TY - JOUR
T1 - Beta Assertive Graphs
T2 - Proofs of Assertions with Quantification
AU - Bellucci, Francesco
AU - Chiffi, Daniele
AU - Pietarinen, Ahti Veikko
N1 - Publisher Copyright:
© Individual authors and College Publications 2021
PY - 2021/3
Y1 - 2021/3
N2 - 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.
AB - 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.
KW - Assertions
KW - Existential/Assertive Graphs
KW - Intuitionistic Logic
KW - Quantifiers
KW - Transformations
KW - Type vs. occurrence-referential notations
UR - https://www.collegepublications.co.uk/ifcolog/?00044
UR - http://www.scopus.com/inward/record.url?scp=85111108312&partnerID=8YFLogxK
M3 - Journal article
AN - SCOPUS:85111108312
SN - 2631-9810
VL - 8
SP - 353
EP - 376
JO - Journal of Applied Logics - IfCoLog Journal of Logics and their Applications
JF - Journal of Applied Logics - IfCoLog Journal of Logics and their Applications
IS - 2
ER -