Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1

In recent years, AI research has started to take seriously the role of context in developing models. This trend is heralded by a departure from logic-based views and their over-emphasis on the formal aspects of thought processes. The departure, however, has not been complete--- some fundamental assumptions of formal logic are still maintained in the new approaches. Similarly, "relevance" remains a major challenge for AI research. This paper outlines an alternative proposal that takes context and relevance as intertwined aspects of thought and intelligence. We argue for a pragmatic approach, which shows better promise than formal logic in dealing with such issues.

Abstract This paper is concerned with a natural deduction system for First Degree Entailment (FDE). First, we exhibit a brief history of FDE and of combined systems whose underlying idea is used in developing the natural deduction system. Then, after presenting the language and a semantics of FDE, we develop a natural deduction system for FDE. Wethen prove soundness and completeness of the system with respect to the semantics. The system neatly represents the four-valued semantics for FDE.

It is often suggested that truth-preservation is insufficient for logical consequence, and that consequence needs to satisfy a further condition of relevance. Premises and conclusion in a valid consequence must be relevant to one another, and truth-preservation is too coarse-grained a notion to guarantee that. Thus logical consequence is the intersection of truth-preservation and relevance.

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