Deductive schemas with uncertain premises using qualitative probability expressions

The new paradigm in the psychology of reasoning redirects the investigation of deduction conceptually and methodologically because the premises and the conclusion of the inferences are assumed to be uncertain. A probabilistic counterpart of the concept of logical validity and a method to assess whether individuals comply with it must be defined. Conceptually, we used de Finetti's coherence as a normative framework to assess individuals' performance. Methodologically, we presented inference schemas whose premises had various levels of probability that contained non-numerical expressions (e.g., “the chances are high”) and, as a control, sure levels. Depending on the inference schemas, from 60% to 80% of the participants produced coherent conclusions when the premises were uncertain. The data also show that (1) except for schemas involving conjunction, performance was consistently lower with certain than uncertain premises, (2) the rate of conjunction fallacy was consistently low (not exceeding 20%, even with sure premises), and (3) participants' interpretation of the conditional agreed with de Finetti's “conditional event” but not with the material conditional.     

Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic

We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cut-elimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.     

脑的逻辑功能的核磁共振成像及其哲学意义

国际上关于逻辑对应的脑物理特征的功能核磁共振成像（fMRI）研究的主要成就在于：建立了高级精神活动和低级神经活动的关联关系的试验方法;提出了高级精神活动的（判断和推理）神经计算模型建立方法。这些研究的实验结果表明,逻辑与神经活动具有确定性的对应关系,它证明了强人工智能的观点：导致意识的可验证的实体（或者称为“意识关联物”）就是意识本身。意识的运行方式是可计算的,思维的本质可以还原为机械性的操作步骤,人工智能和人的天然智能在本质上并无二致。     

直言推理系统

本文构造了直言推理的一个系统MA,内容包括直接推理和三段论,但未考虑单称命题。此系统实际上是名辞逻辑系统MZ的子系统,精确地说是关于类名及其否定和由它们形成的本辞的系统。文中证明了系统MA的可靠性和完全性,并提出了它的自然推理系统。     

Free-Variable Tableaux for Propositional Modal Logics

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.     

Farewell to certitude: Einstein's novelty on Induction and Deduction, Fallibilism

In the late 19th century great changes in theories of light and electricity were in direct conflict with certitude, the view that scientific knowledge is infallible. What is, then, the epistemic status of scientific theory? To resolve this issue Duhem and Poincaré proposed images of fallible knowledge, Instrumentalism and Conventionalism, respectively. Only in 1919–1922, after Einstein's relativity was published, he offered arguments to support Fallibilism, the view that certainty cannot be achieved in science. Though Einstein did not consider Duhem's Instrumentalism, he argued against Poincaré's Conventionalism. Hitherto, Einstein's Fallibilism, as presented at first in a rarely known essay of 1919, was left in the dark. Recently, Howard obscured its meaning. Einstein's essay was never translated into English. In my paper I provide its translation and attempt to shed light on Einstein's view and its context; I also direct attention to Einstein's images of philosophical opportunism in scientific practice.     

The enduring scandal of deduction

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.     

Executable specifications for hypothesis-based reasoning with Prolog and Constraint Handling Rules

Constraint Handling Rules (CHR) is an extension to Prolog which opens up a spectrum of hypothesis-based reasoning in logic programs without additional interpretation overhead. Abduction with integrity constraints is one example of hypothesis-based reasoning which can be implemented directly in Prolog and CHR with a straightforward use of available and efficiently implemented facilities.The present paper clarifies the semantic foundations for this way of doing abduction in CHR and Prolog as well as other examples of hypothesis-based reasoning that is possible, including assumptive logic programming, hypotheses with priority and scope, and nonmonotonic reasoning.Examples are presented as executable code so the paper may also serve the additional purpose of a practical guide for developing such programs, and it is demonstrated that the approach provides a seamless integration with existing constraint solvers.     

An O(n log n)-Space Decision Procedure for the Relevance Logic B+

In previous work we gave a new proof-theoretical method for establishing upper-bounds on the space complexity of the provability problem of modal and other propositional non-classical logics. Here we extend and refine these results to give an O(n log n)-space decision procedure for the basic positive relevance logic B⁺. We compute this upper-bound by first giving a sound and complete, cut-free, labelled sequent system for B⁺, and then establishing bounds on the application of the rules of this system.