In this paper we present a general theory of Π2-rules for systems of intuitionistic and modal logic. We introduce the notions of Π2-rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many Π2-rule systems extending LC=IPC+(p→q)∨(q→p), and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in LC: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all Π2-rules which are admissible are derivable, and (2) show that the problem of admissibility of Π2-rules over LC is decidable.

Abstract This paper presents a novel concept of a polyatomic logic and initiates its systematic study. This approach, inspired by inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalized Blok–Esakia theory, which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of polyatomic logics and the general Blok–Esakia theory for a wide class of interesting translations.

In recent times, heritage has become a key focus of socio-economic concern due to its connections to history, nationalism, and, especially, tourism. It has been highlighted as having major consequences for the sociocultural engagement of local populations. The notion that tourism transforms local heritage through the categorization of local realities has become particularly widespread. However, the specific mechanism through which these transformations occur has remained unformalized in cultural heritage studies. This has resulted in a lack of computational approaches to the issue of heritage categorization, which can connect with other issues of tourism promotion and management. In this article, we conceptualize the cultural dynamics of heritage, such as changes in the categories used to signify heritage, and design and analyze a dedicated computational agent-based model of heritage engagement. Here, we seek to explore the role of communicative strategies, as they relate to social groups, as well as the specific topologies of interaction networks. We attempt to understand what role mediation, such as that produced by online platforms, can play in producing different levels of consensus between tourists. The results suggest that topology plays the greatest role in generating consensus in the evolution of heritage meaning, with different patterns emerging according to the communicative strategies employed. The results have implications for the future direction of the study of heritage and heritage modeling, placing emphasis on the need to analyze tourist communication content, as well as being potentially useful for public policy with regard to heritage management.