Proof Explanation Research Articles

Emerging Proof Productions of Freshmen in Euclidean Geometry Proof Tasks between Conjecturing and Proving

ABSTRACT This study was conducted to examine the effectiveness of proof tasks in the transition from conjecture to proof in Euclidean geometry on freshmen’s proof schemes. In line with this aim, the proof schemes of the freshmen who performed conjecture-proof and theorem-proof tasks were compared. The freshmen were composed of 109 pre-service middle school mathematics teachers who are enrolled in their first year of undergraduate education. The study was modeled as a multiple-case study. Fifty-three freshmen performed conjecture-proof tasks in Case-1, and fifty-six freshmen performed theorem-proof tasks in Case-2. The video recordings, including the written proof reports, reflection papers, and proof explanations of the freshmen, were used as data collection tools in the tasks. The proof schemes were used as construct maps to evaluate the proofs of freshmen and analyzed using Winsteps Rasch software. The proof schemes in freshmen’s proof tasks were evaluated by the Wright Map and supported with direct quotations from the proofs. In the study, it was observed that the proof schemes of freshmen who made proofs based on their own conjectures were mostly empirical and analytical proof schemes, while the proof schemes of freshmen who made proof of the presented theorem were generally external and empirical proof schemes.

Investigating Students’ Proof Reasoning: Analyzing Students’ Oral Proof Explanations and their Written Proofs in High School Geometry

Constructing formal geometry proofs in is an important topic in the mathematics curriculum. But students’ difficulties with proof are well documented. This article focuses on proofs that use triangle congruence postulates in US high school geometry. Examining students’ proof reasoning in one-on-one task-based interviews, we analyzed students’ oral planning/explanations and two-column written proofs. We found that much of the reasoning students conveyed in their oral planning/explanations was not incorporated into their written proofs, resulting in formalization and fatal logical reasoning gaps. We found that the majority of the students utilized sound reasoning in their oral explanations, but struggled to capture that reasoning in their written two-column proofs. We link our findings to learning progression research, further elaborating the van Hiele levels. Our research contributes to understanding the learning and teaching of proof and to the ongoing controversy over the use of two-column proofs in the high school geometry curriculum.

Legal Probabilism—A Qualified Rejection: A Response to Hedden and Colyvan

The best explanation of juridical proof has evolved from probabilism to explanationism. In a recent article, Brian Hedden and Mark Colyvan criticize the evolution from probability to explanation based on a sparse review of some of the pertinent literature, joining a cottage industry of experts from cognate disciplines attempting to rescue probability theory from the corroding effects of legal scholarship. Their major point is to “fault” this transformation “for assuming an outdated and inexhaustive catalogue of ‘interpretations’ of probability. . . which . . . ignores the possibility of integrating the explanatory and other epistemological considerations . . . into a probabilistic framework.” This claim is false. The legal literature has vetted the two decades old literature that Hedden and Colyvan refer to as a new development in epistemology, which in any event has no value in resolving the problems the legal literature on juridical proof addresses. A proper understanding of both the dated literature that Hedden and Colyvan refer to, and what actually are “new developments” in epistemology support the progression from probabilism to explanationism as the best explanation of juridical proof. In addition, H&C make a fundamental error about what the conceptual problem is. Together these errors vitiate their criticisms of explanationism as the best explanation of juridical proof at present. Correcting these mistakes and misapprehensions may assist those in other fields in their understanding of what the actual issues are.

Scenarios, Probability, and Evidence Scholarship, Old and New.

Scenarios, Probability, and Evidence Scholarship, Old and New.

Сравнительный анализ статических методов верификации динамической памяти

The article provides an overview of the existing methods of dynamic memory verification; a comparative analysis is carried out; the applicability for solving problems of control, monitoring, and verification of dynamic memory is evaluated. This article is divided into eight sections. The first section introduces formal verification, followed by a section that discusses dynamic memory management problems. The third section discusses Hoare's calculus resumed by heap transformations to the stack. The fifth and sixth sections introduce the concept of dynamic memory shape analysis and the rotation of pointers. The seventh is on separation logic. The last section discusses possible areas of further research, particularly the recognition at recording level of various instances of objects; automation of proofs; "hot" code, that is, software code that updates itself when the program runs; expanding intuitiveness, for instance, on proof explanations.

