In the introduction, we identified a general prejudice against diagrams in the history of logic and mathematics. Diagrams, in spite of their widespread use, have never been permitted as valid or real proofs. We also identified as one of the main reasons behind this prejudice a general worry that diagrams tend to mislead us. I showed in the main part of this work that the misapplication of diagrams is not intrinsic to the nature of diagrams. Venn diagrams, one of the most well-understood and widely used kinds of diagrams, can be presented as a standard representation system which is sound and complete. Accordingly, as long as we follow the transformation rules of the system, the use of Venn diagrams should be considered a valid or real proof, just as the use of first-order logic is. So, mathematicians' and logicians' worries about the misapplication of diagrams in general cannot be justified. We should not give up using diagrams in a valid proof just because there is a possibility of the misuse of diagrams. What is needed are rules of a system that give us permission to perform certain manipulations. The validity of these rules presupposes the semantics of the system.

As I showed in detail in the second chapter, this is where our predecessors (including Peirce) stopped. They had a strong intuition about how Venn diagrams should be used. However, they were not able to justify their intuition, since they did not have a semantic analysis.

In the previous chapter we introduced a selection of the more popular inference processes which have been proposed. This raises the question of why to prefer one such process over any other. In this chapter we shall consider this question by presenting a number of properties, or as we shall call them, principles, which it might be deemed desirable that an inference process, N, should satisfy.

For the most part these principles could be said to be based on common sense or rationality or ‘consistency’ in the natural language sense of the word. A justification for assuming that adherence to common sense is a desirable property of an inference process comes from the Watts Assumption given in Chapter 5. For if K genuinely does represent all the expert's knowledge then any conclusion the expert draws from K should be the result of applying what, by consensus, we consider correct reasoning, i.e. of common sense.

So our plan now is to present a list of such principles. We shall limit ourselves to the case where Bel is a probability function, although the same criteria could be applied to inference processes for DS-belief functions, possibility functions etc. In what follows N stands for an inference process for L. Here L is to be thought of as variable. If we wish to consider a principle for a particular language L we shall insert ‘for L’.

Equivalence Principle

If K1, K2 ∈ CL are equivalent in the sense that VL(K1) = VL(K2) then N(K1) = N(K2).

In this chapter, we provide a result which characterizes well-formedness of free-choice nets in a very suitable way for verification purposes. All the conditions of the characterization are decidable in polynomial time in the size of the net. The most interesting feature of the result is that it exhibits a tight relation between the well-formedness of a free-choice net and the rank of its incidence matrix. Accordingly, it is known as the Rank Theorem. It will be an extremely useful lemma in the proof of many results of this chapter and of the next ones.

We also provide a characterization of the live and bounded markings of a well-formed free-choice net. Again, the conditions of the characterization can be checked in polynomial time. Together with the Rank Theorem, this result yields a polynomial time algorithm to decide if a given free-choice system is live and bounded.

In the last section of the chapter we use the Rank Theorem to prove the Duality Theorem. This result states that the class of well-formed free-choice nets is invariant under the transformation that interchanges places and transitions and reverses the arcs of the net.

Characterizations of well-formedness

Using the results of Chapter 4 and Chapter 5, it is easy to obtain the following characterization of well-formed free-choice nets.

Proposition 6.1A first characterization of well-formedness

Let N be a connected free-choice net with at least one place and at least one transition.

1. N is structurally live iff every proper siphon contains a proper trap.

2. […]