Nuel Belnap

(? - 2024)

Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermann's strenge Implikation ([1]). Interest in Ackermann's system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermann's system is identical with the structure of modalities in Lewis's S4, and secondly that (Ackermann's apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication.

In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people working in the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In [6] proof tableau formulations were given of the implication/negation fragments of the important zero-order relevant logics E and R and the semirelevant logic RM . The main purpose of this paper then, is to extend results by giving proof tableau formulations of the distribution-free fragments of E, R and RM and of their first order extensions EQ, RQ and RMQ. Where X is one of these logics, we shall follow [13] in calling its distribution-free fragment OX – the ‘O’ standing for ‘ortho’ which is meant to signify the kinship of these logics to quantum logics or ortho-logics as they now tend to be known. Hence in what follows we shall refer to the logics OX simply as relevant ortho-logics

“Flat pre-semantics” lets each parameter of truth (etc.) be considered sepa-rately and equally, and without worrying about grammatical complications. This allows one to become a little clearer on a variety of philosophical-logical points, such as the use fulness of Carnapian tolerance and the deep relativity of truth. A more definite result of thinking in terms of flat pre-semantics lies in the articulation of some instructive ways of categorizing operations on meanings in purely logical terms in relation to various parame- ters of truth (etc.); namely, closing vs. leaving open, local vs. translocal, and anchored vs. unanchored. Basic relations among these categories are established.