This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the (...) other case, identity is a notion used to define other logical concepts. In our previous paper, [16], we investigated whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic with standard semantics a reliable definition of identity is possible. In the present study we have moved to modal logic and realized that here we can distinguish in the formal language between two different equality symbols, the first one shall be interpreted as extensional genuine identity and only applies for objects, the second one applies for non rigid terms and has the characteristic of synonymy. We have also analyzed the hybrid modal logic where we can introduce rigid terms by definition and can express that two worlds are identical by using the nominals and the @ operator. We finish our paper in the kingdom of identity where the only primitives are lambda and equality. Here we show how other logical concepts can be defined in terms of the identity relation. We have found at the end of our walk a possible point of convergence in the logic Equational Hybrid Propositional Type Theory, [14] and [15]. (shrink)

In this paper, we deal with the problem of putting together modal worlds that operate in different logic systems. When evaluating a modal sentence $\Box \varphi $, we argue that it is not sufficient to inspect the truth of $\varphi $ in accessed worlds (possibly in different logics). Instead, ways of transferring more subtle semantic information between logical systems must be established. Thus, we will introduce modal structures that accommodate communication between logic systems by fixing a common lattice $L$ that (...) contains as sublattices the semantics operating in each world. The value of a formula $\Box \varphi $ in a world with lattice $L^{\prime}$ will be defined in terms of the values of $\varphi $ in accessible worlds relativized to $L^{\prime}$ using the common order of $L$. We will investigate natural instances where formulas $\varphi $ can be said to be necessary/possible even though all the accessible world falsify $\varphi $. Further, we will discuss frames that characterize dynamic relations between logic systems: classically increasing, classically decreasing and dialectic frames. Finally, we formalize the semantics of considering worlds operating in classical logic or logic of paradox, exemplifying the kind of issue one should face in this kind of formalization. (shrink)