Relative plausibility and its critics

Within legal scholarship there is a tendency to use (perhaps overuse) “paradigm shift” in ways far removed from the process famously described by Thomas Kuhn. Within the field of evidence, however, a phenomenon very similar to a paradigm shift, in the Kuhnian sense, is occurring. Although not on the scale of the transformation from Newtonian to Einsteinian physics or other tectonic shifts in science, the best understanding of juridical proof is shifting from probabilism to explanationism. For literally hundreds of years, proof at trial was assumed to be probabilistic. This assumption was given sustained scholarly attention and support beginning with the 1968 publication of John Kaplan’s path-breaking article that generated a rich literature explaining virtually all aspects of juridical proof as probabilistic, from the basic nature of relevancy through the processing of information to the final decision about the facts. Although probabilism quickly became the dominant paradigm, some analytical difficulties were detected quite early (“anomalies” or “irritants” in the words of Kuhn), beginning with L. Jonathan Cohen’s demonstration of certain proof paradoxes. These were extended by Ronald Allen, who also demonstrated the incompatibility of Bayesian reasoning with trials and proposed an analytical alternative. Again a complex literature ensued with the defenders of the dominant paradigm attempting to explain away the anomalies or to shield the probabilistic paradigm from their potentially corrosive effects (in what in fact on a very small scale is precisely what Kuhn explained and predicted with respect to paradigm shifts in science). Over the last two decades, these anomalies have become too irritating to ignore, and the strengths of the competing paradigm involving explanatory inferences (referred to as the relative plausibility theory) have become too persuasive to dismiss. Thus the paradigm shift that the field is now experiencing. We provide here a summary of the relative plausibility theory and its improvement on the probabilistic paradigm. As Kuhn noted, not everybody gets on board when paradigms shift; there are holdouts, dissenters, and objectors. Three major efforts to demonstrate the inadequacies of relative plausibility have recently been published. We analyze them here to demonstrate that their objections are either misplaced or unavailing, leaving relative plausibility as the best explanation of juridical proof. It is interesting to note that two of the three critiques that we discuss actually agree on the inadequacies of the probabilistic paradigm (they provide alternatives). The third concedes that explanationism may provide a better overall account of juridical proof but tries to resuscitate a probabilistic interpretation of burdens of proof in light of one particular analytical difficulty (i.e., the conjunction problem, which arises from the fact that proof burdens apply to the individual elements of crimes, civil claims, and defenses rather than a party’s case as a whole). In analyzing the alternative positions proposed by our critics, we demonstrate that their accounts each fail to provide a better explanation than relative plausibility.

Less probabilism and more about explanationism

According to Allen and Pardo, ‘relative plausibility’ is the best explanation of judicial proof; and the process to identify a best explanation would be, in short, holistic and comparative. My work is a plea for clarification of some items that would constitute their theory, before I (or anyone) give a holistic explanation of them and compare their account with alternatives. I raise three issues: the epistemological relevance of the empirical data used, the kind of holism argued and the fact-finder’s cognitive capacities role in their account.

증명책임의 원칙, 발견과 해설

사후 사실심 단계에서 사실을 재현하고 완전히 유죄를 입증하는 것은 불가능하며, 항상 불확실성이라는 문제가 따르게 된다. 미국의 경우 사실확정이 불가능한 경우, 증거법과 헌법에 따라 사실심 판단자가 이용할 수 있는 증거판단에 관한 몇 가지 법칙이 존재한다. 그 몇 가지 법칙이란 세가지 수준의 확실성에 상응하는 바, (i) 아마도 발생하였을 것이다.(ii) 발생했을 고도의 개연성이 존재한다.(iii) 발생하였음이 거의 확실하다.” 등이 그것이다. 입증책임 법칙에 의존하는 것을 정당화하는 공통적 사법정책은 기능적으로 (i) 사실판단에 있어 오류발생가능성에 대하여 총체적으로, 또는 특정유형 오류발생에 대하여 대응하기 위한 것으로, 남은 오류위험에 대하여 특정 당사자에게 위험을 적절하게 배분하기 위한 것이다. 현재 적용되고 해석상 다투어지는 보통법상의 입증책임 원칙은 정확성과 공정성이라는 정책적 가치를 지지, 표준화, 형평화할 수 있는 사실확정을 확실하게 담보하지 못하고 있다. 이러한 부분은 생명과 자유가 심각하게 문제되는 형사소송 또는 민사상 사기로 인한 소송에서 이용되는 중간적 입증책임 원칙의 경우 특히 부작용을 잉태하고 있다. 이러한 문제에 대한 마이클 파르도의 이론은 매우 설득력이 있는데, 그의 이론에 따르면 확률에 기초한 사실심 법관의 판단은 2차원 설명법칙에 따라 행하여 져야 한다는 것으로 동 문제에 대처함에 있어 효과적이다. 2차원 설명법칙에 따르면 연역적 과정이 뒤바뀌게 되는데, 사실 판단자는 만약 사실을 설명하는 가정적 제언이 사실이라면 여러 제언 중 어떠한 제언이 현출된 증거에 가장 부합하는가에 기초하여 사실을 확정하게 된다.

