The BankXX system models the process of perusing and gathering information for argument as a heuristic best-first search for relevant cases, theories, and other domain-specific information. As BankXX searches its heterogeneous and highly interconnected network of domain knowledge, information is incrementally analyzed and amalgamated into a dozen desirable ingredients for argument (called argument pieces), such as citations to cases, applications of legal theories, and references to prototypical factual scenarios. At the conclusion of the search, BankXX outputs the set of argument (...) pieces filled with harvested material relevant to the input problem situation.This research explores the appropriateness of the search paradigm as a framework for harvesting and mining information needed to make legal arguments. In this article, we describe how legal research fits the heuristic search framework and detail how this model is used in BankXX. We describe the BankXX program with emphasis on its representation of legal knowledge and legal argument. We describe the heuristic search mechanism and evaluation functions that drive the program. We give an extended example of the processing of BankXX on the facts of an actual legal case in BankXX's application domain — the good faith question of Chapter 13 personal bankruptcy law. We discuss closely related research on legal knowledge representation and retrieval and the use of search for case retrieval or tasks related to argument creation. Finally we review what we believe are the contributions of this research to the understanding of the diverse disciplines it addresses. (shrink)

In this article we evaluate the BankXX program from several perspectives. BankXX is a case-based legal argument program that retrieves cases and other legal knowledge pertinent to a legal argument through a combination of heuristic search and knowledge-based indexing. The program is described in detail in a companion article in Artificial Intelligence and Law 4: 1--71, 1996. Three perspectives are used to evaluate BankXX:(1) classical information retrieval measures of precision and recall applied against a hand-coded baseline; (2) knowledge-representation and case-based (...) reasoning, where the baseline is provided by the functionality of a well-known case-based argument program, HYPO (Ashley, 1990); and (3) search, in which the performance of BankXX run with various parameter settings, for instance, resource limits, is compared. In this article we report on an extensive series of experiments performed to evaluate the program. We also describe two additional experiments concerning(1) the program's search behavior; and (2) the use of a modified form of precision and recall based on case similarity. Finally we offer some general conclusions that might be drawn from these particular experiments. (shrink)

It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.

The ability to engage in and recognize pretend play begins around 18 months. A major challenge for theories of pretense is explaining how children are able to engage in pretense, and how they are able to recognize pretense in others. According to one major account, the metarepresentational theory, young children possess both production and recognition abilities because they possess the mental state concept, PRETEND. According to a more recent rival account, the Behavioral theory, young children are behaviorists about pretense, and (...) only produce and recognize pretense as a sort of behavior--namely, behaving 'as-if'. We review both the metarepresentational and Behavioral accounts and argue that the Behavioral theory fails to characterize very young children's abilities to produce and to recognize pretense. Among other problems, the Behavioral theory implies that children should frequently mis-recognize regular behavior as pretense, while certain regular forms of pretend play should neither be produced nor recognized. Like other mental states, pretense eludes purely behavioral description. The metarepresentational theory does not suffer these problems and provides a better account of children's pretense. 2016 APA, all rights reserved). (shrink)

Young children’s failures in reasoning about beliefs and desires, and especially about false beliefs, have been much studied. However, there are few accounts of successful belief-desire reasoning in older children or adults. An exception to this is a model in which belief attribution is treated as a process wherein an inhibitory system selects the most likely content for the belief to be attributed from amongst several competing contents [Leslie, A. M., & Polizzi, P. (1998). Developmental Science, 1, 247–254]. We tested (...) this model with an ‘avoidance false belief task’ in which subjects predict the behavior of a character, who wants to avoid an object but who is mistaken about which of three locations it is in. The task has two equally correct answers—in seeking to avoid the location where she mistakenly believes the object to be, the character might equally go to the location where the object actually is, or to the remaining empty location. However, the model predicts that subjects will prefer one of these answers, selecting the object’s actual location over the empty location. This bias was confirmed in a series of five experiments with children aged between 4 and 8 years of age. A sixth experiment ruled out two rival explanations for children’s biased responding. Two further experiments found the opposite bias in adults. These findings support one selection model as an account of belief-desire reasoning in children, and suggest that a different model is needed for adults. The process of selecting contents for mental state attributions shows a developmental shift between 8 years of age and adulthood. (shrink)

Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, (...) class='Hi'>Friedman has proved that the statement “any two countable well orderings are weakly comparable” is equivalent to ATR 0 . The main goal of this paper is to give a detailed exposition of this result. (shrink)

