Traditional logic is informal and even overlapping about the production of terms and sentences. Traditional logic also avoids to describe how terms latively appear in sentences, i.e., how sentences proceed from terms, and are in fact constructed using terms. Similarly, signatures appear latively in terms. Lativity proceeds towards entailment and provability, and clearly, sentences do appear in provability, but provability as a 'statement' should not be seen as a sentence. This creates self-referentiality which often leads to peculiar situations in traditional logic. Traditional model theory leaning on sets and relations is insufficient to describe the structural generality of logic.
Lative logic is the solution. It provides 'individualization' of logic, and thereby also embraces a concept of logics in 'dialogue', with terms, sentences and entailments being transformed when communicated from one logic to the other. This potential dialogue appears in categories of institutions and entailment systems, developed decades ago, but the structure of objects in those categories were not arranged latively enough.
Lative logic is a generalized framework and a universal structure, where specific logics of all kind can be incorporated.
Lative logic uses signatures as a fundamental atomic structure, and category theory is its metalanguage.
The fundamental idea underlying constructions in lative logic builds upon an Algebraic Foundations of Many-Valuedness, and has bearing also on the Foundations of Mathematics.
Applications of lative logic include developments within