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Set theory originated in an attempt to understand and somehow classify small, or negligible, sets of real numbers. Cantor’s early explorations into the realm of the transfinite were motivated by a desire to understand the points of convergence of trigonometric series. The basic ideas quickly became a fundamental part of analysis, in addition to permeating many other areas of mathematics. Since then, set theory has become a way to unify mathematical practice, and the way in which mathematicians grapple with the infinite in all areas of mathematics.
In many areas of mathematics (like partial orderings, groups, or graphs), we write down some axioms and immediately have several different models of these axioms in mind. In the setting of first-order logic, this corresponds to writing down a set Σ of sentences in a language and looking at the elementary class . Since by Proposition 6.5.3, and Cn(Σ) is a theory by Proposition 6.5.4, we can view this situation as looking at the (elementary) class of models of a theory.
This case study incorporates the different plans discussed in Chapter 7 and the diffusion of an environmental innovation. This is a case history demonstrating how change agents working with the EPA developed and implemented Integrated Pest Management (IPM) in schools in 41 states in 18 years and how, for more than a decade, School IPM became a national initiative. The case study first appeared in A Worm in the Teacher’s Apple: Protecting America’s School Children from Pests and Pesticides (Lame 2005)
Our development of a formal definition of computability in the previous chapter might have seemed out of place. We used our generation template and some simple references to propositional connectives and (bounded) quantifiers, but otherwise there was seemingly little connection to logic. In this chapter, we establish that computability and logic are fundamentally intertwined.
Be still and the earth will speak to you … Carl Moon – Navajo Nation, 1904
Matching legal authority to environmental goals. To develop and learn the language of the environmental management profession, with a specific focus on the terms and concepts of environmental law, to memorize the intent and major provisions of each environmental law, and to be able to compare and contrast the laws and their provisions for more effective and efficient diagnosis. Mastering this skill will enable the professional to better understand their legal accountability and the “arena” – i.e., the areas of operation and expectations of their work delimited by legal statutes – in which they are managing. In other words, to be able to respond to opposition and take advantage of opportunities.
Balancing internal and external resources. Environmental management is about people management. In thewe discussed managing employees that work directly for you, be they full-time or part-time employees. But often the environmental manager will also have to manage contract workers, meaning non-full-time employees that are often hired for a specific project or task and are not integrated into the functioning, culture, and fabric of an organization. Some contracts last weeks, while others last years, often filling a gap in the organization’s capacities for that specific project. Contractors can be a great asset by augmenting the capabilities of an organization, but they can also become intractable to manage, resulting in a host of factors for the environmental manager to monitor and manage.
We now embark on a careful study of propositional logic. As described in Chapter 1, in this setting, we start with an arbitrary set P, which we think of as our collection of primitive statements. From here, we build up more complicated statements by repeatedly applying connectives. The corresponding process generates a set of syntactic objects that we call formulas. In order to assign meaning to these formulas, we introduce truth assignments, which are functions on P that propagate upward through formulas of higher complexity.
Policy Entrepreneurship. The effective environmental manager must also be a policy entrepreneur and be able to surf. Policy entrepreneurship allows the environmental manager to either innovate new policies for more effective execution of their role or new practices to better carry out the policies they are accountable to. Surfing is the process by which the manager understands and can navigate the contexts they work within to recognize opportunities and set themselves or their organization up to catch the wave.
Identify, analyze and prioritize how environmental issues and legal trends impact your program, resource and political management. More specifically, how the environmental manager can protect their program by getting their “ducks in a row.” It is critical that the environmental manager know the key issues and legal trends in the profession to effectively accomplish their, and their institution’s, mission effectively and efficiently. Understanding these issues will allow the environmental manager to ensure they are not only complying with the law but also with the expectations and needs of the citizenry they serve.
Leadership. Integrating and applying previous skills while taking calculated risks so as to inspire. Just as the effective environmental manager has to manage programs, resource distribution, and many other factors, so too they have to manage the experts they supervise, contract, or otherwise engage. The experts employed by the environmental manager may themselves be managers but most often they are charged with a specific function that they either have substantial experience doing, have a relevant academic or professional degree or certification for, or often both. The experts often know what the right thing is or at least the right process by which things ought to be done. It is up to the environmental manager to lead them to doing it and to manage the contexts such that their efforts are facilitated – i.e., to enable them by reducing distractions and barriers.
Science is the key foundation of everything EPA does. Science has defined the challenges, pushed the discoveries, it has operated as the foundation to design new solutions … it has been EPA’s professor, our prosecutor and our protector.