The considerable attention to financial interests in clinical research has focused mostly on academic medical centers, even though the majority of clinical research is conducted in community practice settings. To fill this gap, this article maps the practices and policies in 73 community hospitals and several hundred specialized facilities around the country for reviewing clinical investigators’ financial relationships with research sponsors. Community hospitals face a substantially different mix of issues than academic medical centers do because their physician researchers are usually (...) not employees, and more of their studies are later-phase, multicenter trials. Accordingly, community physicians stand to benefit personally from per capita payments, but they tend to have less direct influence over research findings when compared with their academic peers. These and other contrasts require separate guidance for community-based organizations that takes into account their particular needs, circumstances, and institutional capacities. (shrink)

The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three (...) properly intertwined hierarchies with the same limit, lying strictly above “total indescribability” and strictly below “arrowing ω”. The innovation here is presented in Section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy by means of a striking universal second-order property of linear orderings. (shrink)

BackgroundIn the pandemic time, many low- and middle-income countries are experiencing restricted access to COVID-19 vaccines. Access to imported vaccines or ways to produce them locally became the principal source of hope for these countries. But developing a strategy for success in obtaining and allocating vaccines was not easy task. The governments in those countries have faced the difficult decision whether to accept or reject offers of vaccine diplomacy, weighing the price and availability of COVID-19 vaccines against the concerns over (...) their efficacy and safety. We aimed to analyze public opinion regarding the governmental strategies to obtain COVID-19 vaccines in three Central Asian countries, focusing particularly on possible ethical issues.MethodsWe searched for opinions expressed either in Russian or in the respective national languages. We provided data on the debate within three countries, drawn from social media postings and other sources. The opinion data was not restricted by source and time. This allowed collecting a wide range of possible opinions that could be expressed regarding COVID-19 vaccine supply and human participation in the vaccine trial. We recognized ethical issues and possible questions concerning different ethical frameworks. We also considered scientific data and other information, in the process of reasoning.ResultsAs a result, public views on their respective government policies on COVID-19 vaccine supply ranged from strongly negative to slightly positive. We extracted the most important issues from public debates, for our analysis. The first issue involved trade-offs between quantity, speed, price, freedom, efficacy, and safety in the vaccines. The second set of issues arose in connection with the request to site a randomized trial in one of the countries (Uzbekistan). After considering additional evidence, we weighed individual and public risks against the benefits to make specific judgements concerning every issue.ConclusionsWe believe that our analysis would be a helpful example of solving ethical issues that can arise concerning COVID-19 vaccine supply around the world. The public view can be highly critical, helping to spot such issues. An ignoring this view can lead to major problems, which in turn, can become a serious obstacle for the vaccine coverage and epidemics’ control in the countries and regions. (shrink)

Recently, Rueger and Sharp (1996) and Koperski (1998) have been concerned to show that certain procedural accounts of model confirmation are compromised by non-linear dynamics. We suggest that the issues raised are better approached by considering whether chaotic data analysis methods allow for reliable inference from data. We provide a framework and an example of this approach.

Let R Õ [1,n]3k ¥ [1,n]k. We define R = {y Œ [1,n]k:($xŒA3)(R(x,y))}. We say that R is strictly dominating if and only if for all x,yŒ[1,n]k, if R(x,y) then max(x) < max(y).

Reflection, in the sense of [Fr03a] and [Fr03b], is based on the idea that a category of classes has a subclass that is “similar” to the category. Here we present axiomatizations based on the idea that a category of classes that does not form a class has extensionally different subclasses that are “similar”. We present two such similarity principles, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.

We axiomatize EFA in strictly mathematical terms, involving only the ring operations, without extending the language by either exponentiation, finite sets of integers, or polynomials.

We focus on two formal methods contexts which generate investigations into decision problems for finite strings.

To prove this, we fix P(x) to be any polynomial of degree ≥ 1 with a positive and negative value. We define a critical interval to be any nonempty open interval on which P is strictly monotone and where P is not strictly monotone on any larger open interval. Here an open interval may not have endpoints in F, and may be infinite on the left or right or both sides. Obviously, the critical intervals are pairwise disjoint.

We characterize the o-minimal expansions of the ring of real numbers, in mathematically transparent terms. This should help bridge the gap between investigators in o-minimality and mathematicians unfamiliar with model theory, who are concerned with such notions as non oscillatory behavior, tame topology, and analyzable functions. We adapt the characterization to the case of o-minimal expansions of an arbitrary ordered ring.

Let k ≥ 2 and f:Nk Æ [1,k] and n ≥ 1 be such that there is no x1 < ... < xk+1 £ n such that f(x1,...,xk) = f(x1,...,xk+1). Then we want to find g:Nk+1 Æ [1,3] such that there is no x1 < ... < xk+2 £ n such that g(x1,...,xk+1) = g(x2,...,xk+2). This reducees adjacent Ramsey in k dimensions with k colors to adjacent Ramsey in k+1 dimensions with 3 colors.

