## Tuesday, May 22, 2012

### Corollarial and Theorematic

There's a common misunderstanding of logic that takes it to be part and parcel with logical reasoning that the conclusion can't go beyond the premises. It is easy to see why someone would think that, but it's not strictly true even for some quite rigorous kinds of reasoning. What you get when you go beyond the premises will depend on the kind of reasoning in question, or the kind of formal system used to model it. But I did recently come across a nice little bit by C. S. Peirce that brings the point home by pointing out that there is a rigorous formal system, one that most people would once have studied, that draws out necessities while going beyond premises:

There are two kinds of Deduction; and it is truly significant that it should have been left for me to discover this. I first found, and subsequently proved, that every Deduction involves the observation of a Diagram (whether Optical, Tactical, or Acoustic) and having drawn the diagram (for I myself always work with Optical Diagrams) one finds the conclusion to be represented by it. Of course, a diagram is required to comprehend any assertion. My two genera of Deductions are first those in which any Diagram of a state of things in which the premisses are true represents the conclusion to be true and such reasoning I call Corollarial because all the corollaries that different editors have added to Euclid's Elements are of this nature. Second kind. To the Diagram of the truth of the Premisses something else has to be added, which is usually a mere May-be, and then the conclusion appears. I call this Theorematic reasoning because all the most important theorems are of this nature. (to William James, EP 2:502)

The terminology, of course, comes from Euclidean geometry, as traditionally taught. Actually, it turns out that Peirce uses the distinction a lot, and considered it one of his most important logical discoveries.