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.

One finger cannot lift a pebble. Hopi proverb

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.

Navigating the environmental regulatory infrastructure to analyze and utilize governmental relationships. The environmental manager must learn the language of environmental management and how to communicate with their stakeholders, particularly when navigating the complex and often muddled infrastructure of the national to local environmental regulatory infrastructure. Communicating across stakeholders and integrating voices and perspectives from diverse participants, while being open and transparent with all processes, decisions, and action, is a key skill requisite of the effective environmental manager.

Many of our powerful results about first-order logic, such as the Löwenheim–Skolem Theorem and the Łoś-Vaught Test, focused on countable structures in countable languages. Now that we have a well-developed theory of infinite cardinalities, we can extend these results into the uncountable realm. In addition to the satisfaction we obtain through such generalizations, we will be able to argue that some other important theories are complete, and further refine our intuition about the inability of first-order logic to delineate between infinite cardinalities.

Suppose that we have a (first-order) language $ℒ$. As emphasized in , the elements of ${\text{Form}}_{ℒ}$ are just syntactic sequences of symbols, and we only attach meaning to these formulas once we provide an $ℒ$-structure together with a variable assignment. The fundamental separation between syntactic formulas and semantic structures is incredibly important, because it opens up an interesting way to find both commonalities and differences across structures. That is, given two structures with variable assignments $(ℳ{}_1,s_1)$ and $(ℳ{}_2,s_2)$, we can compare the two sets $\{\phi \in {\text{Form}}_{ℒ}:(ℳ{}_1,s_1)\models \phi \}$ and $\{\phi \in {\text{Form}}_{\mathcal{L}}:(\mathcal{M}{}_2,s_2)\models \phi \}$. Although the two structures and variable assignments likely live in different worlds, these two sets both live inside the same set ${\text{Form}}_{ℒ}$. In other words, the syntactic nature of the formulas provides a shared substrate where we can perform comparisons.