This is the initial publication on Concept Calculus, which establishes mutual interpretability between formal systems based on informal commonsense concepts and formal systems for mathematics through abstract set theory. Here we work with axioms for "better than" and "much better than", and the Zermelo and Zermelo Frankel axioms for set theory.

Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.

We now fix A ⊆ Q. We study a fundamental class of digraphs associated with A, which we call the A-digraphs. An A,kdigraph is a digraph (Ak,E), where E is an order invariant subset of A2k in the following sense. For all x,y ∈ A2k, if x,y have the same order type then x ∈ E ↔ y ∈ E.

This paper was referred to in the Introduction in our paper [Fr97a], “The Axiomatization of Set Theory by Separation, Reducibility, and Comprehension.” In [Fr97a], all systems considered used the axiom of Extensionality. This is appropriate in a set theoretic context.

Extrapolating from the work of Mahlo , one can prove that given any pair of countable closed totally bounded subsets of complete separable metric spaces, one subset can be homeomorphically embedded in the other. This sort of topological comparability is reminiscent of the statements concerning comparability of well orderings which Friedman has shown to be equivalent to ATR0 over the weak base system RCA0. The main result of this paper states that topological comparability is also equivalent to ATR0. In (...) Section 1, the pertinent subsystems of second-order arithmetic and results on well orderings are reviewed. Sections 2 and 3 overview the encoding of metric spaces and homeomorphisms in second-order arithmetic. Section 4 contains a proof of the topological comparability result in ATR0. Section 5 contains the reversal, a derivation of ATR0 from the topological comparability result. In Section 6, additional information about the structure of the embeddings is obtained, culminating in an application to closed subsets of the real numbers. (shrink)

We have been recently engaged in this search, and have announced a long series of successively simpler and more convincing examples. See [Fr09-10].

We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main (...) results for the reals. (shrink)

We present two forms of “sentential reflection”, which are shown to be mutually interpretable with Z2 and ZFC, respectively.

1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to (...) 4. Decidability of statements on N. 5. Decidability of statements on shrink)

Russell’s way out of his paradox via the impredicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.

Provide a formal system that is a conservative extension of PA for Π02 sentences, and even a conservative extension of HA, that supports the worry free smooth development of constructive analysis in the style of Errett Bishop.

BRT is always based on a choice of BRT setting. A BRT setting is a pair (V,K), where V is an interesting family of multivariate functions. K is an interesting family of sets. In this talk, we will only consider V,K, where V is an interesting family of multivariate functions from N into N. K is an interesting family of subsets of N.

The equation P = NP concerns algorithms for deciding membership in sets. The consensus is that P ≠ NP, although some prominent experts guess otherwise.

NOTE: This is an expanded version of my lecture at the special session on reverse mathematics, delivered at the Special Session on Reverse Mathematics held at the Atlanta AMS meeting, on January 6, 2005.

I don’t remember if I got as high as 2-390, but I distinctly remember taking my first logic course - as a Freshman - with Hartley Rogers, in Fall 1964 - here in 2-190. Or was it in 2-290?

This distinction between logic and mathematics is subject to various criticisms and can be given various defenses. Nevertheless, the division seems natural enough and is commonly adopted in presentations of the standard foundations for mathematics.

Mathematical Logic had a glorious period in the 1930s, which was briefly rekindled in the 1960s. Any Shock Value, such as it is, has surrounded unprovability from ZFC.

Normal mathematical culture is overwhelmingly concerned with finite structures, finitely generated structures, discrete structures (countably infinite), continuous and piecewise continuous functions between complete separable metric spaces, with lesser consideration of pointwise limits of sequences of such functions, and Borel measurable functions between complete separable metric spaces.

C. To what extent, and in what sense, is the natural hierarchy of logical strengths rep resented by familiar systems ranging from exponential function arithmetic to ZF + j:V Æ V robust?

We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality. They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC. It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without (...) power set. (shrink)

Ketamine, a dissociative anesthetic commonly used by US physicians, has recently been shown to be a powerful anti-depressant and is also capable of eliciting transpersonal experiences that can be transformative. Although currently approved in the US only for use as an anesthetic, physicians there can legally prescribe it off-label to treat various psychological/ psychiatric problems and it has been used for these non-anesthetic purposes in Argentina, Iran, Mexico, Russia, and the UK, as well as in the US. The literature on (...) using ketamine psychotherapeutically is reviewed and two case studies using ketamine-enhanced psychotherapy for treating death anxiety in terminally-ill people are reported. The potential importance of beginning formal research on using KEP during end-of-life for those suffering death anxiety is emphasized. (shrink)

In mathematics, we back up our discoveries with rigorous deductive proofs. Mathematicians develop a keen instinctive sense of what makes a proof rigorous. In logic, we strive for a *theory* of rigorous proofs.