Bayesian Litigation (An Application of Bayess Theorem to Law)

Ron Allen and Mike Pardo have criticized probabilistic theories of legal proof and have presented an alternative “relative plausibility” theory of legal proof. Alas, both Allen and Pardo’s critique and their new approach are wrong. The remainder of this paper will describe their theory of relative plausibility, restate Allen and Pardo’s objections to subjective or Bayesian probability, and then answer each one of these objections. This paper will then explain why a Ramsian or subjective view of proof provides a better explanation of juridical proof than Allen and Pardo’s relative plausibility theory. At the end of the day, the key question is this: which of the two competing theories of juridical proof--i.e. subjective probability or relative plausibility--makes more sense, either as a descriptive or normative manner? In summary, because relative plausibility models can be Dutch-booked, and because juror evaluation of proof is an essentially subjective exercise, this paper shall defend a subjective conception of legal proof. Lastly, I shall describe a simple method for operationalizing subjective verdicts in civil cases.

Relative Plausibility and its Critics

Within legal scholarship there is a tendency to use (perhaps overuse) “paradigm shift” in ways far removed from the process famously described by Thomas Kuhn. Within the field of evidence, however, a phenomenon very similar to a paradigm shift, in the Kuhnian sense, is occurring. Although not on the scale of the transformation from Newtonian to Einsteinian physics or other tectonic shifts in science, the best understanding of juridical proof is shifting from probabilism to explanationism. For literally hundreds of years, proof at trial was assumed to be probabilistic. This assumption was given sustained scholarly attention and support beginning with the 1968 publication of John Kaplan’s path-breaking article that generated a rich literature explaining virtually all aspects of juridical proof as probabilistic, from the basic nature of relevancy through the processing of information to the final decision about the facts. Although probabilism quickly became the dominant paradigm, some analytical difficulties were detected quite early (“anomalies” or “irritants” in the words of Kuhn), beginning with L. Jonathan Cohen’s demonstration of certain proof paradoxes. These were extended by Ronald Allen, who also demonstrated the incompatibility of Bayesian reasoning with trials and proposed an analytical alternative. Again a complex literature ensued with the defenders of the dominant paradigm attempting to explain away the anomalies or to shield the probabilistic paradigm from their potentially corrosive effects (in what in fact on a very small scale is precisely what Kuhn explained and predicted with respect to paradigm shifts in science). Over the last two decades, these anomalies have become too irritating to ignore, and the strengths of the competing paradigm involving explanatory inferences (referred to as the relative plausibility theory) have become too persuasive to dismiss. Thus the paradigm shift that the field is now experiencing. We provide here a summary of the relative plausibility theory and its improvement on the probabilistic paradigm. As Kuhn noted, not everybody gets on board when paradigms shift; there are holdouts, dissenters, and objectors. Three major efforts to demonstrate the inadequacies of relative plausibility have recently been published. We analyze them here to demonstrate that their objections are either misplaced or unavailing, leaving relative plausibility as the best explanation of juridical proof. It is interesting to note that two of the three critiques that we discuss actually agree with the inadequacies of the probabilistic paradigm (they provide alternatives). The third concedes that explanationism may provide a better overall account of juridical proof but tries to resuscitate a probabilistic interpretation of burdens of proof in light of one particular analytical difficulty (i.e., the conjunction problem, which arises from the fact that proof burdens apply to the individual elements of crimes, civil claims, and defenses). In analyzing the alternative positions proposed by our critics, we demonstrate that their accounts each fail to provide a better explanation than relative plausibility.

