Beta Assertive Graphs: Proofs of Assertions with Quantification

Assertive graphs (AGs) modify Peirce’s Alpha part ofExistential Graphs (EGs). They are used to reason about assertions without aneed to resort to any ad hoc sign of assertion. The present paper presents anextension of propositional AGs to the Beta case by introducing two kinds ofnon-interdefinable lines. The absence of polarities in the theory of AGsnecessitates Beta-AGs that resort to such two lines: standard lines that meanthe presence of a certain method of asserting, and barbed lines that mean thepresence of a general method of asserting. New rules of transformations forBeta-AGs are presented by which it is shown how to derive the theorems ofquantificational intuitionistic logic. Generally, Beta-AGs offer a newnon-classical system of quantification by which one can logically analysecomplex assertions by a notation which (i) is free from a separate sign ofassertion, (ii) does not involve explicit polarities, and (iii) specifies atype-referential notation for quantification. These properties stand inimportant contrast both to standard diagrammatic notations and to standard,occurrence-referential quantificational notations.