The UML, in its current state, defines a notation and a meta-model.
The notation is the graphical stuff you see in models; it is the syntax of the modeling language. For instance, class diagram notation defines how items and concepts such as class, association, and multiplicity are represented.
Of course, this leads to the question of what exactly is meant by an association or multiplicity or even a class. Common usage suggests some informal definitions, but many people want more rigor than that.
The idea of rigorous specification and design languages is most prevalent in the field of formal methods. In such techniques, designs and specifications are represented using some derivative of predicate calculus. Such definitions are mathematically rigorous and allow no ambiguity. However, the value of these definitions is by no means universal. Even if you can prove that a program satisfies a mathematical specification, there is no way to prove that the mathematical specification actually meets the real requirements of the system.
Design is all about seeing the key issues in the development. Formal methods often lead to getting bogged down in lots of minor details. Also, formal methods are hard to understand and manipulate, often harder to deal with than programming languages. And you can't even execute them.
Most OO methods have very little rigor; their notation appeals to intuition rather than formal definition. On the whole, this does not seem to have done much harm. These methods may be informal, but many people still find them usefuland it is usefulness that counts.
However, OO methods people are looking for ways to improve the rigor of methods without sacrificing their usefulness. One way to do this is to define a meta-model: a diagram, usually a class diagram, that defines the notation.
Figure 1-1 is a small piece of the UML meta-model that shows the relationship among associations and generalization. (The extract is there just to give you a flavor of what meta-models are like. I'm not even going to try to explain it.)
How much does the meta-model affect the user of the modeling notation? Well, it does help define what is a well-formed modelthat is, one that is syntactically correct. As such, a methods power user should understand the meta-model. However, most users of methods do not need such understanding to get some value out of using the UML notation.
I am not rigorous in this book; rather, I follow the traditional methods path and appeal to your intuition. If you want greater rigor, you should peruse the reference manual or the official documentation.
How strictly should you stick to the modeling language? That depends on the purpose for which you are using it. If you have a CASE tool that generates code, you have to stick to the CASE tool's interpretation of the modeling language in order to get acceptable code. If you are using the diagrams for communication purposes, you have a little more leeway.
If you stray from the official notation, other developers will not fully understand what you are saying. However, there are times when the official notation can get in the way of your needs. I'll admit that in these cases, I'm not at all afraid to bend the language. I believe that the language should bend to help me communicate, rather than the other way around. But I don't do it often, and I'm always aware that a bend is a bad thing if it causes communication problems. In this book, I mention those places where I'm inclined to do a bit of bending.