The Declining Utility of Analyzing Burdens of Persuasion

This brief paper presented at a celebration of the career of Michael Risinger, briefly reviews why the relative plausibility theory is to date the best explanation of juridical proof and the limits of its chief competitor, a probabilistic explanation. It is true that the legal system is structured to generate the most probable outcome as the verdict in cases; nonetheless, the conclusion of the most plausible (and thus most probable) outcome is achieved through plausible reasoning, not probabilistic reasoning, and virtually the entire procedural context supports, indeed forces the parties to pursue, this approach. A recent competitor to relative plausibility has been advanced in the form of fuzzy set theory that purportedly dissolves the “conjunction paradox,” which is taken as evidence of the inadequacies of the relative plausibility theory. This paper demonstrates that the purported intimate connection between the conjunction paradox and relative plausibility does not exist. In part this is because the so-called “conjunction paradox” is a feature of the world rather than our thinking about the world, and any explanation of juridical proof must explain this feature of the world to have any chance of success. The apparent resolution of this paradox by fuzzy theorizing thus either must rest on the view that how we think about the world can change how the world behaves, which is unlikely, or on mistakes in the application of fuzzy sets and fuzzy logic. The latter is shown to be the case. Among other difficulties, the argument that fuzzy theorizing eliminates the conjunction paradox is shown to make at least two fundamental mistakes of mathematical logic. First, it confuses a set theoretic definition of the intersection of sets with a probabilistic measure of the probability of events occurring. Second, it confuses the elements of sets with the sets themselves. Fuzzy sets and logic are thus eliminated as competitors to the relative plausibility theory.

Explanationism All the Way Down

ABSTRACTThe probabilistic account of juridical proof meets insurmountable problems. A better explanation of juridical proof is that it is a form of inference to the best explanation that involves the comparative plausibility of the parties’ stories. In addition, discrete evidentiary matters such as relevance and probative value are also best understood as involving inference to the best explanation rather than being probabilistic.

Proof explanation for a nonmonotonic Semantic Web rules language

In this work, we present the design and implementation of a system for proof explanation in the Semantic Web, based on defeasible reasoning. Trust is a vital feature for Semantic Web. If users (humans and agents) are to use and integrate system answers, they must trust them. Thus, systems should be able to explain their actions, sources, and beliefs. Our system produces automatically proof explanations using a popular logic programming system (XSB), by interpreting the output from the proof’s trace and converting it into a meaningful representation. It also supports an XML representation for agent communication, which is a common scenario in the Semantic Web. In this paper, we present the design and implementation of the system, a RuleML language extension for the representation of a proof explanation, and we give some examples of the system. The system in essence implements a proof layer for nonmonotonic rules on the Semantic Web.

Presenting and Explaining Mizar

The Mizar proof language has both many human-friendly presentation features, and also firm semantical level allowing rigorous proof checking. Both the presentation features and the semantics are important for users, and an ideal Mizar presentation should be both human-friendly (i.e. very close to textbook presentations), and also allowing fast access to the detailed semantics and detailed proof explanations. This poses several questions, problems and choices when presenting original Mizar texts, presenting results of semantic queries over the Mizar library, and also when presenting texts produced directly on the semantical level, e.g. by automated theorem provers. This paper discusses solutions to these problems, and particularly implements an initial system for presenting detailed explanations of atomic Mizar inferences. This is done by the cooperation of the Mizar XML presentation tools, the MML Query system, and automated theorem provers working on the MPTP semantic translation of Mizar.

GrafOLog: a Visual Language for a Logic with Objects

GrafOLog is a visual programming language for the representation of knowledge expressed in a logical form. In this paper we mainly describe our graphical formalism and discuss its semantics. Although very simple, this formalism is very powerful and avoids the use of Prolog variables. We end this paper with a discussion of possible applications of our system: database interfacing, Prolog teaching, debugging and proof explanation. The latter is given by a graphic definition of the inference in GrafOLog.