The notions in an empirical theory are refined forms of concepts in systems' knowledge that directly or indirectly come from observations. Therefore, they capture stable patterns in experience, describe the environment as perceived by the systems accepting the theory, and directly make predictions about the environment so as to guide the systems' actions. On the contrary, the notions in a formal theory do not directly corresponds to mental concepts. Instead, they are primarily conventions maintained within the theory, which can be applied to various concrete situations via different interpretations, in which the formal notions in the theory are mapped into the concrete concepts. When interpreted, a formal theory also guide the actions of the systems using the theory.
An empirical theory is "open", in the sense that its validity is constantly checked when the theory is used by the systems. The truth-value of the statements within the theory are tested each time a prediction is made, and will be adjusted (more or less) according to whether it is confirmed by following observation. These adjustments, as well as their derived conclusions, also change the meaning of the notions of the theory. Therefore, intuitively speaking, we can see the theory as an information system opening to new experience and adapting to it, though all the actual thinking happen in the minds of the users of the theory.
A formal theory is "closed", in the sense that its notions and statements do not adapt to new experience, but are determined by the definitions and axioms of the theory, and remain unchanged. When such a theory is applied to a practical situation, whether the application is successful only change people's beliefs on whether the theory can be interpreted or applied in this way, rather than on the soundness of the theory itself. Consequently, no change in the theory is required.
If a theory is taken as an information system, then a formal theory roughly corresponds to an instructive system, because all of the conclusions of the theory are derived from a fixed core, so are predetermined in principle. On the other hand, an empirical theory roughly corresponds to an intelligent system, because the theory develops in time, according to new evidence, and even the most basic principles can be challenged and revised. They are good for different environments. In a relatively stable environment, a closed theory provides an efficient way to organize the relatively stable knowledge into a system that can be efficiently applied to solve problems. On the other hand, an open theory is needed in situations that are novel in nature, and change from time to time, so that past experience does not always dependable. Generally speaking, empirical theories efficiently organize common experience in specific domains, while formal theories provides efficient tools that can be applied in different domains.
The theory is further developed by defining new notions and proving new theorems, all from the existing ones, according to the rules. Within the theory, the meaning of a notion is its definition, and the truth-value of a statement is determined by whether it is a theorem. In this way, the theory is semantically "closed", as mentioned previously. In principle, the capability of the theory is fully determined by its initial notions and axioms, though it is not trivial to introduce proper definitions and to find proofs for theorems.
When a formal theory is applied to a practical situation, its notions get additional meanings by associated to domain-specific concepts, and its theorems provides domain-specific conclusions whose truth-values are the same as those of the axioms, after the interpretation. In this way, a formal theory allows the system to efficiently find implied statements in an efficient manner.
The internal structure of an empirical theory is very close to a "non-axiomatic system" specified in Section 3.2. Typically, the theory starts from experience on a certain aspect of the environment. New notions are introduced to capture the stable patterns in the experience, and knowledge is collected about these notions, and organized into a "deductive hierarchy", so that the relevant experience can be explained by the theory.
The development of an empirical theory can be roughly described as "axiomization-oriented", in the sense that it is desired to organize the knowledge into a system that is as similar to an axiomatic system as possible. It means to clarify the meaning of the notions to the extent that can be captured by clear and simple definitions, to formulate the knowledge by only keeping the unequivocal and confident beliefs, and to organize the notions and knowledge into a deductive hierarchy. However, since by nature an empirical theory must open to new experience, it will can never be fully axiomatized. Furthermore, no matter how much the notions and judgments have been reorganized, the meaning and truth-value of them are still eventually experience-grounded, rather than defined within the theory.
Even though these two types of theory are structured differently, they are nevertheless closely related to each other. On one hand, an empirical theory can be "locally axiomatized" by treating stable parts of the theory as axiomatic sub-theories. For example, the more mature an empirical theory becomes, the more mathematical tools it tends to use within it. On the other hand, during the building and application of a formal theory, it is empirical knowledge that decide what kind of notions and axioms are needed, and how they are related to the actual experience of the system.
There cannot be a "theory of everything", because the users of a theory are intelligent systems that have restriction in its information processing capability, and have to use the theory when the knowledge and resources are insufficient. Consequently, a theory has to choose a proper scope, which should wide enough to cover the goals to be achieved, while still narrow enough to be manageable. Similarly, the basic notions of the theory will have certain "granularity" that is delicate enough to capture the relevant details, and coarse enough to omit the irrelevant details.
Therefore, each theory describes certain aspects of the environment at a certain level of description, and its notions also correspond to events and patterns observable at that level. When an object is described by several theories at different levels, the one at a lower level usually provides more information about the internal structure of the object, while the one at a higher level usually provides more information about the external relation of the object. Since the meaning of the object is determined both by its internal structure and its external relation, it is hard to say which theory tells more about the object — it depends on the purpose of the theory. Similarly, no theory can be fully "reduced to" or "derived from" another theory at a different level. On the other hand, related theories at different levels do have mutual impact. When their conclusions are consistent, they support each other; when their conclusions are inconsistent, they conflict with each other, though in this case no theory has a privilege just because of its level.
This difference in level of description is sometimes confused with another "level" difference between theories. When a theory is the subject of another theory, the former is usually referred to as at the "object-level", while the latter is at the "meta-level". If somehow a theory and its meta-theory have overlap, we get the a "strange loop", that is, some kind of self-reference. Sometimes this should be avoid, because it leads to paradox. However, in some situations it is desired or inevitable. For example, a theory on philosophy of science must be applicable to itself, as far as it is proposed as a scientific theory.
A scientific theory focusing on a small domain is easier to become well-organized than a theory that deals with a complicated domain, because of the variations of the experience and the limit on the processing capacity of a system. Consequently, natural sciences qualified better than social sciences as "science". Philosophy is the study of the most general domain in our experience. As a result, few philosophical theory is organized so well, that it deserves to be called a "science" in the usual sense of the word. Nevertheless, the difference is a matter of degree.
The conflicts among theories usually cannot be solved by merging them together into a "summarized theory", because their conceptual systems are very often incompatible. For the same reason, the people in different schools often cannot solve their difference by debate. In complex domains, it is unlikely to find a "decisive test" that can settle the competition once for all. Even when certain tests are available, succeeds and failures are often temporary. In certain situation, it is possible for a new theory to "unify" several previous theories, but even when it happens, usually the new theory is based on a class of new notions, rather than simply merge the previous theories together.
In scientific research, the attempt to keep all competing theories alive and the attempt to unify them into a single theory are both valid and valuable for the progress of science.
For an individual researcher and a concrete situation, it is often necessary to select a single theory among the competitors, and to keep it exclusively for a while. This can be the case even when the other theories have their values, because of the restriction on resources and the demand for consistency, both in conceptions and actions. Whenever such a selection is necessary, it is often a compromise among the three desired properties discussed in Section 6.2: Correctness, Concreteness, and Compactness. It is usually the case that one theory is better than another one in one aspect, but worse on another. Since the evaluating of the theories and balancing among the aspects are usually highly dependent on experience and context, there is no objective procedure to follow to make the selection. Instead, the preference tends to be different from person to person, and from situation to situation. However, it does not mean that the selection is arbitrary or irrational — in each system, it is a rational decision, though different systems make different decisions, due to the difference in their goals, actions, and beliefs.