Beta Assertive Graphs: Proofs of Assertions with Quantification

IfCoLoG Journal of Logics and their Applications. 2021. Vol. 8. No. 2. P. 353–376.

Pietarinen A., Bellucci F., Chiffi D.

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

Keywords: transformations, quantifiers, Intuitionistic Logic, Assertive Graphs, Existential Graphs, assertions, Type vs. occurrence-referential notations

Publication based on the results of: Interactive Rationality in Logic, Linguistics and Formal Philosophy (2021)