  The Tarskian classical relevant logic TR arises from Tarski's work on the
foundations of the calculus of relations and on first-order logic restricted to
finitely many variables, presented by Tarski and Givant their book, A
Formalization of Set Theory without Variables, and summarized in first nine
sections. TR is closely related to the well-known logic KR. Every formula of
relevance logic has a corresponding sentence in Tarski's extended first-order
logic of binary relations with operators on the relation symbols. A formula is
in TR (by definition), or in KR (by a theorem), if and only if its
corresponding sentence can be proved in first-order logic, using at most four
variables, from the assumptions that all binary relations are dense and, for
TR, commute under composition, or, for KR, are symmetric. The vocabulary of TR
is the same as the classical relevant logic CR$^*$ proposed by Meyer and
Routley but TR properly contains CR$^*$. The frames characteristic for TR are
the ones that are characteristic for CR$^*$ and satisfy an extra frame
condition. There are formulas in TR (but not in CR$^*$) that correspond to this
frame condition and provide a counterexample to a theorem of T. Kowalski. The
frames characteristic for TR, or KR, are the ones whose complex algebras are
integral dense relation algebras that are commutative, or symmetric,
respectively. For both classes, the number of isomorphism types grows like the
number of isomorphism types of ternary relations. Asymptotic formulas are
obtained for both classes. Similar results apply to a hierarchy of logics
defined by the number of variables used in the first-order proofs of their
corresponding sentences.
