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.
There is abduction in games. Players deliberating about possible future histories take those positions, which according to the standard common knowledge and belief of rationality will never actually be reached, as the surprising facts that need accommodation. The need for such accommodation sets their minds in motion and trigger reasoning from effect to causes. Players are prompted to reason to an antedating action under which such positions would be rendered comprehensible, less surprising, or facile and natural. In games, reasoning abductively means to imaginatively look for where perturbations, such as trembles or quantal responses, could take place. Its conclusion is a conjecture about such perturbations.
Discussions on the scientific pluralism typically involve the unity of science thesis, which has been first advanced by Neo-Positivists in the 1930-ies and later widely criticized in the late 1970-ies. In the present paper the problem of scientific pluralism is examined in the context of modern logic, where it became particularly pertinent after the emergence of non-Classical logics. Usual arguments in favor of a unique choice of “the” logical system are of an extralogical nature. The conception of Universal Logic as a theory of mutual translatability and combination of alternative logical systems allows for a more constructive approach to the issue. Logical pluralism gives rise not only to the ontological pluralism but also to non-Classical mathematics based on various non-Classical logics. Our analysis of ontological pluralism rises the following question: is our mathematics globally Classical and locally non-Classical (i.e. having non-Classical parts) or rather, the other way round, is globally non-Classical and only locally Classical? We conclude that in the context of post-non-Classical science the logical pluralism justifies one’s freedom to chose logical tools in conformity with one’s aims, norms and values.
The identity concept developed in the Homotopy Type theory (HoTT) supports an analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological signicance of this distinction and, nally, provide a homotopical reconstruction of a basic kinematic scheme, which is used in the Classical Mechanics, and discuss its relevance in the Quantum Mechanics.