Informal proof, formal proof, formalism

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened

A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

Norms and Law: Putting the Horse Before the Cart

Law and society scholars have long been fascinated with the interplay of formal legal and informal extralegal procedures. Unfortunately, the fascination has been accompanied by imprecision, and scholars have conceptually conflated two very different mechanisms that extralegally resolve disputes. One set of mechanisms might be described as the shadow of the law, made famous by seminal works by Professors Stewart Macaulay and Marc Galanter, in which social coercion and custom have force because formal legal rights are credible and reasonably defined. The other set of mechanisms, recently explored by economic historians and legal institutionalists, might be described as order without law, borrowing from Professor Robert Ellickson\u27s famous work.1 In this second mechanism, extralegal mechanisms—whether organized shunning, violence, or social disdain—replace legal coercion to bring social order and are an alternative to, not an extension of, formal legal sanctions. One victim of conflating these mechanisms has been our understanding of industry-wide systems of private law and private adjudication, or private legal systems. Recent examinations of private legal systems have chiefly understood those systems as efforts to economize on litigation and dispute-resolution costs, but private legal systems are better understood as mechanisms that economize on enforcement costs. This is not a small mischaracterization. Instead, it reveals a deep misunderstanding of when and why private enforcement systems arise in a modern economy. This Essay provides a taxonomy for the various mechanisms of private ordering. These assorted mechanisms, despite their important differences, have been conflated in large part because there has been a poor understanding of the particular institutional efficiencies and costs of the alternative systems. Specifically, enforcement costs have often been inadequately distinguished from procedural or disputeresolution costs, and this imprecision has produced theories that inaccurately predict when private ordering will thrive and when the costs of private ordering overwhelm corresponding efficiencies. The implications for institutional theory are significant, as confusion in the literature has led to overappreciation of private ordering, underappreciation of social institutions, and Panglossian attitudes toward both lawlessness and legal development

Towards a Formalism-Based Toolkit for Automotive Applications

The success of a number of projects has been shown to be significantly improved by the use of a formalism. However, there remains an open issue: to what extent can a development process based on a singular formal notation and method succeed. The majority of approaches demonstrate a low level of flexibility by attempting to use a single notation to express all of the different aspects encountered in software development. Often, these approaches leave a number of scalability issues open. We prefer a more eclectic approach. In our experience, the use of a formalism-based toolkit with adequate notations for each development phase is a viable solution. Following this principle, any specific notation is used only where and when it is really suitable and not necessarily over the entire software lifecycle. The approach explored in this article is perhaps slowly emerging in practice - we hope to accelerate its adoption. However, the major challenge is still finding the best way to instantiate it for each specific application scenario. In this work, we describe a development process and method for automotive applications which consists of five phases. The process recognizes the need for having adequate (and tailored) notations (Problem Frames, Requirements State Machine Language, and Event-B) for each development phase as well as direct traceability between the documents produced during each phase. This allows for a stepwise verification/validation of the system under development. The ideas for the formal development method have evolved over two significant case studies carried out in the DEPLOY project