This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition.
In a predicate calculus, relevance requires sharing of variables and constants between premises and conclusion.
This can be ensured (along with stronger conditions) by, e.g., placing certain restrictions on the rules of a natural deduction system.
In particular, a Fitch-style natural deduction can be adapted to accommodate relevance by introducing tags at the end of each line of an application of an inference indicating the premises relevant to the conclusion of the inference.
This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional or predicate letters with the consequent cannot be true (or derivable).
The basic idea of relevant implication appears in medieval logic, and some pioneering work was done by Ackermann,[3] Moh,[4] and Church[5] in the 1950s.
Drawing on them, Nuel Belnap and Alan Ross Anderson (with others) wrote the magnum opus of the subject, Entailment: The Logic of Relevance and Necessity in the 1970s (the second volume being published in the nineties).
The early developments in relevance logic focused on the stronger systems.
The standard model theory for relevance logics is the Routley-Meyer ternary-relational semantics developed by Richard Routley and Robert Meyer.
A Routley–Meyer frame F for a propositional language is a quadruple (W,R,*,0), where W is a non-empty set, R is a ternary relation on W, and * is a function from W to W, and
, that assigns a truth value to each atomic proposition relative to each point
By an inductive argument, hereditariness can be shown to extend to complex formulas, using the truth conditions below.
The class of all Routley–Meyer frames satisfying the above conditions validates that relevance logic B.
One can obtain Routley-Meyer frames for other relevance logics by placing appropriate restrictions on R and on *.
The last two conditions validate forms of weakening that relevance logics were originally developed to avoid.
Operational models for negation-free fragments of relevance logics were developed by Alasdair Urquhart in his PhD thesis and in subsequent work.
Since the operational models do not generally interpret negation, this section will consider only languages with a conditional, conjunction, and disjunction.
that maps pairs of points and atomic propositions to truth values, T or F.
The conditional fragment of R is sound and complete with respect to the class of semilattice models.
is valid for the operational models but it is invalid in R. The logic generated by the operational models for R has a complete axiomatic proof system, due Kit Fine and to Gerald Charlwood.
The operational semantics can be adapted to model the conditional of E by adding a non-empty set of worlds
The accessibility relation is required to be reflexive and transitive, to capture the idea that E's conditional has an S4 necessity.
The valuations then map triples of atomic propositions, points, and worlds to truth values.
The second way is to keep the semilattice conditions on frames and add a binary relation,
Urquhart showed that the semilattice logic for R is properly stronger than the positive fragment of R. Lloyd Humberstone provided an enrichment of the operational models that permitted a different truth condition for disjunction.
The resulting class of models generates exactly the positive fragment of R. An operational frame
that maps pairs of points and atomic propositions to truth values, T or F.
The positive fragment of R is sound and complete with respect to the class of these models.
Humberstone's semantics can be adapted to model different logics by dropping or adding frame conditions as follows.
is valid just in case it holds on all interpretations on all de Morgan monoids.