THE JOURNAL OF SYMBOLIC LOGIC Volume 37, Number 1, March 1972 SEMANTICS FOR RELEVANT LOGICS ALASDAIR URQUHART ?1. Introduction. In what follows there is presented a unified semantic treat- ment of certain "paradox-free" systems of entailment, including Church's weak theory of implication (Church [7D and logics akin to the systems E and R of Ander- son and Belnap (Anderson [3], Belnap [6D.1 We shall refer to these systems gene- rally as relevant logics. The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice- interpretedinformally as the semilattice of possible pieces of information. Complete- ness theorems can be given relative to these semantics for the implicational frag- ments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems-in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the "paradoxes of material implication". The systems discussed are formulated in fragments of a first-order language with -* (entailment), &, v, -A, (x) and (3x) primitive, omitting identity but includ- ing a denumerable list of propositional variables (p, q, r, pl, - X etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, Al, - - e are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The seman- tics of the systems are given in informal terms; it is an easy matter to turn the infor- mal descriptions into formal set-theoretical definitions. ?2. Relevant implication. Let us begin with the concept of apiece of information. A piece of information is a set of basic sentences concerning a subject or subjects about which reasoning is being carried out. In physics the basic sentences might consist of statements of experimental results, in mathematics elementary facts about numbers, and so forth. These sets of statements may be finite (e.g. listed on sheets of paper) or infinite (e.g. given by a mechanical listing procedure). It is clear that given any two pieces of information, X and Y, we may put them together to form a new piece of information, X u Y, containing all the information in X Received November 27, 1970; revised June 21, 1971. 1 The author is much indebted to Nuel D. Belnap Jr., discussions with whom resulted in essential corrections to material in ?3, and to Alan Ross Anderson, whose subproof formula- tions of relevant logics in [2] provided the fundamental inspiration for the present semantics. Essentially identical semantics were conceived independently by Richard Routley [15]. 159