I started this work with the following assumption: There is a distinction between diagrammatic and linguistic representation, and Venn diagrams are a nonlinguistic form of representation. By showing that the Venn system is sound and complete, I aimed to provide a legitimate reason why logicians who care about valid reasoning should be interested in nonlinguistic representation systems as well. However, the following objection might undermine the import of my project: How do we know that Venn diagrammatic representation is a nonlinguistic system? After all, it might be another linguistic representation system which is too restricted in its expressiveness to be used in our reasoning. If we accept this criticism, what I have done so far would be reduced to the following: A very limited linguistic system was chosen and proven to be sound and complete. Considering how far symbolic logic has developed, this could not be an interesting or meaningful project at all. Therefore, it seems very important to clarify the assumptions that diagrammatic systems are different from linguistic ones and that the Venn systems are nonlinguistic.

There has been some controversy over how to define diagrams in general, despite the fact that we all seem to have some intuitive understanding of diagrams. For example, all of us seem to make some distinction between Venn diagrams and first-order languages. Everyone would classify Euler circles under the same category as Venn diagrams. Or suppose that both a map and verbal instructions are available for us to locate a certain place.

Diagrams have been widely used in reasoning, for example, in solving problems in physics, mathematics, and logic. Mathematicians, psychologists, philosophers, and logicians have been aware of the value of diagrams and, moreover, there has been an increase in the research on visual representation. Many interesting and important issues have been discussed: the distinction, if any, between linguistic symbols and diagrams, the advantages of diagrams over linguistic symbols, the importance of imagery to human reasoning, diagrammatic knowledge representation (especially in artificial intelligence systems), and so on.

The work presented in this book was mainly motivated by the fact that we use diagrams in our reasoning. Despite the great interest shown in diagrams, nevertheless a negative attitude toward diagrams has been prevalent among logicians and mathematicians. They consider any nonlinguistic form of representation to be a heuristic tool only. No diagram or collection of diagrams is considered a valid proof at all. It is more interesting to note that nobody has shown any legitimate justification for this attitude toward diagrams. Let me call this traditional attitude, that is, that diagrams can be only heuristic tools but not valid proofs, the general prejudice against diagrams. This prejudice has been unquestioned even when proponents of diagrams have worked on the applications of diagrams in many areas and argued for the advantages of diagrams over linguistic symbols. This is why it is quite worthwhile to question the legitimacy of this prejudice, that is, whether this prejudice is well grounded or not.

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. […]