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At the core of the research project of relevance logic (or relevant logic) is the idea of capturing a notion of conditionality that doesn't fall afoul of the so-called fallacies of relevance, or paradoxes of the material and strict conditional.
Other introductions can be found in Read's Relevant Logic, Dunn and Restall's "Relevance Logic" (2002), and Mares' Relevant Logic: a philosophical interpretation (2004).
Dunn has developed a theory of intrinsic and essential properties based on relevant logic.
Relevant logic has been used as the basis for mathematical theories other than set theory.
In 1929, Orlov gave the first axiomatisation of the relevant logic R which is paraconsistent.
Other sorts of modal operators have been added to relevant logic as well.
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Our rationale for doing as much is that one will get the technical details, and the particular philosophical issues that motivated them, from looking at specific logics—specific theories of logical consequence (e.g., relevant logics, substructural logics, non-monotonic logics, dynamic logics, modal logics, theories of quantification, and so on).
In relevant logics, if A → B is a logical truth, then A is relevant to B, in the sense that A and B share at least one propositional variable.
The theory of relevance that is captured by at least some relevant logics can be understood in terms of how the corresponding natural deduction system records the real use of premises.
Relevant logics were pioneered in order to study the relevance of the conclusion with respect to the premises by Anderson and Belnap (1975) in Pittsburgh.
So, in relevant logics, the inference from an arbitrary premise to a logical truth such as q → q may well fail.
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