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.