(如“裙子不是红色的”)的动态表征过程, 包括3个实验。结果发现：(1)在否定加工的初期, 即250 ms间隔时间条件下, 读者表征事件的否定状态; 肯定句已经通达事件的实际状态。(2)在否定加工的中期, 即750 ms间隔时间条件下, 读者仍然表征事件的否定状态, 但已开始对事件的备择选项进行搜索; 肯定句仍然保持对事件实际状态的表征。(3)在否定加工的后期, 即1500 ms间隔时间条件下, 读者表征否定标记与事件的否定状态, 同时完成对备择选项的搜索; 肯定句依旧保持对事件实际状态的表征。据此, 本文提出的锚激活与限制满足模型(anchor-based activation and satisfaction constrained model)得到初步证明。