Real reduced models for relevant logics without ${\rm WI}$.

Abstract

Slaney has provided reduced models (ones in which there is but one world) for a number of relevant logics via certain kinds of frames, as opposed to the conventional Routley-Meyer model structures. This paper does three things: it corrects Slaney's paper, extends his results in a different direction, and draws a moral from the errors it corrects. The corrections to Slaney's paper are very minor, the errors having been more in the nature of slips than of outright mistakes. The semantic structures of Slaney's paper are criticized for not being enough. It is then shown that Slaney's basic results can be used to provide reduced models for most of the same logics (the system E being a notable exception) using the Routley-Meyer model structures which do not suffer from this defect. The basic slip in the original paper was not to close the worlds of the canonical models of some of the systems under all of the primitive rules of inference of that system. The paper ends with a brief discussion of the philosophical significance of insisting that theories (worlds) be closed under certain rules of inference as well as under provable implication. That discussion insists upon the importance of a distinction between primitive/derivable rules of inference and merely admissible rules along the lines of Anderson and Belnap. / Introduction Slaney [8] discusses the motivationally important matter of reduced modeling for relevant logics, duly notes that many important weaker relevant logics have not been provided with reduced modeling, and goes on to offer such for them in terms of frames, as opposed to the conventional model structures of Routley and Meyer [5],[6] for instance. In addition to their motivational importance, reduced models are technically and practically important for the practicing logician. They are simpler and hence easier to use. However we find [8] lacking in some very important respects. In the first place, there are some minor (i.e., easily fixed) technical inaccuracies in the paper. Some of the claims made therein are false as stated, and some Received June 24, 1991; revised September 12, 1991 REDUCED MODELS FOR RELEVANT LOGICS 443 of the proffered proofs are unsound. So in what follows we will revise the claims of [8] as necessary and at least indicate how the incorrect proofs can be made right. (It is very unlikely that the author of [8] was actually mistaken on any of these points. The errors corrected herein were surely just slips.) But our discontent with [8] runs deeper: the models of [8] are not in general really models —in a very clear sense, they are not semantical enough! As the reader will see when we get around to introducing the technical definitions, for a given logic an L-frame is defined in terms of the crucial syntactic notions of 'theorem of L' and 'primitive rule of L Now there may be those who are not purists in these matters and whose sensibilities are not offended by such wanton self-indulgence. We ourselves are very open-minded on this point and would not care to judge them to be moral reprobates. However, it is our duty to warn them that they will pay for their sins immediately in this case. For this illicit conjoining of syntax and semantics begets here a degenerate offspring: frames so defined are useless for many of the tasks to which logicians are wont to appoint them, e.g., proving that a given formula (neither already known to be a theorem nor known to be a nontheorem of L) is valid or invalid as the case may be. It is not seriously being suggested that the results of [8] are unworthy. The extension of metavaluation techniques given there is particularly admirable. But such upright works should be put at the service of holier tasks. So in addition to correcting [8], we will extend it. In particular, we prove that the basic results of [8] can be used to show that almost all of the logics treated there are characterized by real reduced models, i.e., ones based on the traditional Routley-Meyer relational model structures. So without further ado, the technical details. 2 Syntactic and semantic preliminaries The systems to be considered can be fitted with Hilbert-style axiomatizations from the following list of axioms and rules (the side notation designates the corresponding semantic postulate as listed further below): (Al) A->A pi (A2) A&B^A (A3) A&B-+B (A4) (A -* B) & (A -> C) -* .A -* B & C p2.i (A5) A^AvB (A6) B-+AvB (A7) (A^C)&(B-+C)-+.AvB-+C p2.i (A8) A&(BvC)-+(A&B)vC (A9) ^A-*A p4 (A10) A^-^B-».B-*-yA p7 (All) A-+B-+ .C-+A-+ .C->£ pό.i (A12) A-+B->.B-+C->.A-+C pό.ii (A13) (A-+B) & (B^C)-+.A->C p9 (A14) A->.A->A pl2 (A15) A^.B-+A pl3 (A16) (A-+BvC)& (A&B^O^.A^C ? (A17) A-+ .A->B-+B plO (A18) (A-+ .B-+C)-+ .£-• .A-+C pll