Supporting the formal verification of mathematical texts

The formal verification of mathematical texts is one of the most interesting applications for computer systems. In fact, we argue that the expert language of mathematics is the natural choice for achieving efficient mathematician–machine interaction. Our empirical approach, the analysis of carefully authored textbook proofs, forces us to focus on the language and the reasoning pattern that mathematician use when presenting proofs to colleagues and students. Enabling a machine to understand and follow such language and argumentation is seen to be the key to usable and acceptable math assistant systems. In this paper, we first perform an analysis of three textbook proofs by hand; we then describe a computational framework that aims at mechanising such an analysis. The resulting proof-of-concept implementation is capable of processing simple textbook proofs and constitutes promising steps towards a natural mathematician–machine interface for proof development and verification.     

幼儿反事实推理的错误源分析

本研究选取57名3~5岁幼儿,考察了在两种选择和三种选择的反事实推理任务中儿童的表现以及是否出现现实错误。结果如下:无论在两种还是三种选择的反事实任务中,3岁组儿童的得分显著低于4岁和5岁组儿童,而4岁和5岁组儿童之间不存在显著差异;幼儿在反事实推理中的确会出现现实错误,但除此之外,幼儿在三种选择任务中也会选择现实和正确答案之外的第三个选项。由此可见,年幼儿童可能在抑制现实和建构反事实替代情境方面都存在困难,这尚需我们进一步探究。     

What in the world are the ways things might have been?

Robert Stalnaker is an actualist who holds that merely possible worlds are uninstantiated properties that might have been instantiated. Stalnaker also holds that there are no metaphysically impossible worlds: uninstantiated properties that couldn't have been instantiated. These views motivate Stalnaker's "two dimensional" account of the necessary a posteriori on which there is no single proposition that is both necessary and a posteriori. For a (metaphysically) necessary proposition is true in all (metaphysically) possible worlds. If there were necessary a posteriori propositions, that would mean that there were propositions true in all possible worlds but which could only be known to be true by acquiring empirical evidence. Consider such a purported proposition P. The role of empirical evidence for establishing P's truth would have to be to rule out worlds in which P is false. If there were no such worlds to be ruled out, we would not require evidence for P. But by hypothesis, P is necessary and so true in all metaphysically possible worlds. And on Stalnaker's view, the metaphysically possible worlds are all the worlds there are. So there can be no proposition that is true in all possible worlds, but that we require evidence to know. In this way, the motivation for Stalnaker's two dimensional account of the necessary a posteriori rests on his denying that there are metaphysically impossible Worlds. I argue that given his view of what possible worlds are, Stalnaker has no principled reason for denying that there are metaphysically impossible worlds. If I am right, this undercuts Stalnaker's motivation for his two dimensional account of the necessary a posteriori.

Distortion of probability and outcome information in risky decisions

Substantial evidence indicates that information is distorted during decision making, but very few studies have assessed the distortion of probability and outcome information in risky decisions. In two studies involving six binary decisions (e.g., banning blood donations from people who have visited England, because of “mad cow disease”), student and nonstudent participants distorted their evaluations of probability and outcome information in the direction of their preferred decision alternative and used these biased evaluations to update their preferences. Participants also evaluated the utilities of possible outcomes more positively when the outcomes could follow only from the preferred alternative and more negatively when they could follow only from the competing alternative. Such circular reasoning is antithetical to the normative consequentialist principles underlying decision analysis. Presenting numerical information as precise values or as ranges of values did not significantly affect information distortion, apparently because the manipulation did not affect perceived ambiguity as intended.     

A causal model theory of the meaning of cause, enable, and prevent

The verbs cause , enable , and prevent express beliefs about the way the world works. We offer a theory of their meaning in terms of the structure of those beliefs expressed using qualitative properties of causal models, a graphical framework for representing causal structure. We propose that these verbs refer to a causal model relevant to a discourse and that "A causes B" expresses the belief that the causal model includes a link from A to B. "A enables/allows B" entails that the model includes a link from A to B, that A represents a category of events necessary for B, and that an alternative cause of B exists. "A prevents B" entails that the model includes a link from A to B and that A reduces the likelihood of B. This theory is able to account for the results of four experiments as well as a variety of existing data on human reasoning.     

The development of possibility judgment within and across domains

The ability to differentiate possible events from impossible ones is an invaluable skill when reasoning about claims that transcend the perceptual evidence at hand, yet preschool-aged children do not readily make this differentiation when reasoning about physically extraordinary events [Shtulman, A., &; Carey, S. (2007). Improbable or impossible? How children reason about the possibility of extraordinary claims. Child Development, 78, 1015–1032]. The present study sought to determine whether this failure stems from deficits in domain-specific knowledge or deficits in the domain-general procedure by which possibility judgments are made. Participants (48 children aged 4-9 years olds and 16 adults) were asked to judge the possibility of three types of extraordinary events – physical, psychological, and biological – and to justify their judgments of impossibility. Developmental changes in the ability to differentiate improbable events from impossible events were qualitatively similar across domains. Moreover, participants’ propensity to judge improbable events possible was significantly correlated with the quality of their justifications, both within and across domains. These findings suggest that modal development involves a domain-general change in how modal claims are evaluated.     

Evaluating Dialectical Structures

This paper develops concepts and procedures for the evaluation of complex debates. They provide means for answering such questions as whether a thesis has to be considered as proven or disproven in a debate or who carries a burden of proof. While being based on classical logic, this framework represents an (argument-based) approach to non-monotonic, or defeasible reasoning. Debates are analysed as dialectical structures, i.e. argumentation systems with an attack- as well as a support-relationship. The recursive status assignment over the arguments is conditionalised on proponents in a debate. The problem of multiple status assignments arising on circular structures is solved by showing that uniqueness can be guaranteed qua reconstruction of a debate. The notion of burden of proof as well as other discursive aims rational proponents pursue in a debate is defined within the framework.     

Inexact Knowledge with Introspection

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.     

基于非逻辑机制的条件推理模式：P—Q映射模型及其实证检验

研究结合数学分析方法,提出了基于非逻辑机制的条件推理模型：P-Q映射模型。并根据这个模型,对人们在不同命题类型奈件下的推理行为进行了预测。预测结果显示,当推理前提为LH和HL型命题时,基于P-Q映射模型的预测结果与基于条件概率模型的预测结果完全一致。但当推理前提为LL和HH型命题时,两种模型给出的预测结果存在差异。实验结果表明,当前提命题为LL和HH型命题时,被试的条件推理行为与P-Q映射模型的预言完全一致。